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Action Physics
Dr. Jay Maron


Firearms and explosives Scotch, gin, and vodka Science fundamentals
Drag force History of Western Civilization Physics
Driving Opera Chemistry
Aircraft Architecture Astronomy
Bikes Gems Music performance
Nuclear weapons Precious metals Anatomy
Rockets and spacecraft Spices Electromagnetism
Mixed Martial Arts Geography Global economics
Action scientists History of science Metallurgy
Sports Sports leagues Telescopes
Knots Spaceships
Eye training Python programming
Ear training Unix wizardry
Breathing Materials
Novel vehicles Natural disasters


Firearms

Walther PPK/E 9 mm
FN SCAR-H 7.6 mm

Barrett M82 13 mm
M2 Bradley, M242 Bushmaster 25 mm
GAU-8 Avenger 30 mm

A-10 Warthog, GAU-8 Avenger
M1 Abrams 120 mm
M777 howitzer 155 mm

M777 Howitzer
U.S.S. Iowa 406 mm

                Bullet  Bullet   Speed   Energy   Barrel    Gun     Fire   Vehicle
                 diam    mass                                       rate    mass
                  mm      kg      m/s    kJoule   meters     kg     Hertz   tons

Walther PPK         5.6    .0020   530       .277   .083       .560
Walther PPK         5.6    .0030   370       .141   .083       .560
Walther PPK/S       7.65   .0050   318       .240   .083       .630
Walther PPK/E       9.0    .0065   323       .338   .083       .665
M4 Carbine          5.56   .0041   936      1.796   .370      2.88    15.8
FN SCAR-H Rifle     7.62   .011    790      3.506   .400      3.58    10.4           20 round magazine
Barrett M82        13.0    .045    908     18.940   .74      14.0                    10 round magazine
Vidhwansak         20      .13     720     33.7    1.0       26                      20x81 mm. 3 round magazine
RT-20              20      .13     850     47       .92      19.2                    1 round magazine
M621 cannon        20      .102   1005     51.5              45.5     13.3           20x102 mm
M61 Vulcan         20      .102   1050     56.2              92      110             20x102. 6 barrels
Oerlikon KBA       25      .184   1335    164      2.888    112       10
M242 Bushmaster    25      .184   1100    111      2.175    119        8.3    27.6   M2 Bradley
GAU-12 Equalizer   25      .184   1040     99.5             122       70       6.3   Harrier 2. 5 barrels
M230 chain gun     30      .395    805    128                55.9     10.4     5.2   Apache. 30x113 mm
Mk44 Bushmaster 2  30      .395   1080    230      2.41     160        3.3    27.6   M2 Bradley. 30x173 mm
GAU-8 Avenger      30      .395   1070    226      2.30     281       70      11.3   A-10 Warthog. 30x173 mm. 7 barrels
Bushmaster III     35             1180                      218        3.3           35x228 mm
Bushmaster IV      40     1.08                              198        3.3           40x365 mm
Rheinmetall 120   120     8.350   1750  12800      6.6     4500         .1    62     M1 Abrams tank
M777 Howitzer     155    48        827  16400      5.08    4200         .083
Iowa Battleship   406   862        820 290000     20.3   121500         .033  45000
2 bore rifle       33.7    .225    460     23.7     .711      4.5                    Historical big-game rifle
Cannonball 6 lb    87     2.72     438    261      2.4
Cannonball 9 lb    96     4.08     440    395      2.7
Cannonball 12 lb  110     5.44     453    558      2.4
Cannonball 18 lb  125     8.16     524   1120      2.6     2060
Cannonball 24 lb  138    10.89     524   1495      3.0     2500
Cannonball 32 lb  152    14.5      518   1945      3.4     2540
Cannonball 36 lb  158    16.33     450   1653      2.9     3250
Cannonball diameters are calculated from the mass assuming a density of 7.9 g/cm3.
For a pistol or rifle, the "vehicle mass" is the mass of the person wielding it. We use the mass of a typical person.
The "Metal Storm" gun has 36 barrels, 5 bullets per barrel, and fires all bullets in .01 seconds. The bullets are stacked in the barrel end-to-end and fired sequentially.

12 pound cannonballs
24 pound cannonballs


Bullet speed

25 mm
25 mm rocket propelled gernade
Excalibur 155 mm

The energy distribution for a 7.62 mm Hawk bullet is

Bullet energy    .32
Hot gas          .34
Barrel heat      .30
Barrel friction  .02
Unburnt powder   .01
To estimate the velocity of a bullet,
Energy efficiency  =  e  =  .32    (Efficiency for converting powder energy to bullet enery)
Bullet mass        =  M
Powder mass        =  m
Powder energy/mass =  Q  =  5.2 MJoules/kg
Bullet velocity    =  V
Bullet energy      =  E  =  ½ M V2  =  e Q m    (Kinetic energy = Efficiency * Powder energy)

V  =  (2 e Q m / M)2  =  1820 (m/M)½  meters/second

Muzzle break

M777 Howitzer
XD-40 V-10

The muzzle break at the end of the barrel deflects gas sideways to reduce recoil.


Explosives

Medieval-style black powder
Modern smokeless powder

                   MJoules  Speed   Density  C  H  N  O
                     /kg    (km/s)  (g/cm3)

Bombardier beetle      .4                                 Hydroquinone + H2O2 + protein catalyst
Ammonium nitrate      2.0    2.55    1.12    0  4  2  3
Black powder          2.6     .6     1.65                 Used before 1884
Smokeless powder      5.2    6.4     1.4     6  9  1  7   Used after 1884. Nitrocellulose
TNT                   4.7    6.9     1.65    7  5  3  6   Trinitrotoluene
PETN                  5.8    8.35    1.77    5  8  4 12
Dynamite              5.9    7.2     1.48    3  5  3  9   75% Nitroglycerine + stabilizer
Composition 4         6.3    8.04    1.59    3  6  6  6   91% RDX. "Plastic explosive"
PLX                   6.5            1.14    1  3  1  2   95% CH3NO2 + 5% C2H4(NH2)2
Nitroglycerine        7.2    8.1     1.59    3  5  3  9   Unstable
RDX (Hexagen)         7.5    8.7     1.78    3  6  6  6
HMX (Octogen)         8.0    9.1     1.86    4  8  8  8
Dinitrodiazeno.       9.2   10.0     1.98    4  0  8  8
Octanitrocubane      11.2   10.6     1.95    8  0  8 16
Gasoline + Oxygen    10.4                    8 18  0 13
Hydrogen + Oxygen    13.16                   0  2  0  1
Uranium bomb     219000
Hydrogen bomb        10 mil
Antimatter        90000 mil

Speed: Detonation speed
C:     Carbon atoms
H:     Hydrogen atoms
N:     Nitrogen atoms
O:     Oxygen atoms
Nitrocellulose
TNT
RDX
HMX
PETN
Octanitrocubane

Nitrocellulose
TNT
RDX
HMX
PETN
Octanitrocubane

Dinitrodiazenofuroxan
Nitromethane

~808  Qing Xuzi publishes a formula resembling gunpower, consisting of
      6 parts sulfur, 6 parts saltpeter, and 1 part birthwort herb (for carbon).
~850  Incendiary property of gunpower discovered
1132  "Fire lances" used in the siege of De'an, China
1220  al-Rammah of Syria publishes "Military Horsemanship and Ingenious War
        Devices", describes the purification of potassium nitrate by
        adding potassium carbonate with boiling water, to precipitate out
        magnesium carbonate and calcium carbonate.
1241  Mongols use firearms at the Battle of Mohi, Hungary
1338  Battle of Arnemuiden.  First naval battle involving cannons.
1346  Cannons used in the Siege of Calais and the Battle of Crecy
1540  Biringuccio publishes "De la pirotechnia", giving recipes for gunpowder
1610  First flintlock rifle
1661  Boyle publishes "The Sceptical Chymist", a treatise on the
      distinction between chemistry and alchemy.  It contains some of the
      earliest modern ideas of atoms, molecules, and chemical reaction,
      and marks the beginning of the history of modern chemistry.
1669  Phosphorus discovered
1774  Lavoisier appointed to develop the French gunpowder program.  By 1788
         French gunpowder was the best in the world.
1832  Braconnot synthesizes the first nitrocellulose (guncotton)
1846  Nitrocellulose published
1847  Sobrero discovers nitroglycerine
1862  LeConte publishes simple recipes for producing potassium nitrate.
1865  Abel develops a safe synthesis of nitrocellulose
1867  Nobel develops dynamite, the first explosive more powerful than black powder
      It uses diatomaceous earth to stabilize nitroglycerine
1884  Vieille invents smokeless gunpowder (nitrocellulose), which is 3 times
         more powerful than black powder and less of a nuisance on the battlefield.
1902  TNT first used in the military.  TNT is much safer than dynamite
1930  RDX appears in military applications
1942  Napalm developed
1949  Discovery that HMX can be synthesized from RDX
1956  C-4 explosive developed (based on RDX)
1999  Eaton and Zhang synthesize octanitrocubane and heptanitrocubane

Above 550 Celsius, potassium nitrate decomposes. 2 KNO3 ↔ 2 KNO2 + O2.

Black powder           =  .75 KNO3  +  .19 Carbon  +  .06 Sulfur
1 kg TNT equivalent    =   4.184   MJ
Fission bomb           =   9.20e13 J     =  22000 tons of TNT equivalent
Fission bomb           =   420     kg
Fission bomb           =   2.19e11 J/kg
Fusion bomb maximum    =   2.51e13 J/kg   (Maximum theoretical efficiency)
Fusion bomb practical  =   1.0 e13 J/kg   (Practical efficiency achieved in real bombs)

Black powder

Sulfur
Sulfur
Saltpeter
Saltpeter

Charcoal
Icing sugar and KNO3
Mortar and pestle
Mortar and pestle

Potassium nitrate  KNO3     75%       (Saltpeter)
Charcoal           C7H4O    15%
Sulfur             S        10%

Oversimplified equation:  2 KNO3 + 3 C + S  →  K2S + N2 + 3 CO2

Realistic equation:       6 KNO3 + C7H4O + 2 S  →  KCO3 + K2SO4 + K2S + 4 CO2 + 2 CO + 2 H2O + 3 N2
Nitrite (NO3) is the oxidizer and sulfur lowers the ignition temperature.
Fuel air explosives
                   MJoules
                     /kg

Black powder           2.6
Smokeless powder       5.2
HMX (Octogen)          8.0
Gasoline + Oxygen     10.4
Hydrogen + Oxygen     13.16
Uranium bomb      219000
Hydrogen bomb         10 mil
Antimatter         90000 mil

        Mass   Energy    Energy/Mass
         kg      MJ         MJ/kg

MOAB    9800   46000        4.7               8500 kg of fuel

Phosphorus
White phosphorus
White, red, violet, and black phosphorus
Red phosphorus

Violet phosphorus
Black phosphorus
Black phosphorus

Form      Ignition    Density
          (Celsius)

White        30        1.83
Red         240        1.88
Violet      300        2.36
Black                  2.69
Red phosphorus is formed by heating white phosphorus to 250 Celsius or by exposing it to sunlight. Violet phosphorus is formed by heating red phosphorus to 550 Celsius. Black phosphorus is formed by heating white phosphorus at a pressure of 12000 atmospheres. Black phosphorus is least reactive form and it is stable below 550 Celsius.
Matches

Striking surface
P4S3

The safety match was invented in 1844 by Pasch. The match head cannot ignite by itself. Ignitition is achieved by striking it on a rough surface that contains red phosphorus. When the match is struck, potassium chlorate in the match head mixes with red phosphorus in the abrasive to produce a mixture that is easily ignited by friction. Antimony trisulfide is added to increase the burn rate.

Match head                 Fraction             Striking surface   Fraction

Potassium chlorate    KClO3  .50                Red phosphorus      .5
Silicon filler        Si     .4                 Abrasive            .25
Sulfur                S      small              Binder              .16
Antimony3 trisulfide  Sb2S3  small              Neutralizer         .05
Neutralizer                  small              Carbon              .04
Glue                         small
A "strike anywhere" match has phosphorus in the match head in the form of phosphorus sesquisulfide (P4S3) and doesn't need red phosphorus in the striking surface. P4S3 has an ignition temperature of 100 Celsius.
Flint

Before the invention of iron, fires were started by striking flint (quartz) with pyrite to generate sparks. Flintlock rifles work by striking flint with iron. With the discovery of cerium, ferrocerium replaced iron and modern butane lighters use ferrocerium, which is still referred to as "flint".

Cerium        .38      Ignition temperature of 165 Celsius
Lanthanum     .22
Iron          .19
Neodymium2    .04
Praseodymium  .04
Magnesium     .04

Nitrous oxide engine

Nitrous oxide is stored as a cryogenic liquid and injected along with gaoline into the combustion chamber. Upon heating to 300 Celsius the nitrous oxide decomposes into nitrogen and oxygen gas and releases energy. The oxygen fraction in this gas is higher than that in air (1/3 vs. .21) and the higher faction allows for more fuel to be consumed per cylinder firing.

Air density                  =  .00122 g/cm3
Nitrous oxide gas density    =  .00198 g/cm3
Diesel density               =  .832   g/cm3
Gasoline density             =  .745   g/cm3
Diesel energy/mass           =  43.1   MJoules/kg
Gasoline energy/mass         =  43.2   MJoules/kg
Nitrous oxide boiling point  = -88.5   Celsius
Air oxygen fraction          =  .21
Nitrous oxide oxygen fraction=  .33
Nitrous oxide decompose temp =  300    Celsius
Nitrous oxide liquid pressure=   52.4  Bars     Pressure required to liquefy N2O at room temperature

Bombardier beetle

Hydroquinone
P-quinone

Hydroquinone and peroxide are stored in 2 separate compartments are pumped into the reaction chamber where they explode with the help of protein catalysts. The explosion vaporizes 1/5 of the liquid and expels the rest as a boiling drop of water, and the p-quinone in the liquid damages the foe's eyes. The energy of expulsion pumps new material into the reaction chamber and the process repeats at a rate of 500 pulses per second and a total of 70 pulses. The beetle has enough ammunition for 20 barrages.

2 H2O2  →  2 H2O +  O2           (with protein catalyst)
C6H4(OH)2  →  C6H4O2 + H2        (with protein catalyst)
O2 + 2 H2  →  2 H2O

Firing rate                     = 500 pulses/second
Number of pulses in one barrage =  70
Firing time                     = .14 seconds
Number of barrages              =  20

Flame speed

A turbojet engine compresses air before burning it to increase the flame speed and make it burn explosively. A ramjet engine moving supersonically doesn't need a turbine to achieve compression.

Turbojet
Ramjet

Airbus A350 compression ratio  =  52
Air density at sea level       = 1    bar
Air density at 15 km altitude  =  .25 bar
Air density in A350 engine     =  13  bar
From the thermal flame theory of Mallard and Le Chatelier,
Temperature of burnt material    =  Tb
Temperature of unburnt material  =  Tu
Temperature of ignition          =  Ti
Fuel density                     =  Dfuel
Oxygen density                   =  Doxygen
Reaction coefficient             =  C
Reaction rate                    =  R  =  C Dfuel Doxygen
Thermal diffusivity              =  Q  = 1.9⋅10-5 m2/s
Flame speed                      =  V

V2  =  Q C Dfuel Doxygen (Tb - Ti) / (Ti - Tu)

Shocks

Spherical implosion
Mach < 1,    Mach = 1,     Mach > 1

If the pressure front moves supersonically then the front forms a discontinuous shock, where the pressure makes a sudden jump as the shock passes.


Energy boost

Metal powder is often included with explosives.

        Energy/mass    Energy/mass
        not including  including
        oxygen         oxygen
        (MJoule/kg)    (MJoule/kg)

Hydrogen    113.4      12.7
Gasoline     46.0      10.2
Beryllium    64.3      23.2
Aluminum     29.3      15.5                                      
Magnesium    24.5      14.8                                      
Carbon       12.0       3.3
Lithium       6.9       3.2
Iron          6.6       4.6                                      
Copper        2.0       1.6

Fireworks

Li
B
Na
Mg
K
Ca
Fe

Cu
Zn
As
Sr
Sb
Rb
Pb

BaCl (green), CuCl (blue), SrCl (red)
Zero gravity
Bunsen burner, O2 increases rightward
Methane


Oxygen candle

Sodium chlorate

An oxygen candle is a mixture of sodium chlorate and iron powder, which when ignited smolders at 600 Celsius and produces oxygen at a rate of 6.5 man-hours of oxygen per kilogram of mixture. Thermal decomposition releases the oxygen and the burning iron provides the heat. The products of the reaction are NaCl and iron oxide.


Aerodynamic drag

Newton length

The characteristic distance a ball travels before air slows it down is the "Newton length". This distance can be estimated by setting the mass of the ball is equal to the mass of the air the ball passes through.

Mass of a soccer ball              =  M  =  .437  kg
Ball radius                        =  R  =  .110  meters
Ball cross-sectional area          =  A  =  .038  meters2
Ball density                       =  D  =  78.4  kg/meters3
Air density                        =  d  =   1.22 kg/meter3   (Air at sea level)
Ball initial velocity              =  V
Newton length                      =  L
Mass of air the ball passes through=  m  =  A L d

m  =  M

L  =  M / (A d)  =  (4/3) R D / d  =  9.6 meters
The depth of the penalty box is 16.45 meters (18 yards). Any shot taken outside the penalty box slows down substantially before reaching the goal.

Newton was also the first to observe the "Magnus effect", where spin causes a ball to curve.


Balls

The orange boxes depict the size of the court and the Newton length is the distance from the bottom of the court to the ball. Ball sizes are magnified by a factor of 20 relative to the court sizes.

          Diameter  Mass  Drag  Shot   Drag/  Density   Ball   Max    Spin
            (mm)    (g)   (m)   (m)    Shot   (g/cm3)   speed  speed  (1/s)
                                                        (m/s)  (m/s)
Ping pong    40      2.7   1.8    2.74    .64   .081     20    31.2    80
Squash       40     24    15.6    9.75   1.60   .716
Golf         43     46    25.9  200       .13  1.10      80    94.3   296
Badminton    54      5.1   1.8   13.4     .14   .062
Racquetball  57     40    12.8   12.22   1.0    .413
Billiards    59    163    48.7    2.7   18     1.52
Tennis       67     58    13.4   23.77    .56   .368     50    73.2   119
Baseball     74.5  146    27.3   19.4    1.4    .675     40    46.9    86
Whiffle      76     45     8.1                  .196
Football    178    420    13.8   20       .67   .142     20    26.8    18
Rugby       191    435    12.4   20       .62   .119
Bowling     217   7260   160     18.29   8.8   1.36
Soccer      220    432     9.3   16.5     .56   .078     40    59      29
Basketball  239    624    11.4    7.24   1.57   .087
Cannonball  220  14000   945   1000       .94  7.9
"Drag" is the Newton drag length and "Shot" is the typical distance of a shot, unless otherwise specified. "Density" is the density of the ball.

For a billiard ball, rolling friction is greater than air drag.

A bowling pin is 38 cm tall, 12 cm wide, and has a mass of 1.58 kg. A bowling ball has to be sufficiently massive to have a chance of knocking over 10 pins.

Mass of 10 bowling pins  /  Mass of bowling ball  =  2.18

Bullet distance

To estimate the distance a bullet travels before being slowed by drag,

Air density              =  Dair    =   .012 g/cm3
Water density            =  Dwater  =  1.0   g/cm3
Bullet density           =  Dbullet = 11.3   g/cm3
Bullet length            =  Lbullet =  2.0   cm
Bullet distance in water =  Lwater  ≈  Lbullet Dbullet / Dwater ≈ 23  cm
Bullet distance in air   =  Lair    ≈  Lbullet Dbullet / Dair  ≈ 185 meters

Density

         g/cm3                                    g/cm3

Air        .00122  (Sea level)           Silver     10.5
Wood       .7 ± .5                       Lead       11.3
Water     1.00                           Uranium    19.1
Magnesium 1.74                           Tungsten   19.2
Aluminum  2.70                           Gold       19.3
Rock      2.6 ± .3                       Osmium     22.6   (Densest element)
Titanium  4.51
Steel     7.9
Copper    9.0

Kinetic energy penetrator

Massive Ordnance Penetrator
Bunker buster

                         Cartridge  Projectile  Length  Diameter  Warhead  Velocity
                            (kg)      (kg)       (m)     (m)       (kg)     (m/s)

Massive Ordnance Penetrator   -       13608     6.2     .8        2404
PGU-14, armor piercing       .694     .395       .173   .030               1013
PGU-13, explosive            .681     .378       .173   .030               1020
The GAU Avenger armor-piercing shell contains .30 kg of depleted uranium.

The massive ordnamce penetrator typically penetrates 61 meters of Earth.

The PGU-13 and PGU-14 are used by the A-10 Warthog cannon.

The composition of natural uranium is .72% uranium-235 and the rest is uranium-238. Depleted uranium has less than .3% of uranium-235.


Drag force

The drag force on an object moving through a fluid is

Velocity             =  V
Fluid density        =  D  =  1.22 kg/m2   (Air at sea level)
Cross-sectional area =  A
Drag coefficient     =  C  =  1            (typical value)
Drag force           =  F  =  ½ C D A V2
Drag power           =  P  =  ½ C D A V3  =  F V
Terminal velocity    =  Vt
"Terminal velocity" occurs when the drag force equals the gravitational force.
M g  =  ½ C D A Vt2
Suppose we want to estimate the parachute size required for a soft landing. Let a "soft landing" be the speed reached if you jump from a height of 2 meters, which is Vt = 6 m/s. If a skydiver has a mass of 100 kg then the area of the parachute required for this velocity is 46 meters2, which corresponds to a parachute radius of 3.8 meters.
Drag coefficient

               Drag coefficient

Bicycle car         .076        Velomobile
Tesla Model 3       .21         2017
Toyota Prius        .24         2016
Bullet              .30
Typical car         .33         Cars range from 1/4 to 1/2
Sphere              .47
Typical truck       .6
Formula-1 car       .9          The drag coeffient is high to give it downforce
Bicycle + rider    1.0
Skier              1.0
Wire               1.2

Speed records

                       m/s     Mach

Swim                    2.39
Boat, human power       5.14
Aircraft, human power  12.3
Run                    12.4
Boat, wind power       18.2
Bike                   22.9
Car, solar power       24.7
Bike, streamlined      38.7
Land animal            33               Cheetah
Bird, level flight     45               White-throated needletail
Aircraft, electric     69
Helicopter            111       .33
Train, wheels         160       .54
Train, maglev         168       .57
Aircraft, propeller   242       .82
Rocket sled, manned   282       .96
Aircraft, manned      981      3.33
Rocket plane, manned 2016      6.83
Rocket sled          2868      9.7
Scramjet             5901     20
Mach 1 = 295 m/s at high altitude.
Fastest manned aircraft
                  Mach

X-15              6.7      Rocket
Blackbird SR-71   3.5
X-2 Starbuster    3.2
MiG-25 Foxbat     2.83
XB-70 Valkyrie    3.0
MiG-31 Foxhound   2.83
F-15 Eagle        2.5
Aardvark F-111    2.5      Bomber
Sukhoi SU-27      2.35
F-22 Raptor       2.25     Fastest stealth aircraft

Drag power

Cycling power

Fluid density    =  D
Cross section    =  A
Drag coef        =  C
Drag force       =  F  =  ½ C A D V2
Drag power       =  P  =  ½ C A D V3  =  K D V3  =  F V
Drag parameter   =  K  =  ½ C A


                 Speed   Density   Drag force   Drag power    Drag
                 (m/s)   (kg/m3)      (kN)       (kWatt)    parameter

Bike                 10       1.22      .035        .305   .50
Bike                 18       1.22      .103       1.78    .50
Bike, speed record   22.9     1.22      .160       3.66    .50
Bike, streamlined    38.7     1.22      .095       3.66    .104
Porche 911           94.4     1.22     7.00      661      1.29
LaFerrari            96.9     1.22     7.31      708      1.28
Lamborghini SV       97.2     1.22     5.75      559      1.00
Skydive, min speed   40       1.22      .75       30       .77        75 kg
Skydive, max speed  124       1.22      .75      101       .087       75 kg
Airbus A380, max    320        .28  1360      435200     94.9
F-22 Raptor         740        .084  312      231000      6.8
SR-71 Blackbird    1100        .038  302      332000      6.6
Sub, human power      4.1  1000         .434       1.78    .052
Blue Whale           13.9  1000      270        3750      2.8         150 tons, 25 Watts/kg
Virginia nuclear sub 17.4  1000     1724       30000     11.4
The drag coefficient is an assumption and the area is inferred from the drag coefficient.

For the skydiver, the minimum speed is for a maximum cross section (spread eagled) and the maximum speed is for a minimum cross section (dive).

Wiki: Energy efficiency in transportation


Altitude

Airplanes fly at high altitude where the air is thin.

                Altitude   Air density
                  (km)     (kg/m3)

Sea level          0       1.22
Denver (1 mile)    1.6      .85
Mount Everest      9.0      .45
Airbus A380       13.1      .25    Commercial airplane cruising altitude
F-22 Raptor       19.8      .084
SR-71 Blackbird   25.9      .038

Drag coefficient and Mach number

Commercial airplanes fly at Mach .9 because the drag coefficient increases sharply at Mach 1.


Turbulence and Reynolds number

The drag coefficient depends on speed.

Object length    =  L
Velocity         =  V
Fluid viscosity  =  Q                  (Pascal seconds)
                 =  1.8⋅10-5 for air
                 =  1.0⋅10-3 for water
Reynolds number  =  R   =  V L / Q      (A measure of the turbulent intensity)
The drag coefficient of a sphere as a function of Reynolds number is:

Golf balls have dimples to generate turbulence in the airflow, which increases the Reynolds number and decrease the drag coefficient.


Drag coefficient and Reynolds number
Reynolds  Soccer  Golf   Baseball   Tennis
 number
  40000   .49    .48      .49       .6
  45000   .50    .35      .50
  50000   .50    .30      .50
  60000   .50    .24      .50
  90000   .50    .25      .50
 110000   .50    .25      .32
 240000   .49    .26
 300000   .46
 330000   .39
 350000   .20
 375000   .09
 400000   .07
 500000   .07
 800000   .10
1000000   .12             .35
2000000   .15
4000000   .18    .30
Data
Drafting

If the cyclists are in single file then the lead rider has to use more power than the following riders. Cyclists take turns occupying the lead.

A "slingshot pass" is enabled by drafting. The trailing car drops back by a few lengths and then accelerates. The fact that he is in the leading car's slipstream means he has a higher top speed. As the trailing car approaches the lead car it moves the side and passes.


Drag differential equation

For an object experiencing drag,

Drag coefficient  =  C
Velocity          =  V
Fluid density     =  D
Cross section     =  A
Mass              =  M
Drag number       =  Z  =  ½ C D A / M
Drag acceleration =  A  =  -Z V2
Initial position  =  X0 =  0
Initial velocity  =  V0
Time              =  T
The drag differential equation and its solution are
A  =  -Z V2
V  =  V0 / (V0 Z T + 1)
X  =  ln(V0 Z T + 1) / Z

Spin force (Magnus force)

Topspin

1672  Newton is the first to note the Magnus effect while observing tennis players
      at Cambridge College.
1742  Robins, a British mathematician and ballistics researcher, explains deviations
      in musket ball trajectories in terms of the Magnus effect.
1852  The German physicist Magnus describes the Magnus effect.
For a spinning tennis ball,
Velocity    =  V                          =    55 m/s             Swift groundstroke
Radius      =  R                          =  .067 m
Area        =  Area                       = .0141 m2
Mass        =  M                          =  .058 kg
Spin number =  S   =  W R / V             =   .25                 Heavy topspin
Spin rate   =  W   =    V / R             =   205 Hz
Air density =  Dair                       =  1.22 kg/m3
Ball density=  Dball
Drag coef   =  Cdrag                      =    .5                 For a sphere
Spin coef   =  Cspin                      =     1                 For a sphere and for S < .25
Drag force  =  Fdrag = ½ Cdrag Dair Area V2   =  13.0 Newtons
Spin force  =  Fspin = ½ Cspin Dair Area V2 S =   6.5 Newtons
Drag accel  =  Adrag                      =   224 m/s2
Spin accel  =  Aspin                      =   112 m/s2
Gravity     =  Fgrav = M g
For a rolling ball the spin number is S=1.

If the spin force equals the gravity force (Fspin = Fgrav),

V2 S C R-1 Dair/Dball = .0383

Spin vs. gravity

Rolling drag

Force of the wheel normal to ground  =  Fnormal
Rolling friction coefficient         =  Croll
Rolling friction force               =  Froll  =  Croll Fnormal

                             Croll

Railroad                      .00035     Steel wheels on steel rails
Steel ball bearings on steel  .00125
Racing bicycle tires          .0025      8 bars of pressure
Typical bicycle tires         .004
18-wheeler truck tires        .005
Typical car tires             .01
Car tires on sand             .3

Rolling friction coefficient
Wheel diameter          =  D
Wheel sinkage depth     =  Z
Rolling coefficient     =  Croll  ≈  (Z/D)½
1672 Newton is the first to note the Magnus effect while observing tennis players at Cambridge College. 1742 Robins, a British mathematician and ballistics researcher, explains deviations in musket ball trajectories in terms of the Magnus effect. 1852 The German physicist Magnus describes the Magnus effect.
Cars

Fast cars
          0-100kph  400m  400m    Top   Power  Mass   Top
             (s)    (s)   speed  speed  (kw)   (kg)  speed
                          (kph)  (kph)               (m/s)

Porche 918      2.2   9.8  233   340    661   1704   94.4
LaFerrari       2.4   9.7  240   349    708   1255   96.9
Bugatti Veyron  2.5   9.7  224   431    883   1888  119.7
Tesla S         2.6  10.9  198   249    568   2000   69.2
Lamborghini SV  2.6  10.4  218   350    559   1769   97.2
Porche 997 S    2.7  10.9  205   315    390   1570   87.5

Vehicle drag

Leitras velomobile
Loremo
Edison 2
BMW i8

Nissan GTR
Lamborghini Diablo
Ford Escape Hybrid
Hummer H2

                  Drag   Area   Drag   Engine  Mass  Top    Top   100 kph  Year
                  coef          Area                speed  speed   time
                          m2     m2     kWatt   kg   k/hr   m/s   s

Aptera 2            .15   1.27   .190    82    820   137   38.1        2011
Loremo              .20   1.25   .250    45          100   27.8        2009
Edison 2 VLC        .16   1.62   .259          450                     2010
Volkswagen XL1      .189  1.48   .279    55    795   158*        11.9  2011
BMW i8              .26   2.11   .548   260   1539   250*  69.4   4.4  2015
Nissan GTR          .27   2.09   .565   357   1740   314   87.2   3.4  2008
Lamborghini Diablo  .31   1.85   .573   362   1576   325   90.3        1995
Tesla Model S       .24   2.40   .576   568   2000   249   69.2   3.0  2012
Toyota Prius        .24   2.40   .576                                  2016
Chevrolet Volt      .281  2.21   .622                             9.5  2014
Mercedes Benz CLA   .30   2.17   .650                                  2013
Nissan Leaf         .29   2.50   .725                                  2012
Ford Escape hybrid  .40   2.62  1.05                                   2005
Hummer H2           .57   4.32  2.46    242   2900                     2003

*: The top speed is electronically limited
The Saab 900, last of the boxy cars

Energy and power

Power sources
                          MJ/kg   kWatt/kg   kWatts   kg

Diesel                    48        -
Gasoline                  45        -
Electric motor, maximum     -     10        200      19.9     EMRAX268 Brushless AC
Turbofan jet engine               10.0       83.2             GE90-115B Brayton
Supercapacitor, BaTiO3     1.47    8.0
ElectriFly brushless DC            7.78       1.04     .133   ElectriFly GPMG5220
Gasoline engine (BMW)              7.5      690               BMV V10 3L P84/5 2005
Battery, LiFePO4            .39    3.3
Model aircraft engine       -      2.8
Battery, lithium-ion        .95    1.5                        Typical commercial battery
Fuel cell, Honda            -      1.0
Battery, Li-S              1.21     .67
Typical diesel V8 turbo     -       .65
Battery, aluminum-air      4.68     .13
Solar cell, space station   -       .077
Solar cell, multilayer      -       .065
Solar cell, polycrystal     -       .040
Nuclear battery, Pu-238  2265000    .0051      .285  56       Teledyne Pu-238 GPHS-RTG

Power/Mass ratio
                     Engine   Engine  Engine  Vehicle  Vehicle   Year
                     (kWatt)   (hp)   (kW/kg)  (ton)   (kW/kg)

Human, sustained          .42     .6   .0067   .062    .0068            Alberto Contador
Human, maximum           2.0     3     .025    .080    .025
Sled dog, sustained       .16     .2   .004    .04     .004    -30000
Horse, sustained          .7     1     .004    .5      .004     -4000
Horse, maximum          11      15     .022    .5      .022     -4000
Car, first prototype      .56     .8   .0021   .265    .0021     1886   Benz Motorwagon. .95L
Car, Ford Model T       15      20             .54     .028      1908   2.9 Litres
Car, Mazda RX-8        184     247    1.5     1.34     .137      2003
Car, Toyota Prius       73      98            1.38     .053      2010   1.8 L
Car, Honda Accord      202     271            1.63     .124      2011
Car, Chevy Corv. C6    321     430            1.44     .223      2005
Car, Porche 911 GT2    390     523            1.44     .271      2007
Car, Lamborghini       493     661            1.55     .318      2009   Murcielago
Car, BMW 7             327     439            2.25     .145      2006   760Li 6 L V12
Car, Ferrari FXX       597     801            1.155    .517      2005
Car, Formula-1         690     925    7.5      .600   1.15       2005   Williams. BMW engine
Car, Hummer H1         224     300            3.56     .063      2006   6.6 L V8
Car, electric           30      40            1.038    .029      2008   Th!nk City
Car, electric           26      35             .635    .041      2008   Tata Nano. .62 L
Car, electric           47      63            1.08     .044      2009   Mitsubishi i MiEV
Car, electric, Tesla   215     288            1.235    .174      2011   Roadster
Car, drag racer       5960    8000            1.043   5.72       2008   John Force Racing
Tank, M1 Abrams       1120    1500           55.7      .020      1980
Motorcycle, Kawasaki    26      35             .182    .143      1987   KLR650 .65 L
Motorcycle, Suzuki      50      67             .194    .258      2004   V-Strom 650 .65 L
Motorcycle, Honda      177     237             .148   1.19       2005   Honda RC211V
Plane, Wright Bros       9      12             .274    .033      1903
Plane, Junkers Jumo    647     867    1.10    6.7      .097      1934
Plane, B-50 Bomber    3210    4300    1.83   38.4      .084      1944
Plane, B-29 Bomber    2540    3400    2.09   33.8      .075      1937
Plane, 747-400     4x44700  4x59900   5.67  178.8     1.00       1971
Plane, 777         2x83200 4x111526  10.0   134.8     1.23       1992
Space shuttle      3x53700   72000  153      68.6     2.35       1981   H2 + O2 rocket
Train, steam, 1829     015             .0035  4.32     .0035     1829   Stephenson's Rocket
Cargo ship           80100  108920     .030  55400     .0014     2006   Emma Maersk
Snowmobile, Polaris    115     154             .221    .523      2009
Model aircraft         930       1.2  2.8                               O.S. Engines 49-PL Type 2
Typical V8 turbo       250     340     .65     .38
Electric motor, max                  10
Nuclear, Pu-238                        .00051                           Galileo spacecraft
1 horsepower = 745.7 Watts
Engine efficiency
Gasoline engine      .15
Diesel engine        .20
Human muscles        .22
Electric car engine  .80
Biomass plant        .25
Natural gas plant    .35
Solar cell           .20     Crystalline type
Solar cell           .40     Multilayer type
Turboprop, Mach .4   .80     Turboprops work up to Mach .5
Turbojet, Mach .4    .40
Turbofan, Mach .4    .68
Turbojet, Mach .9    .77
Turbofan, Mach .9    .90
For an electric vehicle the overall efficiency is similar to that of a diesel engine.
Overall efficiency  =  Power plant efficiency  *  Vehicle efficiency  =  .35 * .80 =  .28

Energy loss
Energy fraction    High speed    Low speed    Bicycle
                     car           car       at 10 m/s

Air drag            .14            .08       .16
Rolling drag        .08            .10       .03
Engine loss         .72            .75       .78
Drivetrain loss     .06            .07       .01
For a cyclist, "engine loss" is the cyclist's muscle energy loss.

For a bicycle, the "drivetrain" is the chain, gears, and derailleurs. The loss from derailleurs is greater than the loss from gears, which is why sprinting bikes have only one gear.

Wiki: Bicycle performance


Fuel efficiency
                       Speed   l/km   l/km/   Passengers
                        m/s           person

Walk                    1.4     .0065  .0065   1         60 Watts
Run                             .009   .009    1
Bike                    4.4     .0032  .0032   1
Bike, aerodynamic      13.9     .0005  .0005   1
Car, solar power                .067   .067    1
Car, electric, Tesla            .015   .004    4
Car, electric, GEM NER 10.8     .012   .003    4
Car, electric, GE EV1           .026   .006    4
Car, electric, Volt             .026   .006    4
Car, VW Bluemotion              .038   .010    4
Car, Honda Insight              .049   .012    4
Car, Toyota Prius               .051   .013    4
Car, Cadillac Wagon             .17    .028    6         6.2L engine
Car, Bugatti Veyron             .24    .12     2
Train, Switzerland              .17    .0026  65
Train, Japan                    .65    .011   59
Plane, Dieselis        44.4     .019   .010    2
Plane, Pipistrel Sinus 62.5     .048   .024    2
Plane, Tecnam Sierra   65.8     .072   .036    2
Plane, DynAero MCR-4S  61.1     .088   .022    4         100 hp
Plane, Boeing 747-400                 3.1    660
Plane, Concorde                      16.6    128
Plane, Airbus A380                    3.0    835
Ship, Queen Elizabeth        300       .17  1777
Ship, Cargo            12.8 1070       -       -         Emma Maersk. 170000 tons
Helicopter, Sikorsky   72.2    1.43    .12    12         Model S-76
1 litre gasoline = 31.7 MJoules
U.S. transportation averages
                  MJ/km/    Passengers
                  person    per vehicle

Train, Switzerland  .085       65
Train, Japan        .35        59
Car, electric      1.2          1.5
Train, city        1.60        30.9
Train, intercity   1.65        24.5
Motorcycle         1.61         1.16
Air                1.85        99.3
Car, gasoline      2.32         1.55
Bus                2.78         9.2
Taxi              10.3          1.55

Freight
                  MJ/km/ton

Ship, U.S. local    .16
Ship, ocean cargo   .22     Emma Maersk. 170000 tons
Train               .21
Truck              2.43
Air                6.9

Formula 1

If everything seems under control, you're just not going fast enough. -- Mario Andretti

I will always be puzzled by the human predilection for piloting vehicles at unsafe velocities -- Data


The car

Car minimum mass           =  702 kg        Includes the driver and not the fuel
Engine volume              =  1.6 litres    Turbocharged. 2 energy recovery systems allowed
Energy recovery max power  =  120 kWatts
Energy recovery max energy =  2 Megajoules/lap
Engine typical power       =  670 kWatts  =  900 horsepower
Engine cylinders           =  6
Engine max frequency       =  15000 RPM
Engine intake              =  450 litres/second
Fuel consumption           =  .75 litres/km
Fuel maximum               =  150 litres
Forward gears              =  8
Reverse gears              =  1
Gear shift time            =  .05 seconds
Lateral accelertion        =  6 g's
Formula1 1g downforce speed=  128 km/h       Speed for which the downforce is 1 g
Formula1 2g downforce speed=  190 km/h       Speed for which the downforce is 2 g
Indycar 1g downforce speed =  190 km/h
Rear tire max width        =  380 mm
Front tire max width       =  245 mm
Tire life                  =  300 km
Brake max temperature      = 1000 Celsius
Deceleration from 100 to 0 kph = 15 meters
Deceleration from 200 to 0 kph = 65 meters    (2.9 seconds)
Time to 100 kph            = 2.4 seconds
Time to 200 kph            = 4.4 seconds
Time to 300 kph            = 8.4 seconds
Max forward acceleration   = 1.45 g
Max breaking acceleration  = 6 g
Max lateral acceleration   = 6 g
Drag at 250 kph            = 1 g

Budget


Timeline
1950  Formula-1 begins. Safety precautions were nonexistent and death was considered
      an acceptable risk for winning races.
1958  Constructor's championship established
1958  First race won by a rear-engine car. Within 2 years all cars had rear engines.
1966  Aerodynamic features are required to be immobile (no air brakes).
1977  First turbocharged car.
1978  The Lotus 79 is introduced, which used ground effect to accelerate air
      under the body of the car, generating downforce. It was also the first
      instance of computer-aided design. It was unbeatable until the introduction
      of the Brabham Fancar.
1978  The Brabham "Fancar" is introduced, which used a fan to extract air from
      underneath the car and enhance downforce. It won the race decisively.
      The rules committee judged it legal for the rest of the season but the
      team diplomatically
      Wiki
1982  Active suspension introduced.
1983  Ground effect banned. The car underside must be flat.
1983  Cars with more than 4 wheels banned.
1989  Turbochargers banned.
1993  Continuously variable transmission banned before it ever appears.
1994  Electronic performance-enhancing technology banned, such as active suspension,
      traction control, launch control, anti-lock breaking, and 4-wheel steering.
      (4-wheel steering was never implemented)
1999  Flexible wings banned.
2001  Traction control allowed because it was unpoliceable.
2001  Beryllium alloys in chassis or engines banned.
2002  Team orders banned after Rubens Barrichello hands victory to Michael
      Schumacher at final corner of the Austrian Grand Prix.
2004  Automatic transmission banned.
2007  Tuned mass damper system banned.
2008  Traction control banned. All teams must use a standard electrontrol unit.
2009  Kinetic energy recovery systems allowed.

Circuits

Catalunya
Suzuka
Magny Cours


Points
Place   Points          Place   Points

  1       25              6       8
  2       18              7       6
  3       15              8       4
  4       12              9       2
  5       10             10       1

Electric cars

Tesla Model S

Electric cars outperform gasoline cars in all aspects except range. They're simpler, more powerful, quieter, and more flexible, plus in time the cost of batteries will decrease and the range problem will be solved.

The range of an electric car depends on the battery price and is typically .03 kilometers per battery dollar. The value for battery energy/$ increases with time and in the future range won't be a problem. The battery tends to be around 1/3 the cost of the car. For a typical car,

Battery energy/mass =  em           =  .60 MJoules/kg
Battery energy/$    =  e$           =.0070 MJoules/$
Battery mass        =  Mb           =  100 kg
Battery energy      =  E  =  Mb em  =   60 MJoules
Battery cost        =  P  =  E /e$  = 8570 $
Range               =  X            =  273 km            At a city speed of 17 m/s
Range/$             =  X$ =  X/P    = .032 km/$

The range is limited by air drag and rolling drag. Rolling drag dominates at low speed and air drag dominates at high speed, and at the critical "drag speed" Vd they are equal, typically around 17 m/s. For a typical car,

Car mass                   =  M           = 1200 kg
Gravity constant           =  g           =  9.8 m/s2
Tire rolling drag coeff    =  Cr          =.0075
Rolling drag force         =  Fr = Cr M g =   88 Newtons

Air drag coefficient       =  Ca          =  .25
Air density                =  D           = 1.22 kg/meter3
Air drag cross-section     =  A           =  2.0 m2
Car velocity               =  V           =   17 m/s      (City speed. 38 mph)
Air drag force             =  Fa = ½CaADV2 =  88 Newtons

Total drag force           =  F  = Fr + Fa = 176 Newtons
Drag speed                 =  Vd           =  17 m/s     Speed for which air drag equals rolling drag
Car electrical efficiency  =  Q            = .80
Battery energy             =  E            =  60 MJoules
Work done from drag        =  EQ = F X     =  Cr M g [1 + (V/Vd)2] X
Range                      =  X  = EQ/(CrMg)/[1+(V/Vd)2] =  272 km
The range is determined by equating the work from drag with the energy delivered by the battery.   E Q = F X.

The drag speed Vd is determined by setting Fr = Fa.

Drag speed  =  Vd  =  [Cr M g / (½ Ca D A)]½  =  4.01 [Cr M /(Ca A)]½  =  17.0 meters/second

Electric car batteries

Mitsubishi i-MiEV
Nissan Leaf

The battery is the dominant cost in an electric car.

                  Engine  Battery  Battery  Battery  Battery  Battery  Battery  Battery  Car    Car   Car
                  power    power   energy    mass                       cost             mass  range  cost
                  kWatt    kWatt   MJoule     kg      MJ/kg    kW/kg      $      MJ/$    kg     km     $

Tesla S P85D         568    397     306       540     .57      .74     44000    .0070    2239   426  115000
Ford Focus Electric  107     92      82.8     295     .281     .31     12000    .0069    1674   122   22495
Nissan Leaf           80     80      76.7     218     .35      .37      5500    .0139    1493   172   22360
Mitsubishi i-MiEV     47     47      58       201     .288     .23                       1080   100   16345

Electric car vs. electric aircraft

An electric aircraft uses 7 times more energy than an electric car in terms of energy/distance/mass. Electric aircraft have a cruising speed of 50 m/s and electric cars in cities move at around 15 m/s.

Typical values for electric aircraft and cars are:

Electric aircraft speed                     =  50 m/s
Electric aircraft power/mass                =  50 Watts/kg
Electric aircraft energy/distance/mass=  eair= 1.0 Joules/m/kg
Electric aircraft flying time         =  T  =3600 seconds
Electric aircraft range               =  X  = 180 km
Gravity constant                      =  g  = 9.8 m/s2
Electric car mass                     =  M
Electric car rolling drag coefficient =  Cr = .0075
Electric car rolling drag             =  Fr =  Cr M g
Electric car total drag               =  F  =  2 Fr        (Assume rolling drag = air drag)
Electric car energy/distance/mass     =  ecar=  2 Cr g  =  .147 Joules/m/kg
Aircraft energy / Car energy          =  eair / ecar  =  6.8

Energy source

Tesla Roadster

The performance of a car depends on its power source. Lithium batteries have a substantially lower value for energy/mass than gasoline.

                         Energy/Mass   Power/mass  Energy/$   Recharge    Max
                          MJoule/kg     kWatt/kg   MJoule/$   time       charges

Diesel fuel                   48          -        41          -
Lithium battery                 .60        .75       .007     hour       104
Lithium-ion supercapacitor      .054     15                   seconds    105
Supercapacitor                  .016      8          .00005   seconds    106
Aluminum electrolyte capacitor  .010     10          .0001    seconds     ∞
Electric motors can reach a power/mass of 10 kWatts/kg.
Supercapacitors are a rapidly-improving technology and lithium batteries are a mature technology.
Motors

Electric motors and gasoline motors have a similar power/mass.

                        MJ/kg  kWatt/kg  kWatts  kg

Supercapacitor, Li-ion    .054  15
Electric motor, maximum   -     10       200     19.9    EMRAX268 Brushless AC
Turbofan jet engine       -     10.0      83.2    8.32   GE90-115B Brayton
Electric motor, DC        -      7.8       1.04    .133  ElectriFly GPMG5220 brushless DC
Gasoline engine (BMW)            7.5     690             BMV V10 3L P84/5 2005
Model aircraft engine     -      2.8
Battery, lithium-ion      .75    1.5
Fuel cell, Honda          -      1.0
Typical diesel V8 turbo   -       .65
Solar cell, space station -       .077

Rolling drag

Typical car tires have a rolling drag coefficient of .01 and specialized tires can achieve lower values.

Mass of car              =  M
Gravity coefficient      =  g  =  9.8 meters/second2
Car downward force       =  Fg =  M g
Rolling drag coefficient =  Cr
Rolling drag             =  Fr =  C Fg
The tires with the lowest rolling drag coefficient are:
Tire                  Rolling drag coefficient

Bridgestone B381          .00615
Michelin Symmetry         .00650
Michelin Tiger Paw        .00683
Bridgestone Dueller       .00700
BFGoodrich Rugged Trail   .00709
Michelin LTX              .00754
Goodyear Integrity        .00758

Railroad                  .00035       Steel wheels on steel rails
Racing bicycle tires      .0025        8 bars of pressure
Typical bicycle tires     .004
18-wheeler truck tires    .005
Typical car tire          .01
Data
Transport energy cost

The cost of transport depends on the drag force per person.

Drag force              =  F
Distance traveled       =  X
Energy expended         =  E  =  F X
Number of people        =  N
Energy/distance/person  =  Z  =  E/X/N  =  F/N
Drag speed              =  Vd         (Speed for which rolling drag = air drag)
For typical vehicles,
                            Mass   Cr    Area    Cd  Drag speed   People   Force/person  Force/person
                             kg            m2           m/s                 at 15 m/s      at 30 m/s

Electric bike                  20  .003      .7   1.0     2.6      1          99         387
Electric scooter              120  .005     1.0   1.0     4.0      1         147         550
Electric car, lightweight     600  .0075    2.0    .20   14.3      1         105         269
Electric car, middleweight   1200  .0075    2.5    .25   15.7      1         180         437
Electric car, heavyweight    1800  .0075    3.0    .3    15.9      1         262         632
Bus                         10000  .005     8.0    .6    15.7     60          23.1        56.0
Subway car                  34000  .00035  10.0    .6     6.3    100           9.7        34.4
18-wheel truck              36000  .005     8.0    .6    24.6      1        2422        4397
Aircraft, 747              220128   -        -     -       -     480                     625
The force/person for the 747 aircraft is for a cruising speed of Mach .9 and an altitude of 12 km.

We assume that each person adds 80 kg to the mass of the vehicle.

Buses use 5 times less energy as cars but only if they are full

Within cities, cars have faster travel times than buses, especially if parking is abundant or the cars are self-driving. Buses are more suited to inter-city transport.

Trains use 2 times less energy than buses but they are highly inflexible.


Acceleration

Acceleration depends on the size of the battery, and supercapacitors can add an extra boost. For a typical car that accelerates from 0 to 100 km/h (37.8 m/s) in 8 seconds, the size of the battery required is:

Car mass          =  M             =  1200 kg
Target speed      =  V             =  27.8 m/s           (100 km/h. Speed at end of acceleration)
Kinetic energy    =  E  =  ½ M V2  =  464000 Joules
Time              =  T             =     8 seconds   (Time to accelerate from rest to speed V)
Engine efficiency =  Q             =    .8
Power             =  P  =  E/T     = 58000 Watts
Battey power/mass =  p             =   750 Watts/kg
Battery mass      =  MB =  P / (Q p)   =   112 kg
Battery cost/mass =  c             =    86 $/kg
Battery cost      =  C             =  9600 $

Recovering breaking energy

Supercapacitors are ideal for recovering breaking energy because they can be charged/discharged more times than batteries. To capture the energy from breaking from freeway speed, on order of 27 kg of supercapacotors are required.

Car mass                   =  M   =  1200 kg
Car velocity               =  V   =  27.8 m/s
Car kinetic energy         =  E   =464000 Joules
Supercapacitor energy/mass =  e   = 16000 Joules/kg
Supercapacitor mass        =  E/e =    29 kg

Flying cars

Fantrainer
Fantrainer
Terra Fugia

Optica
Ducted fan

The principal challenge for flying cars is noise. There is no such thing as a quiet flying car.

Fixed wing flight is at least 6 times more efficient than helicopter flight.

The larger the propeller the less noise. The sound power of a propeller scales as the 5th power of tip speed. A flying car should have a propeller as large as possible. A single large propeller is better than multiple small propellers.

A ducted (shielded) propeller is substantially quieter than an unshielded propeller, and is more efficient in producing thrust.

Electric aircraft are substantially simpler and safer than gasoline aircraft.

Nominal configuration for a quiet flying car:

A single large ducted fan mounted on the rear
50 kWatt engine
Gyrofans and gyroscopes for stability
Wings that fold for driving
Telescoping wing section within the main wing (wings should be as long as possible)
A cockpit with a low cross-section, like a velobike. The passengers sit behind the pilot
Thin tires for a low cross section
A 10 kg vehicle parachute for emergency landing
2 kg parachutes for passengers
A total mass in the range of 500 kg


Friction

Fcontact  =  Contact force between the object and a surface (usually gravity)
Ffriction =  Maximum friction force transverse to the surface of contact.
C        =  Coefficient of friction, usually with a magnitude of ~ 1.0.

Ffriction  =  C Fcontact
The larger the contact force the larger the maximum friction force.
      Coefficient of friction
Ice           .05
Tires        1
When two surfaces first come together there is an instant of large surface force, which allows for a large friction force.
Agassi returning a Sampras serve. At T=0:07 Agassi's feet hit the ground simultaneous with when he reads the serve.

Maximum drag racing acceleration
Mass                                             =  M
Contact force between the car and the road       =  Fcontact   =  M g
Maximum friction force that the road can provide =  Ffriction  =  C Fcontact
Maximum acceleration that friction can provide   =  A  =  Ffriction / M
                                                       =  C Fcontact / M
                                                       =  C g M / M
                                                       =  C g
This clip shows the magnitude and direction of the acceleration while a Formula-1 car navigates a racetrack.
Formula-1 lap

Villeneuve vs. Arnoux At 0:49 Arnoux breaks before he hits the turn.


Maximum cornering acceleration

For maximum cornering acceleration, the same equations apply as for the maximum drag racing acceleration. It doesn't matter in which direction the acceleration is.

Maximum cornering acceleration  =  C g

Friction on a ramp

Suppose an object with mass m rests on a ramp inclined by an angle theta. The gravitational force on the object is

F = m g
The force between the object and the surface is equal to the component of the gravitational force perpendicular to the surface.
Fcontact = Fgrav * cos(θ)
The force of gravity parallel to the ramp surface is
Framp = Fgrav sin(θ)
Th maximum friction force that the ramp can exert is
Ffriction = C Fcontact
This is balanced by the gravitational force along the ramp
Ffriction = Framp

Fgrav sin(θ) = C Fgrav cos(θ)

C = tan(θ)
This is a handy way to measure the coefficient of friction. Tilt the ramp until the object slides and measure the angle.
Flight


Lift

Air density            =  D
Velocity               =  V
Wing area              =  Awing
Wing drag coefficient  =  Cwing
Drag force on the wing =  Fdrag = ½ CWing Awing D V2


             Cwing

F-4 Phantom   .021     (subsonic)
Cessna 310    .027
Airbus A380   .027
Boeing 747    .031
F-4 Phantom   .044     (supersonic)

Lift-to-drag ratio
Flift  =  Lift force (upward)
Fdrag  =  Drag force (rearward)
Qlift  =  Lift-to-drag coefficient  =  Flift / Fdrag

              Qlift

U-2            23     High-altitude spy plane
Albatross      20     Largest bird
Gossamer       20     Gossamer albatross, human-powered aircraft  
Hang glider    15
Tern           12
Herring Gull   10
Airbus A380     7.5
Concorde        7.1
Boeing 747      7
Cessna 150      7
Parachute       5
Sparrow         4
Wingsuit        2.5
Flying lemur    ?     Most capable gliding mammal.  2 kg max
Flying squirrel 2.0

Gliding

A glider is an airplane without an engine. The more efficient the glider, the smaller the glide angle. The minimum glide angle is determined by the wing lift/drag coefficient.

Wing lift/drag coefficient =  Qlift  =  Flift / Fdrag
Glider horizontal velocity =  Vx
Glider vertical velocity   =  Vz
Drag force                 =  Fdrag
Gravitational force        =  Fgrav
Lift force                 =  Flift  =  Fgrav
Drag power                 =  Pdrag  =  Fdrag Vx
Power from gravit          =  Pgrav  =  Fgrav Vz
If the glider descends at constant velocity,
Pdrag  =  Pgrav
The goal of a glider is to maximize the glide ratio Vx / Vz.
Vx / Vz  =  (Pdrag / Fdrag)  /  (Pgrav / Fgrav)
         =  Fgrav / Fdrag
         =  Qlift
The glide ratio is equal to the lift coefficient Qlift.

Level flight

D    =  Air density
Awing =  Wing area
Cwing =  Wing drag coefficient
Fdrag =  Drag force on the wing   =  ½ Cwing D Awing V^2
Qwing =  Wing lift coefficient    =  Flift / Fdrag
Flift =  Lift force from the wing =  Qwing Fdrag
M    =  Aircraft mass
Feng  =  Engine force
Fgrav =  Gravity force            =  M g
Pdrag =  Drag power               =  Fdrag V  =  ½ Cwing D Awing V3
V    =  Cruising speed
Agility= Power-to-weight ratio    =  Pdrag / M  =  V g / Q      (derived below)
For flight at constant velocity,
Feng  =  Fdrag              Horizontal force balance

Flift =  Fgrav              Vertical force balance

Agility =  Pdrag   / M
        =  Fdrag V / M
        =  Flift V / M / Q
        =  M g  V / M / Q
        =  V g / Q
We can use this equation to solve for the minimum agility required to fly.
Pdrag  =  M g V / Q  =  ½ Cwing D Awing V3

Agility  =  g3/2 M½ Q-3/2 (½ C D A)
If we assume that mass scales as size cubed and wing area scales as size squared, then
Awing   ~  M2/3

Agility ~  g3/2 M1/6 Q-3/2 C D

Aircraft data

Cessna 150
Boeing 747
Airbus 380

SR-71 Blackbird
U-2 spy plane
Concorde
Concorde temperature at Mach 2

         Vcruise  Vmax  Mass  Takeoff  Ceiling  Density  Force  Wing   Len   Wing   Range
           m/s   m/s   ton    ton      km      kg/m3     kN     m2     m     m      km

Cessna 150    42   56     .60     .73  4.3   .79      1.34   15     7.3  10.1    778
Boeing 747   254  274  178.1   377.8  11.0   .36   1128     525    70.6  64.4  14200
Boeing 787-9 251  262  128.9   254.0  13.1   .26    640     360.5  62.8  60.1  14140
Airbus A380  243  262  276.8   575    13.1   .26   1360     845    72.2  79.8  15200
Concorde     599  605   78.7   190.5  18.3   .115   560     358.2  61.7  25.6   7223
F-22 Raptor  544  740   19.7    38.0  19.8   .091   312      78.0  18.9  13.6   2960
U-2          192  224    6.49   18.1  21.3   .071    84.5    92.9  19.2  31.4  10308
SR-71        954  983   30.6    78.0  25.9   .034   302     170    32.7  16.9   5400
Mach 1 = 298 m/s.

Altitude

Commercial airplanes fly at high altitude where the air is thin. The thinner the air, the less the drag force and the less the energy required to travel a given distance.

                Altitude   Density
                  (km)     (kg/m3)

Sea level          0       1.22
Cessna 150         3.0      .79
Boeing 747        11.0      .36
Airbus A380       13.1      .26
Concorde          18.3      .115
F-22 Raptor       19.8      .091
U-2               21.3      .071
SR-71 Blackbird   25.9      .034

Solar powered aircraft
                Cruise  Max  Ceiling  Mass  Cruise  Motor  Solar  Cells  Battery
                 m/s    m/s    kW     tons    kw     kW    cells   m2     tons
                                                            kW

Aquila           35.8          27.4     .40   5.0                          .2
Solar Impulse 2  25.0   38.9   12      2.3           52     66    269.5    .633

The Loon balloon is 15 meters wide, 12 meters, tall, and .076 mm thick. The solar panels generate 100 Watts and the payload is 10 kg. It is too large to be self-propelled and relies and buoyancy modulation and air currents to maneuver.


History
1961  Piggott accomplishes the first human-powered flight, covering a distance
      of 650 meters.
1977  The "Gossomer Condor 2" flies 2172 meters in a figure-eight and wins
      the Kremer Prize.  It was built by Paul MacCready and piloted by amateur
      cyclist and hang-glider pilot Bryan Allen. 
      It cruised at 5.0 m/s with a power of 260 Watts.
1988  The MIT Daedalus 88 piloted by Kanellos Kanellopoulos flies from Crete
      to Santorini (115.11 km), setting the distance record, which still stands.
Human-powered helicopters can only reach a height of 3 meters and can only hover for 20 seconds.

Agility
               Mass    Power   Agility
               (kg)    (kW)   (Watts/kg)

Human             75    2500     33
BMW i8          1485     170    114
Cessna 150       600      75    125
Airbus A380   276000   49000    178
Formula-1 car    642     619    964
SR-71          30600   33000   1078
F-22 Raptor    19700   33000   1675
If you put a wing on a BMW i8, it would be able to go fast enough to take off.
Wing shape
Xwing =  Length of the wing, from the fuselage to the tip
Ywing =  Wing dimension in the direction of flight,
        measured along the point of attachment with the fuselage
Awing =  Wing area
Rwing =  Wing aspect ratio   =  Xwing / Ywing
Qlift =  Wing lift-drag ratio


         QLift  Rwing     Wing     Xwing
                           area
                           (m2)        (m)
U-2         23     10.6                        High-altitude spy plane
Albatros    20                       1.7       Largest bird
Gossamer    20             41.34    14.6       Gossamer albatross, human-powered aircraft  
Airbus A380  7.5    7.5   845       36.3
Concorde     7.1          358.2     11.4
Boeing 747   7      7.9   525       29.3
Cessna 150   7             15        4.5
Wingsuit     2.5    1       2        1.0
QLift tends to be proportional to Rwing.

Wingtip vortex

A wingtip creates a vortex as it moves. Birds fly in a "V" formation to use the updraft from their neighbor's wingtip vortex.


Flight on other worlds

The minimum agility required to fly scales as

Agility  ~  g3/2 M1/6 Q-3/2 Cwing D
We can normalize the Earth to 1 and estimate the minimum agility for other planets. For example,
MarsAgility / EarthAgility  =  (MarsGravity / EarthGravity)3/2 * (MarsDensity / EarthDensity)


        Gravity  Atmosphere   Agility     Power/     Maximum
                  density     normalized   mass      mass for
        (m/s^2)   (kg/m^3)    to Earth   (Watts/kg)  flight (kg)

Earth     9.78      1.22       1.0        400              20
Mars      3.8        .020      1.89       756                .44
Titan     1.35      5.3         .025       10        >1000000
Venus     8.87     67           .12        48        >1000000
Pandora   7.8       1.46        .65       261             265
For the "Power/mass" column we assume that the power required for human flight is 400 Watts and estimate the power required for flight on other planets.

On Titan you can fly with a wingsuit. A creature as massive as a whale can fly.

"Pandora" is the fictional moon from the film "Avatar".

The largest flying birds on the Earth have a mass of 20 kg. We can use the agility scaling to estimate the maximum mass for flight on other planets.

Agility ~  g3/2 M1/6 Q-3/2 Cwing D

M       ~  g-9  D3

Downforce

The wing on a Formula-1 car is an upside-down aircraft wing that generates downforce, to help with friction.

M   =  Mass
V   =  Velocity
A   =  Acceleration (in any direction)
Cfri=  Friction coefficient
C  =  Wing coefficient for downforce
Fgrav=  Gravitational force on the car
                =  M g
F  =  Downforce from the wing
    =  M g C V2
Ffri=  Maximum friction force
    =  Cfri (Fgrav + F)
    =  Cfri M g (1 + C V2)
A formula-1 car generates 1 g of downforce at 50 m/s, hence C = 1/502. At the top speed of 100 m/s the downforce is 4 g. The maximum accelerations incurred by the driver are of order 5 g.

The maximum cornering speed for a circle of radius R is:

Ffri  =  M V2/R  =  M g Cfri (1 + V2/Cfri2)

V2 = g R Cfri / (1 - R/Cfri2)

Orville and Wilbur Wright

Orville Wright
Wilbur Wright

The Wright brothers designed, manufactured, and sold their own line of bicycles and used the profits to fund their aircraft research.

They began by designing wings and gliders.

They were the first to use wind tunnels to test wings. Their wing design eclipsed the European designs.

They innovated the design of steering and stability systems

They advanced the design of propellers.

First flight
82nd flight: 2.75 miles and 304 seconds

Orville was injured in a crash and a passenger was killed
The original aircraft


Angle of attack

The angle of attack is the angle of the plane's noze with respect to level fight. As the angle of attack increases the lift increases, with an accompanying increase in drag. If the angle of attack is too high then lift drops and the plane stalls.


Engines


Turboprop

In a turboprop, incoming air is compressed by a fan and then mixed with fuel. The exploding air powers a crankshaft that turns the propeller. This is the most efficient engine up to Mach .5.


Turbofan
In a turbofan the exploding fuel+air mixture powers a fan. Some of the incoming air is reacted with fuel and most of it is bypassed, channelled instead to the fan and propelled aftward.

Turbofans are the most efficient kind of engine from Mach .5 to 1.0. All commercial aircraft that fly close to Mach 1 are turbofans.


Turbojet

A turbojet uses most of its intake air for combustion and most of the thrust comes from jet action rather than from a fan. Such engines can go beyond Mach 1.


Afterburner

F/A-18 Hornet
SR-71 Blackbird
SR-71 Blackbird engine test

A turbofan in afterburner mode injects fuel aft of the fan. The fuel explodes and adds thrust. This enables an aircraft to go beyond Mach 1.


Ramjet

If the aircraft is moving faster than Mach 1 then the incoming air doesn't need to be compressed with a fan. The ram pressure from slowing down the air in the combustion chamber is enough.

Ramjets work up to Mach 6, at which point the incoming compressed air becomes too hot.

Ramjets are simpler than turbofans because they don't have any moving parts. They are often used for missiles.


Scramjet

NASA X-23
Turbofan, ramjet, scramjet

In a scramjet the incoming air is compressed enough to make it explodable but not enough to slow it down below Mach 1. Such an engine sidesteps the heating concern of a ramjet and can go up to Mach 15.


SR-71 Blackbird engine


de Laval nozzle

In a de Laval nozzle, incomming subsonic air accelerates as it approaches the nozzle. The nozzle is shaped so that the air becomes supersonic at the narrowest point, after which it continues accelerating as it expands aftward.


Specific impulse

Exhaust velocity  =  V
Gravity constant  =  g
Specific impulse  =  I  =  V/g

Air compression

A jet engine compresses air before mixing it with fuel. For adiabatic compression,

Adiabatic index =  A  =  1.4      (for air)
Density         =  D
Pressure        =  P  ≈  DA  ≈  D T
Temperature     =  T  ≈  DA-1
The larger the air density the more efficient the engine. Increased density comes with increased temperature and the performance of an engine is determined by the quality of the high-temperature alloys.

The following table shows the properties of an adiabatically compressed gas in relative units and in Kelvin. In practice the gas compression is not adiabatic and the pressure and temperature values are larger.

Density  Pressure   Temperature   Temperature
                                   (Kelvin)
   1      1             1            250         Ambient air at 10 km altitude
   2      2.64          1.32         330
   3      4.66          1.55         388
   4      6.96          1.74         435
   6     12.3           2.05         512
   8     18.4           2.30         574
  12     32.4           2.70         675
  16     48.5           3.03         758
  24     85.6           3.57         891
  32    128.0           4.0         1000
Aircraft engines:
                                   Pressure ratio

Airbus A350   Rolls-Royce Trent XWB     52
Boeing 747    General Electric CF6      42
Boeing 777    General Electric GE90     42
Airbus A380   Rolls-Royce Trent 900     39
F-15          General Electric F110     30
Concorde      Rolls-Royce 593           15.5

Bird flight

Flapping the wings propels the bird upward and then the bird glides downward until the next flap. Forward propulsion comes more from gravitational descent than from flapping.


Flight lab


Wings

Construct a wing and a wind tunnel and measure the following:

Wing drag force       =  Fdrag  =  Rearward force on the wing
Wing lift force       =  Flift  =  Upward lift force on the wing
Wing lift coefficient =  Qlift  =  Flift / Fdrag
The larger the value of Qlift the better. You can measure the lift coefficient by measuring wing forces as above or you can measure the glide ratio, discussed below.

You will have to adjust the wing's "angle of attack" to get the optimal lift coefficient.


Gliders

Build a glider using any materials available and try to minimze the glide angle, which is defined as the change in height divided by the change in lateral distance. Try different kinds of designs and see if you can optimize the design.

To measure the glide ratio, launch the glider with zero initial velocity from a height close to the ceiling and measure how far it travels before hitting the floor.

Z  =  Glide ratio  =  Distance traveled / Initial height
The larger the glide ratio the better.

The glide ratio is equal to the lift coefficient.

Z  =  Qlift

Propellers

First electric helicopter, 2011

Construct a propeller powered by a rubber band and construct a helicopter to support the propeller. Place the helicopter on the ground and engage the propeller and measure the maximum height that the helicopter reaches.

Maximum height of the helicopter  =  H
The larger the value of H the better.
Propeller aircraft

The propeller provides forward force and the wing provides upward force.

Combine a glider with a rubber-band-powered propeller to form a propeller aircraft. Release it from at rest on the ground and measure the distance it flies.

Distance the plane flies  =  X
The larger the value of X the better.

Combat aircraft

F-22 Raptor
F-35 Lightning
F-15 Eagle

F-15 Eagle cockpit
F-16 Falcon
MiG-25 Foxbat

               Speed  Mass  Takeoff  Ceiling  Thrust  Range  Cost  Number Year Stealth
               Mach   ton     ton      km       kN     km     M$

SR-71 Blackbird  3.3   30.6   78.0     25.9    302    5400          32   1966
MiG-25 Foxbat    2.83  20.0   36.7     20.7    200.2  1730        1186   1970
MiG-31 Foxhound  2.83  21.8   46.2     20.6    304    1450         519   1981
F-22A Raptor     2.51  19.7   38.0     19.8    312    2960   150   195   2005   *
F-15 Eagle       2.5   12.7   30.8     20.0    211.4  4000    28   192   1976
MiG-29 Fulcrum   2.25  11.0   20.0     18.0    162.8  1430    29  1600   1982
Su-35            2.25  18.4   34.5     18.0    284    3600    40    48   1988
F-4 Phantom II   2.23  13.8   28.0     18.3           1500        5195   1958
Chengdu J-10     2.2    9.8   19.3     18.0    130    1850    28   400   2005
F-16 Falcon      2.0    8.6   19.2     15.2    127    1200    15   957   1978
Chengdu J-7      2.0    5.3    9.1     17.5     64.7   850        2400   1966
Dassault Rafale  1.8   10.3   24.5     15.2    151.2  3700    79   152   2001
Euro Typhoon     1.75  11.0   23.5     19.8    180    2900    90   478   2003
F-35A Lightning  1.61  13.2   31.8     15.2    191    2220    85    77   2006   *
B-52              .99  83.2  220       15.0    608   14080    84   744   1952
B-2 Bomber        .95  71.7  170.6     15.2    308   11100   740    21   1997   *
A-10C Warthog     .83  11.3   23.0     13.7     80.6  1200    19   291   1972
Drone RQ-180          ~15              18.3          ~2200               2015   *
Drone X-47B       .95   6.4   20.2     12.2           3890           2   2011   *  Carrier
Drone Avenger     .70          8.3     15.2     17.8  2900    12     3   2009   *
Drone RQ-4        .60   6.8   14.6     18.3     34   22800   131    42   1998
Drone Reaper      .34   2.2    4.8     15.2      5.0  1852    17   163   2007
Drone RQ-170                           15                           20   2007   *

India HAL AMCA   2.5   14.0   36.0     18.0    250    2800     ?     0   2023   *
India HAL FGFA   2.3   18.0   35.0     20.0    352    3500     ?     0  >2020   *
Mitsubishi F-3   2.25   9.7     ?        ?      98.1  3200     ?     1   2024   *
Chengdu J-20     2.0   19.4   36.3       ?     359.8     ?   110     4   2018   *
Sukhoi PAK FA    2.0   18.0   35.0     20.0    334    3500    50     6   2018   *
Shenyang J-31    1.8   17.6   25.0       ?     200    4000     ?     0   2018   *

Mach 1 = 295 m/s
5th generation fighters: F-22, F-35, X-2, HAL AMCA, J-20, J-31, Sukhoi PAK FA

An aircraft moving at Mach 2 and turning with a radius of 1.2 km has a g force of 7 g's.


Drones

X-47B
X-47B

RQ-170 Sentinel
MQ-9 Reaper


Missiles

Air to air missiles

F-22 and the AIM-120
AIM-9
Astra
Predator and Hellfire
Helfire in a transparent case

                Mach   Range  Missile  Warhead  Year  Engine
                        km      kg       kg

Russia  R-37      6      400    600      60    1989   Solid rocket
Japan   AAM-4     5      100    224       ?    1999   Ramjet
India   Astra     4.5+   110    154      15    2010   Solid rocket
EU      Meteor    4+     200    185       ?    2012   Ramjet
Russia  R-77-PD   4      200    175      22.5  1994   Ramjet
USA     AIM-120D  4      180    152      18    2008   Solid rocket
Israel  Derby-IR  4      100    118      23           Solid rocket
Israel  Rafael    4       50    118      23    1990   Solid rocket
France  MICA      4       50    112      12    1996   Solid rocket
Israel  Python 5  4       20    105      11           Solid rocket
Russia  K-100     3.3    400    748      50    2010   Solid rocket
UK      ASRAAM    3+      50     88      10    1998   Solid rocket
Germany IRIS-T    3       25     87.4          2005   Solid rocket
USA     AIM-9X    2.5+    35     86       9    2003   Solid rocket
USA     Hellfire  1.3      8     49       9    1984   Solid rocket  AGM-114

Ground to air missiles

David's Sling
Terminal High Altitude Area Defense (THAAD)

SM-3
SM-3
Chu-SAM
RIM-174

                 Mach   Range  Missile  Warhead  Year  Engine     Stages   Anti
                         km      kg       kg                              missile

USA     SM-3      15.2   2500   1500       0    2009   Solid rocket  4       *
Israel  Arrow      9      150   1300     150    2000   Solid rocket  2
USA     THAAD      8.24   200    900       0    2008   Solid rocket          *
USA     David      7.5    300                   2016   Solid rocket          *
Russia  S-400      6.8    400   1835     180    2007   Solid rocket          *
India   Prithvi    5     2000   5600            2006   Solid, liquid 2       *
India   AAD Ashwin 4.5    200   1200       0    2007   Solid rocket  1
Taiwan  Sky Bow 2  4.5    150   1135      90    1998   Solid rocket
China   HQ-9       4.2    200   1300     180    1997   Solid rocket  2
USA     Patriot 3  4.1     35    700      90    2000   Solid rocket          *
China   KS-1       4.1     50    900     100    2006   Solid rocket          *
USA     RIM-174    3.5    460   1500      64    2013   Solid rocket  2
India   Barak 8    2      100    275      60    1015   Solid rocket  2
Japan   Chu-SAM                  570      73    2003   Solid rocket
Korea   KM-SAM             40    400            2015   Solid rocket

Ground to ground missiles

Tomahawk
Tomahawk

                Mach   Range  Missile  Warhead  Year  Engine        Launch
                        km      kg       kg                         platform

USA     Tomahawk   .84  2500   1600     450    1983   Turbofan      Ground
USA     AGM-129    .75  3700   1300     130    1990   Turbofan      B-52 Bomber
USA     AGM-86     .73  2400   1430    1361    1980   Turbofan      B-52 Bomber

Hypersonic missiles

HTV-2
X-51
DARPA Falcon HTV-3

                   Speed   Mass  Payload  Range  Year
                   mach    tons   tons     km

USA      SR-72         6                          Future. Successor to the SR-71 Blackbird
USA      HSSW          6                    900   Future. High Speed Strike Weaspon
USA      HTV-2        20           5500   17000   2 Test flights
USA      X-41          8           450            Future
USA      X-51          5.1  1.8             740   2013    Tested. 21 km altitude. Will become the HSSW
Russia   Object 4202  10                          Tested
India    HSTDV        12                          Future
China    Wu-14        10                          2014   7 tests.  also called the DZ-ZF
The SR-72 has two engines: a ramjet for below Mach 3 and a ramjet/scramjet for above Mach 3. The engines share an intake and thrust nozzle.
Intercontinental ballistic missiles

First ICBM: SM-65 Atlas, completed in 1958
Titan 2
Peacekeeper
Minuteman 3
Minuteman 3

Trident 2
Peacekeeper
Minuteman 3

                     Payload  Paylod   Range  Mass    Launch   Year
                     (tons)   (Mtons)  (km)   (tons)

USA     Titan 2               9        15000   154     Silo    1962   Inactive
USA     Minuteman 3            .9      13000    35.3   Silo    1970
USA     Trident 2              .95     11300    58.5   Sub     1987
USA     Titan                 3.75     10200   151.1   Silo    1959   Inactive
USA     Peacekeeper           3         9600    96.8   Silo    1983   Inactive
Russia  RS-24                 1.2      12000     49    Road    2007
Russia  Voevoda         8.7   8        11000    211.4  Silo    1986
Russia  Layner                         11000     40    Sub     2011
Russia  RS-28 Sarmat   10              10000   >100    Silo    2020   Liquid rocket
Russia  Bulava                 .9      10000     36.8  Sub     2005
France  M51.1                 1        10000    52     Sub     2006
China   DF-5B                 8        15000    183    Silo    2015
China   DF-5A                 4        15000    183    Silo    1983
China   JL-2                  6        12000     42    Sub     2001
China   DF-5                  5        12000    183    Silo    1971
China   DF-31A                3        12000     42    Road
China   DF-31                 1         8000     42    Road    1999
China   DF-4                  3.3       7000     82    Silo    1974
India   Surya          15              16000     70    Road    2022
India   Agni-VI        10              12000     70    Road    2017
India   Agni-V          6               8000     50    Road    2012
India   K-4             2.5             3500     17    Sub     2016   Solid. Arihant nuclear sub
India   K-15           ~6.5              750      1.0  Sub     2010   Solid. 2 stages. Arihant nuclear sub
Israel  Jericho 3        .75           11500     30    Road    2008
N. Kor. Taepodong-2                     6000     79.2  Pad     2006
Pakis.  Shaheen 3                       2750           Road    2015   Solid. 2 stages.
Pakis.  Shaheen 2                       2000     25    Road    2014   Solid. 2 stages.
Pakis.  Ghauri 2        1.2             1800     17.8  Road
Pakis.  Ghauri 1         .7             1500     15.8  Road    2003   Liquid. 1 stage.
Iran    Shabab 3        1.0             1930                   2003
Payload in "tons" represents the mass of the payload.
Payload in "Mtons" is the nuclear detonation payload in terms of tons of TNT.
Helicopter drones

The flight time of a drone is determined by:
*) The battery energy/mass.
*) The power/mass required to hover.
*) The ratio of the battery mass to the drone mass.

Typical parameters for a drone are:

Drone mass         =  M          =  1.0 kg
Battery mass       =  Mbat        =  .5 kg           (The battery is the most vital component)
Battery energy     =  E          =  .38 MJoules
Battery energy/mass=  ebat= E/Mbat=  .75 MJoules/kg   (Upper range for lithium batteries)
Drone energy/mass  =  e  =  E/M  =  .38 MJoules/kg
Drone power/mass   =  p  =  P/M  =   60 Watts/kg    (Practical minimum to hover. Independent of mass)
Drone power        =  P  =  p M  =   60 Watts       (Power required to hover)
Flight time        =  T  =  E/P  = 6250 seconds  =  104 minutes
The flight time in terms of component parameters is
T  =  (ebat/p) * (Mbat/M)


                Drone  Battery  Drone  Battery  Battery  Drone  Drone  Flight  Price  Wireless
                mass   energy           mass             power          time           range
                 kg    MJoule   MJ/kg    kg      MJ/kg   Watt    W/kg  minutes   $      km

Mota Jetjat Nano .011    .00160  .145                     3.3    303      8      40    .02
ByRobot Fighter  .030    .0040   .133                     6.7    222     10     120    .1
Blade mQX        .0751   .0067   .089                    11.2    149     10     115
XDrone Zepto     .082    .0067   .081                     4.6     56     24      40    .05
Walkera QR Y100  .146    .0213   .146   .0413    .52     17.8    122     20     100    .1
MJX Bugs 3       .485    .0480   .099                    44.4     92     18     100    .5
DJI Mavic Pro    .725    .157    .217   .24      .65    109      150     24    1000   7.0
XK Detect X380C 1.18     .216    .183                   120      102     30     522   1.0
Xiaomi Mi       1.376    .319    .214                   197      143     27     380   1.0
DJI Phantom 4   1.38     .293    .212   .426     .69    174      126     28    1000   5.0
XK X500         1.8      .288    .160                   160       89     30     303   1.0
JYU Spider X    2.1      .360    .171   .812     .44    200       95     30     155   4.0
MD4-1000        2.65    1.039    .392                   197       74     88    2000    .5
DJI Inspire     2.935    .360    .123   .67      .54    333      114     18    2000   7.0
Altura Zenith   3.5     1.327    .379                   491      140     45    2000   1.0
Walkera QR X800 3.9      .799    .205  1.134     .70    222       57     60    2700   2.0
AEE F100        6.0     1.598    .266                   380       63     70   58000  10.0
Chaos HL48      6.8     1.758    .259                   651       96     45   20000  20.0
MD4-3000       10.4     2.80     .269                  1037      100     45           4.0
Ehang 184     200      51.8      .259                 37500      188     23  300000   3.5
Airbus EFan                                              60
The minimum power requirement for quadcopter flight is of order 60 Watts/kg.
1 MJ = 1 MJoule = 106 Joules

Electric power outperformes gasoline power in all categories except energy density. Electric motors are lighter, simpler, cheaper, more flexible, and more reliable than combustion motors.


Fixed-wing drones
            Drone  Battery  Drone  Drone  Drone  Flight  Cost  Wireless  Wing  Wing  Cruise   Max   Flight
            mass   energy    E/M   power          time          range    area  span  speed   speed  range
             kg    MJoule   MJ/kg  Watts  W/kg   minutes  $       km      m2    m     m/s     m/s    km

Sky Hawk       .355  .0346  .097    19.2  54.1     30     400    1.0           1.0
Chaipirinha    .62   .088   .142                          154                   .85
AgDrone       2.25   .639   .284   161    71.7     66            4.5            .124  46.7     82    50
Trimble UX5   2.5    .320   .128   107    42.7     50   10000    2.5     .34    .10   80             60
Airbus E-Fan 450  104.4     .232 29000    64.4     60                          9.5    44.4     61.1 160
Sun Flyer   1225

Hover efficiency

The goal of a propeller is to maximize the power/mass ratio.

A propeller test consists of measuring the lift force, the power, and the tip speed (F,P,V), and this can be used to determine the rotor quality parameters (Q,q,W).

Mass             =  M
Rotor radius     =  R
Rotor lift/drag  =  Q  =  F / Fd
Air density      =  D  =  1.22 kg/meter3
Rotor lift param =  W  =  F D-1 R-2 V-2
Rotor tip speed  =  V
Rotor lift force =  F  =  D W R2 V2
Rotor drag force =  Fd
Rotor power      =  P   =  Fd V  =  F V / Q
Rotor quality    =  q   =  Q W½ D½
                        =  F3/2 P-1 R-1
Rotor force/power=  Z   =  F / P
                        =  Q / V
                        =  D½ W½ Q R F
                        =  q R F


Rotor           Radius    Z    Force   Power  Rotors  Blades  Freq   V     q    W     Q
                  m      N/W     N       W            /rotor   Hz   m/s

10 inch           .127   .078     2.94    37.5  1       2     72    57.5  1.06  .045  4.48
10 inch           .127   .056     9.8    176    1       2    131   104.5  1.37  .046  5.85
 9x47SF           .114   .129     1.06     8    1       2     50    35.8  1.17  .052  4.62
 9x47SF           .114   .0427   11.5    269    1       2    166.7 119.4  1.27  .051  5.10
10x47SF           .127   .186      .696    4    1       2     33.3  26.6  1.22  .050  4.95
10x47SF           .127   .0405   16.8    414    1       2    166.7 133.0  1.31  .048  5.39
11x47SF           .140   .161      .960    6    1       2     33.3  29.3  1.13  .047  4.72
11x47SF           .140   .0381   23.1    607    1       2    166.7 146.6  1.45  .045  5.59
12x38SF           .152   .160     1.20     7    1       2     33.3  31.8  1.15  .042  5.09
12x38SF           .152   .0381   28.4    745    1       2    166.7 159.2  1.34  .040  6.07
13 inch           .165   .081     5.81    72    1       2                 1.18
Walkera QR X800   .193   .172     9.56    55.5  4       2                 2.76
Chaos HL48        .203   .102     8.33    81.4  8       3                 1.45
Ehang             .80    .052   245     4688    8       2                 1.02
Phurba           1.6     .115  1200    10435    4       2                 1.4
For drones, the force and power are for one propeller.
Propeller data
#1 #2
Propellers
    Radius   Mass
      m      grams

     .12     11
     .127     6.5
     .171    39.7
     .193    40

Power

For a typical drone,

Drone mass             =  M
Battery mass           =  Mbat
Payload mass           =  Mpay
Climbing speed         =  Vcli
Max horizontal speed   =  Vmax
Hover constant         =  H            =  60 Watts/kg       Power/mass required to hover
Hover power            =  Phov=  H M
Hover power for payload=  Ppay=  H Mpay
Gravity constant       =  g            =  9.8 m/s
Power to climb         =  Pcli=  M g V
Drag coefficient       =  C
Air density            =  D            = 1.22 kg/meter3
Drone cross-section    =  A
Drag power             =  Pdrag= ½ C D A V3
If the climbing power is equal to the hover power,
V = H / g  =  6 meters/second
If the climbing power is equal to the drag power,
M g V = ½ C D A V3

V  =  [2Mg/(CDA)]½  =  4.0 [M/(CA)]½ meters/second
              Drone  Battery  Load   Hover  Load  Climb   Power   P/M  Climb Speed  Thrust
               kg      kg      kg    Watt   Watt  Watt    Watt    W/kg  m/s   m/s   Newtons

Nihui NH-010     .0168  .0047    -       6.7                 6.7  1426   -     -
Walkera QR Y100  .146   .0413    -      17.8         35.5   53.3  1291   -     -
DJI Mavic Pro    .725   .24      -     109                 109     454   5    18
DJI Phantom 4   1.38    .426     .462  174    58     81    232     545   6    20
JYU Spider X    2.1     .812    2.3    200   219    103    419     516   5     8
MD4-1000        2.65            1.2    197    89    195    286           7.5  12     118
DJI Inspire 1   2.935   .67     1.7    333   193    144    526     785   5    22
AEE F100        6.0             2.5    300                 380                28
Chaos HL48      6.8             6.8    651   651          1302
MD4-3000       10.4             4.6   1037         1019   1496          10           280
Ehang         200             100    37500                                    28

Drone   kg     Drone mass without payload
Battery kg     Battery mass
Load    kg     Maximum payload for hovering
Hover   Watt   Power required to hover without payload
Load    Watt   Power used for the payload only
Climb   Watt   Climbing power  =  Mass * ClimbVelocity * Gravityconstant
Power   Watt   Total power  =  Hover power + Load power
P/M     W/kg   Power/Mass for the battery
Climb   m/s    Maximum climbing speed
Speed   m/s    Maximum horizontal speed
Thrust  Newton Maximum thrust
The JYU Spider X has the largest value for (payload mass) / (drone mass). The battery power/mass is
Battery power/mass  =  Phov ⋅ (M + Mpay) / M / Mbat  =  516 Watts/kg

Noise

Noise power is proportional to the rotor tip speed to the fifth power. The more rotors and the more blades per rotor, the less noise produced.


Flying electric car

We give a sample design for a flying electric car based on current technological capabilities. We choose a design with 4 rotors mounted on a cross above the car, with a forward, aft, left, and right rotor. The forward-aft rotor distance is larger than the left-right rotor distance. The left-right cross section is a wing, and it has an interior telescoping sections that extend the length of the wing when flying. The elements of a 1-person flying car are:

                    kg

Passenger
Cabin
Battery
Motors
Rotors
Cross structure
Wing
Car axels
Wheels

Battery energy/mass =  e        =  .8 MJoules/kg
Battery power/mass  =  p        =  2000 Watts/kg
Rotor quality       =  q        =  1.25
Mass                =  M        =  400 kg
Gravity constant    =  g
Force               =  F  = Mg  =  4000 Newtons
Number of rotors    =  N
Rotor radius        =  R        =  1.5 meters
Hover force/power   =  Z  =qR(F/N)=  .0593
Hover power         =  P        =  67450 Watts
Hover power/mass    =  p        =  169 Watts/kg
Rotors         =    4
Blades/Rotor   =    3
Rotor radius   =  1.6 meters
Payload        =  100 kg             One person plus luggage
Fuselage mass  =  125 kg
Battery mass   =   75 kg
Total mass     =  300 kg
Hover          =  100 Watts/kg
Hover power    =30000 Watts
Total power    =60000 Watts
Battery        =  800 Watts/kg
Battery mass   =   75 kg

Beam strength

The maximum force on a beam is:

Beam length     =  L
Beam height     =  H
Beam width      =  W
Shear strength  =  S
Density         =  D
Mass            =  M
Breaking force  =  F  =  (2/3) W H2 S / X




Supercapacitors

Capacitance      =  C            (Farads)
Voltage          =  V            (Volts)
Total energy     =  E0 =  ½ C V2  (Joules)
Effective energy =  E  =  ¼ C V2  (Joules)
Max current      =  I            (Amperes)
Max power        =  P  =  I V    (Watts)
Not all of the energy in a capacitor is harnessable because the voltage diminishes as the charge diminishes, hence the effective energy is less than the total energy. In the data table we use the effective energy.
                Mass     Energy      E/M    Power   P/M   Price  Energy/$    C     Voltage
                 kg      kJoule     kJ/kg   kWatt  kW/kg    $    kJoule/$  Farads   Volts

PM-5R0V105-R      .000454   .0062   13.8                    5.7   .0011       1      5.0
Maxwell BCAP0350  .060      .638    10.6     .459  7.65    16     .040      350      2.7
Adafruit          .135      .984     7.3                   20     .049      630      2.5
BMOD0006E160B02  5.2      37.1       7.1    2.08    .40  1170     .032        5.8  160
XLM-62R1137-R   15       125.3       8.4  124.2    8.3   1396     .090      130     62.1

Drone power system

One has to choose a wise balance for the masses of the motor, battery, fuselage, and payload. The properties of the electrical components are:

                    Energy/Mass  Power/mass  Energy/$  Power/$  $/Mass
                     MJoule/kg    kWatt/kg   MJoule/$  kWatt/$   $/kg

Electric motor          -         10.0        -        .062     160
Lithium-ion battery     .75        1.5        .009     .0142    106
Lithium supercapacitor  .008       8          .0010    .09       90
Aluminum capacitor      .0011    100
If the battery and motor have equal power then the battery has a larger mass than the motor.
Mass of motor            =  Mmot
Mass of battery          =  Mbat
Power                    =  P             (Same for both the motor and the battery)
Power/mass of motor      =  pmot  =  P/Mmot  =   8.0 kWatt/kg
Power/mass of battery    =  pbat  =  P/Mbat  =   1.5 kWatt/kg
Battery mass / Motor mass=  R    =Mbat/Mmot  =  pmot/pbat  =  5.3
The "sports prowess" of a drone is the drone power divided by the minimum hover power. To fly, this number must be larger than 1.
Drone mass               =  Mdro
Motor mass               =  Mmot
Motor power/mass         =  pmot =  8000 Watts/kg
Hover minimum power/mass =  phov =    60 Watts/kg
Drone power              =  Pdro =  pmot Mmot
Hover minimum power      =  Phov =  phov Mdro
Sports prowess           =  S   =  Pdro/Phov  =  (pmot/phov) * (Mmot/Mdro)  =  80 Mmot/Mdro
If S=1 then Mmot/Mdro = 1/80 and the motor constitutes a negligible fraction of the drone mass. One can afford to increase the motor mass to make a sports drone with S >> 1.

If the motor and battery generate equal power then the sports prowess is

S  =  (pbat/phov) * (Mbat/Mdro)  =  25 Mbat/Mdro
If Mbat/Mdro = ½ then S=12.5, well above the minimum required to hover.

Suppose a drone has a mass of 1 kg. A squash racquet can have a mass of as little as .12 kg. The fuselage mass can be much less than this because a drone doesn't need to be as tough as a squash racquet, hence the fuselage mass is negligible compared to the drone mass. An example configuration is:

              kg

Battery       .5
Motors        .1   To match the battery and motor power, set motor mass / battery mass = 1/5
Rotors       <.05
Fuselage      .1
Camera        .3
Drone total  1.0
Supercapacitors can generate a larger power/mass than batteries and are useful for extreme bursts of power, however their energy density is low compared to batteries and so the burst is short. If the supercapacitor and battery have equal power then
Battery power/mass         =  pbat  =  1.5 kWatts/kg
Supercapacitor power/mass  =  psup  =  8.0 kWatts/kg
Battery power              =  P
Battery mass               =  Mbat  =  P / pbat
Supercapacitor mass        =  Msup  =  P / psup
Supercapacitor/Battery mass=  R     =Msup/ Mbat  =  pbat/psup  =  .19
The supercapacitor is substantially ligher than the battery. By adding a lightweight supercapacitor you can double the power. Since drones already have abundant power, the added mass of the supercapacitor usually makes this not worth it.

If a battery and an aluminum capacitor have equal powers,

Aluminum capacitor mass  /  Battery mass  =  .015
If a battery or supercapacitor is operating at full power then the time required to expend all the energy is
Mass          =  M
Energy        =  E
Power         =  P
Energy/Mass   =  e  =  E/M
Power/Mass    =  p  =  P/M
Discharge time=  T  =  E/P  =  e/p

                     Energy/Mass  Power/Mass   Discharge time   Mass
                      MJoule/kg    kWatt/kg       seconds        kg

Lithium battery         .75          1.5          500           1.0
Supercapacitor          .008         8.0            1.0          .19
Aluminum capacitor      .0011      100               .011        .015
"Mass" is the mass required to provide equal power as a lithium battery of equal mass.
World War 2 bombers

Avro Lancaster
B-29 Superfortress
Heinkel He 177

Handley Page Halifax
B-17 Flying Fortress
B-17 Flying Fortress

focke-Wulf Condor
Mitsubishi Ki-67
Mitsubishi G4M

Yokosuka Ginga
Tupolev Tu-2

                            Max    Mass   Max   Bombs  Max   Engine   Range    #    Year
                           speed          mass         alt                   Built
                            kph    ton    ton    ton   km    kWatt     km

UK       Avro Lancaster        454  16.6   32.7  10.0   6.5   4x 954   4073   7377  1942
USA      B-29 Superfortress    574  33.8   60.6   9.0   9.7   4x1640   5230   3970  1944
Germany  Heinkel He 177        565  16.8   32.0   7.2   8.0   2x2133   1540   1169  1942
UK       Short Stirling        454  21.3   31.8   6.4   5.0   4x1025   3750   2371  1939
UK       Handley Page Halifax  454  17.7   24.7   5.9   7.3   4x1205   3000   6176  1940
Germany  Fokke-Wulf Condor     360  17.0   24.5   5.4   6.0   4x 895   3560    276  1937
Soviet   Tupolev Tu-2          528   7.6   11.8   3.8   9.0   2x1380   2020   2257  1942
USA      B-17 Flying Fortress  462  16.4   29.7   3.6  10.5   4x 895   3219  12731  1938
Japan    Mitsubishi Ki-67      537   8.6   13.8   1.6   9.5   2x1417   3800    767  1942
Soviet   Petlyakov Pe-2        580   5.9    8.9   1.6   8.8   2x 903   1160  11427  1941
Japan    Yokosuka P1Y Ginga    547   7.3   13.5   1.0   9.4   2x1361   5370   1102  1944
Japan    Mitsubishi G4M        428   6.7   12.9   1.0   8.5   2x1141   2852   2435  1941

Curtis LeMay: Flying fighters is fun. Flying bombers is important.

World War 2 heavy fighters

A-20 Havoc
F7F Tigercat
P-38 Lightning

P-61
P-38
Airspeed chart

Fairey Firefly
Beaufighter
Mosquito
Fairey Fulmar
Defiant

Messerschmitt 410
Heinkel He-219
Junkers Ju-88

Do-217
Me-110

Kawasaki Ki-45
J1N

Gloster Meteor
Me-262 Swallow
Heinkel He-162

                       Max   Climb  Mass   Max   Bombs  Max   Engine   Range   #   Year
                      speed                mass         alt                  Built
                       kph    m/s   ton    ton    ton   km    kWatt     km

USA    P51 Black Widow  589  12.9  10.6   16.2   2.9   10.6  2x1680   982    706  1944
USA    A-20 Havoc       546  10.2   6.8   12.3    .9    7.2  2x1200  1690   7478  1941
USA    F7F Tigercat     740  23     7.4   11.7    .9   12.3  2x1566  1900    364  1944
USA    P-38 Lightning   667  24.1   5.8    9.8   2.3   13.0  2x1193        10037  1941
UK     Fairey Firefly   509   8.8   4.4    6.4    .9    8.5  1x1290  2090   1702  1943
UK     Mosquito         668  14.5   6.5   11.0   1.8   11.0  2x1103  2400   7781  1941
UK     Beaufighter      515   8.2   7.1   11.5    .3    5.8  2x1200  2816   5928  1940
UK     Fairie Fulmar    438         3.2    4.6    .1    8.3  1x 970  1255    600  1940
UK     Defiant          489   9.0   2.8    3.9   0      9.2  1x 768   749   1064  1939
Japan  Dragon Slayer    540  11.7   4.0    5.5   0     10.0  2x 783         1701  1941  Ki-45
Japan  Flying Dragon    537   6.9   8.6   13.8   1.6    9.5  2x1417  3800    767  1942  Ki-67
Japan  J1N Moonlight    507   8.7   4.5    8.2   0           2x 840  2545    479  1942
Ger.   Hornet           624   9.3   6.2   10.8   1.0   10.0  2x1287  2300   1189  1943
Ger.   Flying Pencil    557   3.5   9.1   16.7   4.0    7.4  2x1287  2145   1925  1941  Do-217
Ger.   Heinkel He-219   616               13.6   0      9.3  2x1324  1540    300  1943
Ger.   Junkers Ju-88    360        11.1   12.7   0      5.5  2x1044  1580  15183  1939
Ger.   Me-110           595  12.5          7.8   0     11.0  2x1085   900   6170  1937
SU     Petlyakov Pe-3   530  12.5   5.9    8.0    .7    9.1  2x 820  1500    360  1941
UK     Gloster Meteor   965  35.6   4.8    7.1    .9   13.1   Jet     965   3947  1944
Ger.   Me-262 Swallow   900 ~25     3.8    7.1   1.0   11.5   Jet    1050   1430  1944
Ger.   Heinkel He-162   840  23.4   1.7    2.8   0     12.0   Jet     975    320  1945

Me-262 Swallow jet  =  2x 8.8 kNewtons
Heinkel He-162 jet  =  1x 7.8 kNewtons
Gloster Meteor jet  =  2x16.0 kNewtons

World War 2 light fighters

P-39 Airacobra
P-40 Warhawk
P-43 Lancer

P-47 Thunderbolt
P-51 Mustang
P-63 Kingcobra

F2A Buffalo
F4F
F4U

F6F Hellcat
F8F Bearcat

Ki-27
Ki-43
Ki-44

Ki-61
Ki-84
Ki-100

A5M
Mitsubishi A6M Zero
A6M2

J2M
N1K

Hawker Tempest
Hawker Hurricane
Hawker Typhoon

Submarine Seafire
Submarine Spitfire

Fw-190
Bf-109

YaK-1
Yak-7
Yak-9
Polykarpov I-16

MiG-3
LaGG-3
La-5
La-7

                       Max   Climb  Mass   Max   Bombs  Max   Engine   Range   #    Year
                      speed                mass         alt                  Built
                       kph    m/s   ton    ton    ton   km    kWatt     km

USA    P-39 Airacobra   626  19.3   3.0    3.8    .2   10.7  1x 894   840   9588  1941
USA    P-63 Kingcobra   660  12.7   3.1    4.9    .7   13.1  1x1340   725   3303  1943
USA    F2A Buffalo      517  12.4   2.1    3.2   0     10.1  1x 890  1553    509  1939
USA    P-40 Warhawk     580  11.0   2.8    4.0    .9    8.8  1x 858  1100  13738  1939
USA    P-51 Mustang     703  16.3   3.5    5.5    .5   12.8  1x1111  2755 >15000  1942
USA    F4F Wildcat      515  11.2   2.7    4.0   0     10.4  1x 900  1337   7885  1940
USA    F6F Hellcat      629  17.8   4.2    7.0   1.8   11.4  1x1491  1520  12275  1943
USA    F8F Bearcat      730  23.2   3.2    6.1    .5   12.4  1x1678  1778   1265  1945
USA    P-43 Lancer      573  13.0   2.7    3.8   0     11.0  1x 895  1046    272  1941
USA    P-47 Thunderbolt 713  16.2   4.5    7.9   1.1   13.1  1x1938  1290  15677  1942
USA    F4U Corsair      717  22.1   4.2    5.6   1.8   12.6  1x1775  1617  12571  1942
Japan  Zero             534  15.7   1.7    2.8    .3   10.0  1x 700  3104  10939  1940
Japan  N1K Strong Wind  658  20.3   2.7    4.9    .5   10.8  1x1380  1716   1532  1943
Japan  Ki-84 "Gale"     686  18.3   2.7    4.2    .7   11.8  1x1522  2168   3514  1943
Japan  Ki-61            580  15.2   2.6    3.5    .5   11.6  1x 864   580   3078  1942
Japan  Ki-100           580  13.9   2.5    3.5   0     11.0  1x1120  2200    396  1945
Japan  A5M              440         1.2    1.8   0      9.8  1x 585  1200   1094  1936
Japan  A6M2             436  12.4   1.9    2.9    .1   10.0  1x 709  1782    327  1942
Japan  J2M Thunderbolt  655  23.4   2.8    3.2    .1   11.4  1x1379   560    671  1942
Japan  Ki-27            470  15.3   1.1    1.8    .1   12.2  1x 485   627   3368  1937
Japan  Ki-43            530         1.9    2.9    .5   11.2  1x 858  1760   5919  1941
Japan  Ki-44            605  19.5   2.1    3.0   0     11.2  1x1133         1225  1942
UK     Hawker Hurricane 547  14.1   2.6    4.0    .5   11.0  1x 883   965  14583  1943
UK     Hawker Tempest   700  23.9   4.2    6.2    .9   11.1  1x1625  1190   1702  1944
UK     Hawker Typhoon   663  13.6   4.0    6.0    .9   10.7  1x1685   821   3317  1941
UK   Submarine Seafire  578  13.4   2.8    3.5          9.8  1x1182   825   2334  1942
UK   Submarine Spitfire 595  13.2   2.3    3.0   0     11.1  1x1096   756  20351  1938
Ger.   Fw-190           685  17.0   3.5    4.8    .5   12.0  1x1287   835 >20000  1941
Ger.   Bf-109           640  17.0   2.2    3.4    .3   12.0  1x1085   850  34826  1937
SU     MiG-3            640  13.0   2.7    3.4    .2   12.0  1x 993   820   3172  1941
SU     Yak-1            592  15.4   2.4    2.9   0     10.0  1x 880   700   8700  1940
SU     Yak-3            655  18.5   2.1    2.7   0     10.7  1x 970   650   4848  1944
SU     Yak-7            571  12.0   2.4    2.9   0      9.5  1x 780   643   6399  1942
SU     Yak-9            672  16.7   2.5    3.2   0     10.6  1x1120   675  16769  1942
SU     LaGG-3           575  14.9   2.2    3.2    .2    9.7  1x 924  1000   6528  1941
SU     La-5             648  16.7   2.6    3.4    .2   11.0  1x1385   765   9920  1942
SU     La-7             661  15.7   3.3           .2   10.4  1x1230   665   5753  1944
SU     Polykarpov I-16  525  14.7   1.5    2.1    .5   14.7  1x 820   700   8644  1934

World War 2 aircraft carriers

U.S. Essex Class
U.S. Independence Class

Shokaku Class
Hiyo Class
Chitose Class

Unryu Class
Zuiho Class

       Class        Speed   Power  Length  Displace  Planes     #     Year
                     kph    MWatt    m       kton             built

USA    Essex         60.6   110     263      47       100      24     1942
USA    Independence  58      75     190      11        33       9     1942
Japan  Shokaku       63.9   120     257.5    32.1      72       2     1941
Japan  Hiyo          47.2    42     219.3    24.2      53       3     1944
Japan  Unryu         63     113     227.4    17.8      65       3     1944
Japan  Chitose       53.5    42.4   192.5    15.5      30       2     1944
Japan  Zuiho         52      39     205.5    11.4      30       2     1940

Military

World military budget

             B$/yr   % GDP                        B$/yr   % GDP

World        1676     2.3           Japan          40.9   1.0
USA           597     3.3           Germany        39.4   1.2
China         215     1.9           South Korea    36.4   2.6
Saudi Arabia   87.2  13.7           Brazil         24.6   1.4
Russia         66.4   5.4           Italy          23.8   1.3
UK             55.5   2.0           Australia      23.6   1.9
India          51.3   2.3           UAE            22.8   5.7
France         50.9   2.1           Israel         16.1   5.4

Military equipment

Ronald Reagan, Kitty Hawk, and Abraham Lincoln (front to back)

Virginia class nuclear submarine
Virginia class: the "North Dakota"

      Nuclear  Diesel  Aircraft   Military   F-22  F-35  B-2  Combat   Nuclear
       subs     subs   carriers  Satellites                  aircraft  devices

Total     148   228       20     320        184         20  20089    15913
USA        72             10     123        184   71    20   3680     7100
Russia     45    18        1      74                         1337     7700
UK         11              1       7              18*         278      225
France     10              1       8                          395      300
China       9    46        1      68                         2571      260
India       1    13        2       5                          928      110
Japan            17                4               5*         777       TC
Israel            3                8              33*         440       80
Italy             6        2       6               8*         258
Germany           4                7                          245
S. Korea         12                                           587       TC
Egypt                                                         569
N. Korea                                                      563        8
Taiwan            2                1                          485
Pakistan          5                                           460      120
Iran              3                                           337
Turkey           14                1               6*         335
S. Arabia                                                     313       TC
Syria                                                         277
Greece            8                                           244
Ukraine                                                       203
UAE                                2                          175
Spain             4                                           166
Australia         6                1              72*         146
Myanmar                                                       155       TC
Thailand                   1                                  143
Sweden            5                                           134
Singapore         4                                           126
Argentina         3                                           123       TC
Kazakhstan                                                    122
Algeria           4                                           120
Poland            5                                           113       TC
Finland                                                       107
Canada            4                1                           95
Chile             4                1
Netherlands       4                               10*
Norway            6                                4*
Mexico                             1                                    TC
Spain                              2
Brazil            5        1                                            TC
Malaysia          2
Portugal          2
Romania           1
Vietnam           1
Colombia          2
Ecuador           2
Indonesia         2
Peru              6
S. Africa         3
Venezuela         2

*:  On order
TC: Does not possess nuclear devices but is technologically capable of building them
The aircraft with stealth technology are the F-22, F-35, and B-2.
For the "combat aircraft" column, only countries with at least 100 combat aircraft are listed.
Data
Air Force
       F-22 F-35 F-15 F-16 F-18 F-4 F-5 F-2 MiG31 MiG25 MiG29 MiG29 MiG21 Su35 Su30  J7 J10

USA     195 121  449  983   885         561
Russia                                       152         252              48
China                                                                     24        728 240
Japan        42* 154             71      64
S. Korea     40*  58  169        71 158
India                                          5               108   245       241
Singapore         40   60            27
Taiwan                115            23
Thailand               53            30
Indonesia              16             2
Malaysia                      8      18
Philip.
Total                4500
*: On order

Submarines

Virginia class nuclear submarine
Virginia class: the "North Dakota"

                  Speed  Power   Mass  Depth  Len   Wid   Hei  Drag  Year   #   Power
                  km/h   MWatts  Mkg     m     m     m     m

USA     Virginia     46   30      7.9   240  115    10.0             2004  13   Nuclear
USA     Ohio         46   45     18.8   240  170    13    10.8  .73  1981  18   Nuclear
USA     Los Angeles  37   26      6.9   290  110    10     9.4  .63  1976  36   Nuclear
Russia  Akula        65   32     13.8   520  113.3  13.6   9.7  .55  1986  10   Nuclear
Russia  Oscar        59   73.1   19.4        155    18.2   9   1.02  1981   5   Nuclear
Russia  Borey        56          24     450  170    13.5  10         2010   3   Nuclear
Russia  Delta 4      44.4               320                          1981   7   Nuclear
UK      Vanguard     46   41     15.9        149.9  12.8  12         1993   4   Nuclear
France  Triomphant   46          14.3   400  138    12.5  10.6       1997   4   Nuclear
India   Arihant      44   83     ~7     300  112    11    10   1.73  2016   1   Nuclear
China   Type 93      55.6         7.0        110    11     7.5       2006   5   Nuclear
China   Type 95                                                      2015   5   Nuclear
China   Type 94                  11          135    12.5             2007   4   Nuclear

India   Shishumar    41   10.6    1.85  260   64.4                   1986   4   Diesel electric. German
Germany Type 212     37    2.85   1.83  700   57.2   7     6         2002  10   AIP, Fuel cell
India   Kalvari      37           1.87        61.7   6.2   5.8       2016   1   AIP, Fuel cell
Japan   Soryu        37    6.0    4.2         84.0   9.1   8.5       2009   9   AIP, Stirling
Japan   Oyashio      37    5.78   4.0         81.7   8.9   7.4  .20  1998  11   Diesel electric
Australia  Collins   37    5.37   3.41 >180   77.4   7.8   7    .23  1996   6   Diesel electric
India   Sindhughosh  31   10.2    3.08  300          9.9   6.6  .36  1986   9   Diesel electric. Kilo class
Taiwan  Hai Lung     22           2.66  300   66.9   8.4   6.7       1987   2   Diesel electric. Dutch Zwaardvis class
"Depth" refers to the "Test depth", which is typically 2/3 of the crush depth.
"Drag" is the drag coefficient defined below.
Submarine speed
Height           =  H
Width            =  W
Cross section    =  A  =  π H W / 4
Water density    =  D  =  1025 km/meter3       Typical for seawater
Speed            =  V
Drag coefficient =  C
Propeller power  =  P  =  ½ C D A V3

Submarines by country

      Nuclear  Diesel  Aircraft
       subs     subs   carriers

Total     148   228       20
USA        72             10
Russia     45    18        1
UK         11              1
France     10              1
China       9    46        1
India       1    13        2
Japan            17
Turkey           14
S. Korea         12
Greece            8
Italy             6        2
Norway            6
Peru              6
Australia         6
Pakistan          5
Sweden            5
Brazil            5        1
Poland            5
Germany           4
Singapore         4
Spain             4
Algeria           4
Canada            4
Chile             4
Netherlands       4
S. Africa         3
Israel            3
Iran              3
Argentina         3
Taiwan            2
Malaysia          2
Portugal          2
Colombia          2
Ecuador           2
Indonesia         2
Venezuela         2
Romania           1
Vietnam           1

Torpedoes
                  Speed   Range   Mass  Warhead  Length  Diam   Depth  Year  Fuel
                  km/h     km     ton     ton    meter   meter  meter

UK      Spearfish  150     54     1.85    .30     7.0    .533          1992  Otto fuel 2
China   Yu-6       120.4   45                            .533          2012  Otto fuel 2
USA     Mark 48    102     38     1.68    .29     5.8    .53     800   2008  Otto fuel 2
Germany DM2A4       92.6   50             .26     6.6    .533                Silver zinc battery
USA     Mark 54     74.1           .276   .044    2.72   .324          2004  Otto fuel 2
S Korea White Shark 63     30     1.1             2.7    .48           2004
Inaia   Varunastra  74     40      1.5    .25      7.0   .533    400   2016  Electric

Speed of a battery-powered submarine

Lithium batteries deliver a high power/mass ratio and have the potential to outperform nuclear power for stealthiness. We give sample parameters for a battery-powered submarine using the properties of the battery in the Tesla Model S. This is the most powerful battery among road cars.

Battery energy/mass       =  e    =  .57  MJoule/kg
Battery power/mass        =  p    =  .74  kWatt/kg
Battery energy/dollar     =  Q    =  .007 MJoule/$
Battery dollar/mass       =  q    =  81.4 $/kg
Battery density           =  Dbat = 2500  kg/meter3
Seawater density          =  Dwat = 1025  kg/meter3
Submarine drag coefficient=  C    =  .33       Typical value
Submarine speed           =  V
Submarine max speed       =  Vmax
Submarine cross section   =  A
Battery cross section     =  a    =  .8 A      Assume the battery fills most of the compartment
Battery compartment length=  L    =   10  meters
Battery mass              =  M    =  L a Dbat
Battery power             =  Pbat =  p M  =  p L a Dbat
Drag power                =  Pdrag=  ½ C Dwat A V3
The maximum speed occurs when the battery maximum power is equal to the drag power.
Pdrag = Pbat   ↔   p a Dbat L = ½ C Dwat A V3max

Z  =  V3max / L  =  p (a/A) (Dbat/Dwat) / (½ C)  ≈  740 ⋅ .8 ⋅ 2.5  ≈  8900

Vmax  =  (L Z)1/3  ≈  20.7 L1/3 m/s  =  44.6 m/s  =  161 km/h
The speed doesn't depend on the cross sectional area of the submarine.

The range of the submarine as a function of speed is:

Battery endurance time  =  T  =  e/p  =  770 seconds  =  12.8 minutes      Time to drain the battery when at maximum power
Range at speed Vmax     =  R  =  T Vmax  =  34.3 km
Drag power              =  P  =  Pbat V3 /V3max
Travel time at speed V  =  t  =  T Pbat/P =  T V3max / V3
Range at speed V        =  r  =  t V  =  T V3max / V2  =  R V2max / V2
A lithium battery has an endurance of 12.8 minutes.

A submarine moving at 1/10 its maximum speed travels 100 times as far as it does when at maximum speed. At this speed the range is more than 1000 km.


Rockets

Index:   
Orbit    Fuel    Aircraft launch    Future rockets    Power in space     Ion drives     Fission rocket     Fusion rocket     Thermal rocket     Electromagnetic sled    Maneuvers     Chemical rocket engines    Orbital launch systems    Aircraft

Thrust

Hydrogen and oxygen are stored in liquid form and combined in the rocket.

A rocket generates thrust by burning fuel and channeling the exhaust with a rocket cone.


Orbit

To reach orbit you need a velocity of 7.8 km/s. A one-stage rocket isn't enough and so multiple stages are used.

Saturn V
Saturn V stage separation
Ariane 5


Fuel

The faster the exhaust the faster the rocket.

The fuel that generates the fastest exhaust is hydrogen+oxygen and this is usually used for the upper stages.

The first stage usually uses kerosene+oxygen. Liquid hydrogen usually isn't used because its density is too low.

SpaceX recently developed the first methane+oxygen rocket, which is a substantial improvement over kerosene+oxygen.

Fuel     Exhaust    Fuel    Fuel boiling
          speed    density     point
         (km/s)   (g/cm^3)     (K)

Hydrogen    4.4     .07       20.3   Complex because of the low boiling point of hydrogen
Methane     3.7     .42      111.7   New technology
Kerosene    3.3     .80      410     Simple because kerosene is a liquid at room temperature
Solid fuel  2.7    1.2         -     Simple and cheap
Kerosene is a liquid consisting of hydrocarbon chains with 6 to 16 carbon atoms per chain. It is similar to gasoline.
High-altitude launch

Stratolaunch
Pegasus
Pegasus

Vulcan Aerospace is developing the Stratolaunch aircraft to launch rockets from high altitude, yielding several advantages over ground launch.

                         Speed (km/s)

Earth rotation at equator   .46
Stratolaunch aircraft       .27
Earth orbit speed          7.8
The rocket will be launched from near the equator so that it benefits from both the speed of the equator and from the speed of the aircraft.

The Stratolaunch will fly at 14 km where the air has 1/4 the density as at sea level, giving it an edge over ground launch. A rocket launched from the ground has to respect the atmosphere by first going up before it can go sideways. A rocket launched from an aircraft can go sideways immediately, for a big savings in fuel.

The stratolaunch aircraft has 6 engines for a total thrust of 150 tons and it can carry a 230 ton rocket.


Ramjet launch

Turbofan, ramjet, and scramjet

In the future, a ramjet aircraft will be built to launch rockets from a speed of Mach 5 and an altitude of 100 km. Since ramjets only work above Mach 1, a detachable solid rocket booster will have to be used to get the aircraft to Mach 1. The aircraft fuselage will be a shell that will contain the rocket, which will be deployed from the aft of the aircraft.

Ramjets work up to Mach 5 and scramjets are required above this speed. Ramjets are a mature technology and scramjets are an emerging technology.

                                     km/s     Mach

Earth rotation at equator              .46    1.6
Stratolaunch aircraft                  .27     .9
Ramjet aircraft                       1.50    5.0
Earth orbit speed                     7.8    20.4
Exhaust speed of HOX fuel             4.4    14.9
Effective exhaust speed of a ramjet  12      40.7     (from specific impulse)

Payload

We compare the payloads of an Airbus A380 aircraft and a Falcon 9 rocket.

Airbus empty mass         = 276.8 tons
Airbus engine mass        =  25.1 tons
Airbus passenger mass     = 100   tons
Airbus fuel mass          = 200   tons
Airbus max takeoff        = 650   tons
Falcon total mass         = 570   tons
Falcon stage 1 full mass  = 506   tons
Falcon stage 1 engine mass=   5.7 tons
Falcon stage 2 full mass  =  52   tons
Falcon stage 2 engine mass=    .6 tons
Falcon stage 2 empty mass =   3.1 tons
Falcon payload            =  13.2 tons
Airbus payload fraction   =    .2       =  100 tons of passengers / 500 tons of aircraft
Falcon payload fraction   =    .023     = 13.2 tons of payload    / 570 tons of rocket
Airbus payload cost       =   4   $/kg
Falcon payload cost       =4100   $/kg

Atmospheric reentry

Space shuttle
Apollo mission
Mars rover

Reentry spacecraft:

Space shuttle empty mass =  78.0 tons,  7 crew.    Launch rocket = 2030 tons
SpaceX Dragon V2         =   4.2 tons,  7 crew
Soyuz reentry module     =   2.9 tons,  3 crew
ISRO Reentry Vehicle     =   3.7 tons,  3 crew
The space shuttle was senseless because there's no point in bringing unnecessary mass back to the Earth. A reentry spacecraft can be as lightweight as 1 ton/person.

If the rocket fails during launch and the crew are in a lightweight reentry spacecraft, they have the potential to survive.

SpaceX Dragon

Soyuz

ISRO reentry vehicle
Space shuttle and Soyuz


Moon
Earth
Mars
Moon
Ceres
Sizes to scale.

The moon has ice which can be turned into hydrogen+oxygen rocket fuel using solar power. Ice can be brought into space more cheaply from the moon than from the Earth.

Ceres is the largest asteroid in the asteroid belt and it has abundant ice.

     Orbit speed   Atmosphere       Distance from
       (km/s)      density (kg/m3)     sun (AU)

Earth    7.8         1.22             1.00
Mars     3.6          .020            1.52
Moon     1.68        0                1.00
Ceres     .36        0                2.77
A one-stage rocket can easily escape the moon or Mars. Two stages are required for the Earth.
Future rockets

The exhaust speed depends on the energy/mass of the fuel.

Rocket type    Exhaust speed   Exhaust speed
                  (km/s)       / speed of light

Antimatter        150000       .5         React matter with antimatter
Fission fragment   12000       .039       Nuclear fission fragments as exhaust
Fusion              4900       .0163      Nuclear fusion of Deuterium + Lithium6
Ion drive            200       .00067     Uses electric power to accelerate ions
Hydrogen + oxygen      4.4     .000015
Methane  + oxygen      3.7     .000012
Kerosene + oxygen      3.3     .000011
Chemical rockets and ion drives are proven technologies. All the other rockets could be built with present technology except for the antimatter rocket. In the distant future, antimatter rockets will be possible.
Nuclear battery

Radioactive Plutonium-238
Solar panels on the space station

Power in space can be obtained from solar cells or from a nuclear battery. Solar cells work best at Earth orbit but they're not useful beyond Mars. Nuclear batteries work everywhere.

In a nuclear battery, radioactivity produces heat and a thermoelectric generator converts the heat to electricity.

The Voyager missions are powered by Plutonium-238 nuclear batteries, which is why they are still functioning 30 years after their launch. Current plutonium-powered missions include Cassini, Galileo, New Horizons, and Ulysses.

Plutonium-238 and Strontium-90 are the isotopes used for nuclear batteries in space, and Curium-244 can be used as well. The possible power sources are:

Power source     Generator      Watts  Halflife   Cost
                                 /kg   (years)   (M$/kg)

Solar cell       Optic           300     -       .003   Power generated at Earth orbit
Curium-244       Thermo + Optic   40    18.1     .17
Curium-244       Thermo           20    18.1     .17
Strontium-90     Thermo            4    28.8     .01    Product of nuclear reactors
Plutonium-238    Thermo            5.4  87.7     .3     Scarce isotope
Plutonium-238    Stirling          4.1  87.7     .3     Scarce isotope
Nuclear reactor  Stirling        200      -       ?     Data for the SAFE-400 reactor
The numbers for Watts/kg are for the total system, including the isotope, the shielding, and the generator.
Generators

Photoelectric cell
Thermoelectric generator
Stirling engine
Stirling engine

The following methods can convert thermal power to electric power.

Isotope        Generator  Electrical   Fuel      Total       Temperature
                          efficiency   fraction  efficiency  (Kelvin)

Plutonium-238  Thermo         .07        .14      .0098       1050
Plutonium-238  Photo          .07        .14      .0098       1050
Plutonium-238  Stirling       .26        .038     .0099       1050
Strontium-90   Thermo         .06        .1       .006         800
Strontium-90   Photo          .06        .1       .006         800

Electrical efficiency:  Efficiency for converting heat to electricity
Fuel fraction:          Fuel mass / System mass
Total efficiency:       Electrical efficiency * Fuel fraction
The higher the temperature, the more efficient a thermoelectric or optoelectric generator is.

A thermoelectric generator and an optoelectric generator can work in tandem to produce a greater efficiency than either alone.


Isotopes that are useful for generating power
              Watts   GJoules  Halflife  Decay   Decay   Cost   Produce  Stockpile
               /kg     /kg     (years)   (MeV)   mode   (M$/kg) (kg/yr)   (kg)

Cobalt-60      27300   4533     5.27     2.82    Beta,γ    1.3
Curium-244      4013   2293    18.1      5.80    Alpha      .17
Tritium         1540    598    12.3       .0186  Beta     30       .4
Caesium-137      864    824    30.2      1.17    Beta       .01   Huge     Huge
Plutonium-238    818   2265    87.7      5.59    Alpha    10      1         17
Strontium-90     648    589    28.8       .55    Beta       .01   Huge     Huge
The numbers for Watts/kg and GJoules/kg are for the pure isotope and don't include the surrounding system. The energy density of gasoline is .046 GJoules/kg.

Strontium-90 and Caesium-137 are generated en masse as fission products in fission reactors.

For an isotope:

Atomic mass unit         =  Mamu  =  1.661⋅10-27 kg
# of nucleons in nucleus =  N
Mass of nucleus          =  Mnuc N Mamu
1 MeV                    =          1.602⋅10-13 Joules    (1 Mega electron Volt)
Nucleus decay energy     =  Edecay
Nucleus energy/mass      =  S    =  Edecay / Mnuc
Decay half life          =  T
Heat power per kg        =  Qheat =  Edecay / T / Mnuc
Electric power per kg    =  Qelec
Efficiency               =  ε     =  Qelec / Qheat    (for converting heat to electric energy)
Fuel mass                =  Mfuel
System mass              =  Msystem
Fuel fraction            =  ffuel =  Mfuel / Msystem
System power per kg      =  Qsys  =  ε ffuel Qheat

Pebble bed nuclear reactor

A pebble bed nuclear reactor doesn't melt down if the cooling system fails because it's engineered to turn off if it overheats. It's also designed so that adding and removing fuel pebbles is easy. The reactor is easy to build and it can be operated in space.


Ion drives

Chang-Diaz
Franklin Chang Diaz

An ion drive uses electric power from a nuclear battery to accelerate ions. The values given in the table are for the Chang Diaz ion drive.

V  =  Ion speed                       =      50 km/s
M  =  Mass of ion drive               =    1000 kg
m  =  Mass of ions ejected per second = .000096 kg/s
Po =  Power consumed by the ion drive =  200000 Watts
Q  =  Efficiency of the drive         =      .6         For converting electric to ion power
P  =  Power delivered to the ion beam =  120000 Watts    =  Q Po  =  .5 m V^2
F  =  Force generated by the ion beam =     4.8 Newtons  =  m V
A  =  Acceleration of spacecraft      =   .0048 m/s2     =  F / M  =  2 P / (M V)
Agi=  Agility  =  Power/Mass          =     120 Watts/kg =  P / M
At fixed ion speed, the acceleration is determined by the power-to-mass ratio of the power source.
A  =  (2/V) * (P/M)

At fixed power there is a tradeoff between F and V:

P  =  .5 F V
The ion speed V can be customized. It should be at least as large as 10 km/s otherwise you might as well use a hydrogen+oxygen rocket. Increasing V decreases the fuel used, decreases the rocket force, and increases the travel time.
Ion spacecraft

Suppose a spacecraft consists of

Ion Drive mass              =  Mdrive = 1000 kg   Chang-Diaz VF-200 design
Solar cell mass             =  Mcell  = 1000 kg   To power the ion drive
Argon mass                  =  Margon = 1000 kg   Ions for the ion drive
Scientific equipment mass   =  Mequip = 1000 kg
Spacecraft total mass       =  Mship  = 4000 kg
Solar cell power/mass       =  Q  =  300 Watts/kg
Solar cell power            =  P  =  Mcell Q
Ion drive operation time    =  T  =  107 seconds
Ion drive efficiency        =  e  =  .60
Ion velocity                =  V  =  60000 (Mcell/Margon)½
Ion energy                  =  E  =  P T e  =  ½ Margon V2
Gravity constant            =  6.674e-11 Newton meters2 / kg2
Earth-sun distance          =  1.496e11 meters
Sun mass                    =  1.989e30 kg
Earth acceleration          =  .00593  meters/second2
Spacecraft recoil velocity  =  V Margon / Mship  = 15 km/s
Spacecraft acceleration     =  .0015
5 km/s corresponds to 1 AU/year. Using a gravity assist from Jupiter, an ion spacecraft can get anywhere in the solar system within 10 years.
Fission fragment rocket

Fission produces 2 fragments
Fission fragment rocket

When uranium fissions it produces 2 high-speed fragments, which can be herded with magnetic fields to produce thrust.

The characteristic speed of the fragments is 12000 km/s = .039 C. See the appendix for an expanded discussion.

The fuel shold have a critical mass that is as small as possible and the half life should be at least 20 years. The best candidate is Californium-251.

               Critical  Diameter  Halflife
                 mass      (cm)    (Myears)
                 (kg)
Californium-252   2.73      6.9      .0000026
Californium-251   5         8.5      .000290
Californium-249   6         9        .000351
Neptunium-236     7         8.7      .154
Curium-247        7.0       9.9    15.6
Curium-243        8        10.5      .000029
Plutonium-238     9.5       9.7      .000088
Plutonium-239    10         9.9      .024
Curium-245       10        11.5      .0085
Americium-242    11        12        .000141
Plutonium-241    12        10.5      .000014
Uranium-233      15        11        .159
Uranium-235      52        17     704
Neptunium-237    60        18       2.14
Plutonium-240    40        15        .0066

Fusion drive
Hydrogen bombs fuse deuterium and tritium to produce energy. The maximum efficiency for converting mass to energy is .00027, and in practice the efficiency is half this.

If we assume that all the energy goes into kinetic energy of exhaust, the exhaust speed is

Kinetic energy  =  .5 M V2  =  .000135 M C2

V  =  4900 km/s  =  .0163 C
If hydrogen bombs are used for propulsion then the spaceship has to be large to absorb the recoil.
Thermal rocket

Nuclear thermal rocket

A thermal rocket uses solar or nuclear power to heat a propellant. In space, ice is available in bulk and so either ice or hydrogen can be used for propellant.

     Exhaust speed (km/s)

H2        9.0
H2O       1.9
In space, thin reflective material can be used to construct a large low-mass mirror to focus sunlight. Such a rocket will be able to move large objects such as asteroids. If an asteroid has its own ice then it's especially easy to move.
Future orbital launch systems

The Stratolaunch aircraft is subsonic. A supersonic ramjet such as the SR-71 can move at Mach 5 and can launch a rocket from higher altitude than the Stratolaunch.

Launch method        Speed   Altitude  Air density
                     (km/s)    (km)    (kg/m3)

Ground                 0        0       1.22      Conventional ground launch
Subsonic aircraft       .3     14        .26      Stratolaunch aircraft
SR-71 Blackbird        1.1     26        .038     Fastest existing ramjet
Supersonic ramjet      1.5     30        .03      Maximum speed for a ramjet
Electromagnetic sled   3.0      7        .4
"Speed" refers to the initial speed of the launch vehicle and "Altitude" refers to the initial altitude of the launch vehicle after it has been accelerated by the launch system.

Future launch systems will use either a supersonic ramjet or an electromagnetic sled.


Electromagnetic sled launch

The Holloman Air Force Base does hypersonic research using a sled that can reach a speed of 2.88 km/s.

A launch sled can convert electrial power to sled kinetic energy with an efficiency of 90%.

Example values:

Sled acceleration    =  A  =  50 m/s2   (5 g's.  Maximum acceleration for humans)
Sled final velocity  =  V  =  3.0 km/s
Length of the track  =  X  =  90 km
Time spent on track  =  T  =  60 seconds

V2 = 2 A X                X = .5 A T2
If we launch inanimate equipment at an acceleration of 500 m/s2 then the track length is 9 km.

If a sled is moving at 3 km/s then a centripetal acceleration of 5 g corresponds to a radius of curvature of 180 km. The last half of the track has to be straight.

The sled only needs to reach an altitude of ~ 40 km. The rocket can do the rest. If it is launched from Everest then it needs to gain an altitude of ~ 30 km. The vertical velocity required to gain 30 km of altitude is .78 km/s. If the horizontal velocity is 3.0 km/s then the launch slope is .25.

A sled can use a heavy heat shield, which isn't possible with a rocket.


Mountains

A sled launch track can use a mountain for altitude and launch angle. Possible mountains include:

Peak          Height   Earth    Airmass  Mountain range
               (m)    rotation  (tons)
                       (km/s)
Equator           0    .465     10.1     Sea level
Huascaran      6768    .458      4.1     Huascaran
Yerupaja       6634    .457      4.2     Huascaran
Everest        8848    .41       3.1     Himalayas, Everest
Kangchenjunga  8586    .41       3.3     Himalayas, Everest
Aconcagua      6962    .391      4.0     Aconcagua
K2             8611    .37       3.2     Himalayas, Karakoram
Huascaran is the tallest peak that is close to the equator.

"Airmass" is the mass of air per meter2 above the given height.

The rocket has to have a mass of at least 100 tons for the airmass to not matter.


Rocket speed

As a rocket burns through fuel it gets lighter. The "Tsoilkovsky rocket equation" relates the final rocket speed to the exhaust speed.

T   =  Time
M(T)=  Mass of rocket as a function of time
Mi  =  Initial mass of rocket
Mf  =  Final mass of rocket after burning its fuel
Ve  =  Rocket exhaust speed
V(T)=  Rocket speed as a function of time.  V(0)=0.
Vf  =  Final rocket speed after burning its fuel
F   =  Force generated by the rocket
    =  - Ve dM/dT

dV/dT =  F/M  =  -(Ve/M) * dM/dT
V(T)  =  V ln(Mi/M)
Vf    =  V ln(Mi/Mf)        Tsoilkovsky rocket equation

Oberth maneuver

The Oberth maneuver uses a planet's gravity to magnify a rocket impulse.

Suppose a spacecraft is on a highly elliptical orbit, with a perigee slightly larger than the Earth's radius and an apogee vastly larger than the Earth's radius.

Gravity constant                =  G  =  6.67e-11 Newton meters2/kg2
Mass of Earth                   =  M  =  5.97e24 kg
Earth radius                    =  R  =  6371 km/s
Perigee radius                  =  R1                     Slightly larger than R
Apogee radius                   =  R2                     R1 << R2
Escape velocity                 =  Vesc=  11.2 km/s
Rocket speed at perigee         =  V1  =  Vesc
Rocket speed at apogee          =  0
Circular orbit speed at perigee =  Vcirc=   7.2 km/s  =  G M / R1
Circular orbit speed at apogee  =  0
Rocket speed change at perigee  =  Vroc =  16.6 km/s      Calculated below
Final exit speed from planet    =  Vexit=  25.4 km/s      Final speed after far from the planet
At apogee the energy is
E  =  Kinetic energy  +  Gravitational energy
   =         0        +         0
At perigee the energy is
E  =  Kinetic energy  +  Gravitational energy
   =     .5 m V12     -     G M m / R1

V12 =  2 G M / R1
     =  2 Vcirc2
     =  Vesc2
V1 is equal to the "Escape speed", the speed required to escape the planet. The escape speed is independent of the direction of the velocity.

The escape velocity can also be obtained from the gravitational potential energy.

.5 m Vesc2 = G M m / R1     →    Vesc2 = 2 G M / R1
IF the rocket fires at perigee and increases its speed by Vroc, the energy becomes
E  =  .5 m (V1 + Vroc)2  -  G M m / R1
   =  .5 m (Vesc + Vroc)2  -  .5 m Vesc2
   =  .5 m (Vroc2 + 2 Vroc Vesc)
The rocket is now on a hyperbolic orbit and will escape the Earth, As it recedes from the Earth it will approaches a constant velocity Vexit. When far from the Earth, the energy is
E  =  .5 m (Vroc2 + 2 Vroc Vesc)
   =  .5 m Vexit2

Vexit=  (Vroc2 + 2 Vroc Vesc)1/2  >  Vroc
If the spacecraft starts in an elliptical orbit and changes its speed by Vroc at perigee, it departs the Earth at speed Vexit, which is larger than Vroc. This is the "Oberth effect".

If a rocket changes its velocity by 5 km/s at perigee, it departs the Earth with a velocity of

Vexit=  (52 + 2 * 5 * 11.2)1/2
    =  11.7 km/s
This gets you to Mars in about 4 months.

X axis:  Change in velocity at perigee (Vroc)
Y axis:  Departure velocity from the planet.  Vexit = (Vroc2 + 2 Vroc Vesc)
Each curve corresponds to a different planet.
      Escape velocity (km/s)
Moon         2.38
Mars         5.03
Earth       11.2
Saturn      35.5
Jupiter     59.5
Sun        618

Rocket power and the Oberth maneuver

The Oberth maneuver requires a rocket with a large thrust-to-mass ratio. The Oberth effect is most useful when the rocket fires at Perigee, meaning the rocket has only a limited time to burn through its fuel. This restricts the rocket types that can be used for an Oberth maneuver. Chemical rockets deliver the most power, which makes them the rocket of choice for Oberth maneuvers. Nuclear rockets have a heating challenge. Ion drives and mirror-based rockets are low-thrust and can't be used for the Oberth maneuver. The rocket engine with the largest force/mass is the Vulcain-2. For this rocket,

Planet radius           =  R  =  6371 km for the Earth
Escape velocity         =  Ves=  11.2 km/s for the Earth
Oberth time             =  T  =   9.5 minutes for the Earth  =  R / Ve
                              =       Time that the rocket is near perigee
Rocket exhaust speed    =  Vex=   4.2 km/s
Rocket force            =  F  =  1359 kiloNewtons
Rocket engine mass      =  m  =  1800 kg
Rocket force/mass       =  Z  =   755 Newtons/kg  =  F / m
Fuel mass burnt         =  M  =  T Z m / Vex  =  102 m       Fuel mass burnt during one Oberth time
Oberth velocity         =  Vob=  16.6 km/s  =  3.9 Vex  =  [ln(M/m) - ln(2)] Vex  =  ln(.5 T Z / Vex) Vex
                                                                                 =  [ln(T) - 2.4] Vex

Momentum conservation:    M Vex  =  F T

During one Oberth time, a Vulcain-2 rocket burns 102 times its mass in fuel. The Oberth time for the Earth is long enough so that a chemical rocket can comfortably burn through all its fuel.

To calculate the Oberth velocity, we use the Tsoilkovsky rocket equation and assume that the final mass of the spaceship is twice the mass of the rocket engine.

        Escape  Radius   Oberth    Oberth     Exit
        (km/s)          time (s)  velocity  velocity
                                   (km/s)    (km/s)
Mercury   4.3     .38     563       16.5     20.4
Venus    10.5     .95     576       16.6     25.0
Earth    11.2    1.00     569       16.6     25.4
Moon      2.38    .27     723       17.6     19.8
Mars      5.03    .53     671       17.3     21.7
Jupiter  59.5   10.9     1167       19.6     52.1
Saturn   35.5    9.0     1615       20.9     43.9
Uranus   21.3    3.97    1187       19.7     35.0
Neptune  23.5    3.86    1046       19.1     35.6
Pluto     1.23    .184    953       18.7     19.9
Sun     618    109.2     1126       19.4    156.2
"Exit velocity" is the maximum exit velocity from the planet using the Oberth maneuver. It is also equal to the maximum "capture velocity" for using the Oberth maneuver to be captured by a planet.
Space mirrors
Mylar density      =  1.39 g/cm3
Aluminum density   =  2.70 g/cm3
Mylar thickness    =  .025 mm
Aluminum thickness =  .010 mm
Area density       =  62 tons/km2  =  .062 kg/m2

Appendix

Rocket engines

Hydrogen + Oxygen rocket

                         Sea level Vacuum                 Thrust
                  Fuel    Exhaust  Exhaust  Mass  Thrust  /mass
                           km/s     km/s      kg    kN    N/kg
Waxwing           Solid             2.72      87    29.4   345
Atlas V           Solid             2.70          1270           40.8 tons with fuel
P230              Solid             2.80          6472           268 tons with fuel. Ariane rocket
Shuttle booster   Solid    2.42     2.68         12500   21200   590 tons with fuel
Merlin 1D         Kerosine 2.76     3.05     630   801    1300   Falcon rocket. Diameter 1.676 m
Merlin 2          Kerosine          3.16          8540           In development by SpaceX. Falcon Heavy
Raptor            Methane           3.7           8200           In development by SpaceX
Snecma HM7B       HOX               4.3      165    64.8   400   Ariane rocket
RL-10A            HOX               4.42     167    99.1   606   Atlas V. Diameter = 2.13 meters
RL10B-2           HOX               4.547    277   110     406   Atlas V and Delta IV rockets

Mitsubishi LE-5B  HOX               4.38     285   137.2   490
Mitsubishi LE-7A  HOX               4.31    1800  1098     620
Vulcain 2         HOX               4.20    1800  1359     755   Ariane rocket. Diameter = 1.76 m
Shuttle engine    HOX      3.56     4.44    3500  1700     496
RS-68             HOX               4.02    6600  3370     520   Most powerful HOX rocket

HOX      = liquid hydrogan + liquid oxygen
Kerosine = kerosine        + liquid oxygen
Solid    = aluminum        + ammonium perchlorate (N H4 Cl O4)
Methane  = methane         + liquid oxygen

Rockets for reaching low Earth orbit

Saturn V
Ariane 5
Ariane 5

Stratolaunch
Pegasus
Pegasus

                        Stage 1             Stage 2          Stage 3
                     Mass  Thrust Exh   Mass Thrust Exh  Mass Thrust  Exh   Payload  Payload
                     kkg     kN   km/s  kkg    kN   km/s  kkg   kN    km/s  kkg      $/kg
Space Shuttle        1710  25000  ~2.6  530  5100   4.44    ?  5100   4.44   93.
SpaceX Falcon 9       506   6672  ~2.9   52   801   3.35    -     -    -     13.15   4109
SpaceX Falcon Heavy  1400  17000  ~2.9 ~480  5600   3.05    ?   445   3.35   53.     2200
Saturn V             2800  34000   2.58 710  4400   4.13  230  1000   4.13  118.00   9915
Ariane                777  12940   2.80   ?  1340   4.22    ?    64.7 4.37   16.    10500
Pegasus                23.1                                                   .443
Stratolaunch            ?   1500   n/a  230     ?   ?       ?     ?    ?      6.12

Earth rotation at equator   = 463 m/s.
Earth escape speed          = 11.186
Earth orbit speed at 160 km = 7.58 km/s

Falcon 9 stage 2 empty mass = 3.1 tons
Falcon 9 Sea level thrust = 5885 kN
Space shuttle: The space shuttle orbiter has a mass of 68.6 and a payload of 24.4 tons.
Saturn V:      Largest payload ever achieved. Launched the moon missions.
Pegasus:       Air launch
Stratolaunch:  A 6-engine airplane launches the "Pegasus II" rocket.
The Stratolaunch airplane is moving at ~ .3 km/s when it launches the rocket, and the launch can occur at the equator where the Earth's rotation speed is .46 km/s. This gives the rocket a total initial speed of .76 km/s.


Aircraft

SR-71 Blackbird
Concorde

                                Engine   Engine    Empty  Max    Cargo
                Speed  Ceiling  thrust    mass     mass  takeoff mass
                (Mach)   km     (tons)   (tons)    (tons) (tons) (tons)
Blackbird SR-71  3.3    25.9  2 x 14.8  2 x 2.7    30.6   78           Spy
F-15 Eagle       2.5    20.0  2 x 11.3  2 x 1.70   12.7   30.8         Fighter
F-22 Raptor      2.25   19.8  2 x 15.9  2 x 1.77   19.7   38           Stealth Fighter
Concorde         2.02   18.3  4 x 17.2  4 x 3.18   78.7  187           128 passengers
Airbus A380       .96   13.1  4 x 38.2  4 x 6.27  276.8  650           853 passengers
Boeing C-5 Galaxy .8          4 x 19.4  4 x 3.63  172.4  381    122.5  Cargo
Boeing 747-8F     .86   13.0  4 x 30.2  4 x 5.6          448    134.2  Cargo
Antonov 224       .75         4 x 23.4  4 x 4.1     175  405    150    Cargo
Antonov 225       .7          6 x 23.4  6 x 4.1     285  640    250    Cargo
Stratolaunch                  6 x 25.5                   540    230    Orbital launch platform
The Stratolaunch (in development) is designed to launch rockets into space.
Air drag
Drag force  =  .5 * AirDensity * CrossSection * Velocity^2

M = Rocket Mass   / 400 tons
A = Acceleration  / 10 m/s^2           Acceleration in units of g's
D = Air Density   / 1 kg/m^3           Density = 1.28 kg/m^3 at sea level
C = Cross section / 10 m^2             The Falcon 9 rocket has a cross section of 10 m^2
V = Velocity      / 300 m/s            Velocity in units of "Mach"
In these units the drag equation is
10 A M ~ D C V^2

For a falcon 9 rocket, M=1 and C=1.  If the rocket is at sea level, D ~ 1.
If the drag acceleration is 1g, then V ~ 3 (Mach 3). This sets the speed limit for rockets in the lower atmosphere.
Rocket fuel

Fuel            Exhaust  Density   Boil  kNewtons  kNewtons  kNewtons  Diameter  Mass    Rocket engine used
                (km/s)   (g/cm^3)  (K)   /meter^2    /ton              (meters)  (kg)    for data

Liquid hydrogen  4.2      .07      20.3    559        755     1359       1.76    1800    Vulcain-2
Liquid methane   3.7      .42     111.7    493          ?     8200       4.6        ?    Raptor
Kerosine         3.3      .80     410      361       1270      801       1.676    630    Merlin-1D
Solid fuel       2.7     1.2        -      673          ?     1270       1.55       ?    Atlas V booster
Kerosine ramjet           .80     410        9.0        5.5     14.8     1.45       2.7  SR-71 Blackbird
Hydrogen, methane, and kerosine are all reacted with liquid oxygen that is carried by the rocket. Solid fuel contains its own oxidizer.

For the kerosine ramjet, kerosine is reacted with oxygen from the air.

"kNewtons/meter^2" is the thrust/area of the rocket.

"kNewtons/kg" is the thrust-to-mass ratio of the rocket engine.

The density of liquid oxygen is 1.14 g/cm^3 and the boiling point is 90.2 Kelvin.


Electrolysis of water into H2 and O2

Electricity can split H2O into H2 and O2, which can be used for rocket fuel. the maximum efficiency of this process is 0.83.

Energy to split H2O into H2 and O2              =  E  =  1.317e7 Joules/kg
Max efficiency to split H2O into H2 and O2      =  e  =  .83
Solar cell power per mass                       =  Sp  =  300 Watts/kg
Solar cost per mass                             =  Sc  = 3000 $/kg
Time for a 1 kg solar cell to form 1 kg of fuel = T  =  .61 days  =  E / e / Sp

Speed of HOX rocket exhaust

We can calculate the maximum speed of HOX rocket exhaust from the energy required to split H2O.

V  =  Maximum speed of rocket exhaust for a HOX rocket

1.317e7 Joules/kg  =  ½ V2

V = 5.132 km/s
In practice, the best HOX rockets have an exhaust speed of 4.4 km/s.


Fission fragment rocket

Mean energies for the fission of Uranium-235, in MeV:

Fission fragment kinetic energy          169.1
Prompt neutrons                            4.8
Prompt gamma rays                          7.0
Delayed beta rays                          6.5
Delayed gamma rays                         6.3
Captured neutrons                          8.8
Total energy generated as heat           202.5
Prompt antineutrinos                       8.8
Total energy including antineutrinos     211.3
Energy of the original U-235 nucleus  218900

1 MeV  =  10^6 eV  =  1.6*10^-13 Joules
1 Atomic mass unit  =  1.6605*10^-27 kg  =  931.494 MeV/C^2
Mass of Uranium-235 = 235.04 atomic mass units
Only the kinetic energy of the fission fragments is harnessable by a rocket.

C = Speed of light
Mt= Mass of original nucleus
E = Kinetic energy of the fission fragments
F = Fraction of the mass of the original nucleus that is
    converted into kinetic energy.
  = E / (Mt C^2)
  = 169 MeV / (235.04 * 931.49)
  = .000772
Vt= Characteristic speed of the fission fragments

.5 Mt Vt^2 ~ F Mt C^2

Vt = .0393 C
Distribution of fragment masses

Fission tends to produce two fragments, one heavier than the other. The distribution is similar for all fissionable nuclei.

E  =  Total kinetic energy in fission fragments  ~  169 MeV
F  =  Fraction of the mass of the original nucleus that is converted into kinetic energy.
   =  .000772
M  =  Mass of heavy fragment  ~  .40 * Mass of original nucleus
m  =  Mass of light fragment  ~  .58 * Mass of original nucleus
V  =  Velocity of heavy fragment
v  =  Velocity of light fragment

Conservation of momentum:  M V = m v
Conservation of energy:    E = .5 M V^2 + .5 m v^2

M^2 V^2 (M + m)  =  2 E M m

V^2 =  2 F C^2 m / M
v^2 =  2 F C^2 M / m

V  =  .0326 C
v  =  .0473 C


               Critical mass   Half life
Americium-242       .5         141 years          Costs ~ 10^6 $/kg
Californium-251     .9         898 years
Curium-245         1.1        8500 years
Plutonium-239      5.6      241000 years
Uranium-235       11.0         704 million years
For a fission fragment rocket, the lower the critical mass the better. All of the above isotopes produce similar energy when fissioned.
Fusion drive

Hydrogen bombs use the following reactions.

Neutron    +  Lithium6  ->  Tritium  +  Helium4  +   4.874 MeV
Deuterium  +  Tritium   ->  Helium4  +  Neutron  +  17.56  MeV
Leaving out the neutron catalyst, this is
Deuterium  +  Lithium6  ->  Helium4  +  Helium4  +  22.43  MeV

Nucleons = 8

Energy / Nucleon  =  22.434/8
                  =  2.80  MeV/Nucleon

f  =  Fraction of mass converted to energy
   =  (2.80 MeV/Nucleon)  /  (939 MeV/Nucleon)
   =  .00298
The theoretical limit for the efficiency of a hydrogen bomb is
f = .00027
In practice, f is half this.
Thermal rockets

A thermal rocket uses a power source to heat the propellant. The power can come from either a nuclear reactor or from sunlight focused by mirrors.

Propellant   Exhaust speed
             (km/s)
H2             9
H2O            1.9

                   Energy
Hydrogen + Oxygen  1.4e10 Joules/ton
Uranium-235        8.0e16 Joules/ton
Solar energy       1.4e15 Joules.  1 km^2 collector operating for 10^6 seconds at 1 A.U.
A mirror-based thermal rocket offers a means for using H2O as propellant. Such a rocket can potentially move large asteroids.

The solar energy collected by a 1km mirror at 1 A.U. over a time of 10^6 seconds (2 weeks) is

Energy  ~  1400 Watts/m^2 * 10^6 m^3 * 10^6 seconds  ~  1.4e15
The mass of the mirror is
                            Surface area    Thickness    Density
Mirror mass  ~  8*10^5 kg  --------------  -----------  ----------
                               1 km^2        10^-4 m     8 g/cm^3
A solar thermal rocket capable of delivering ~ 10^16 Watts can be built from a ~ 10 meter metallic asteroid.

If a thermal rocket can operate at a temperature high enough to dissociate H2 into elemental hydrogen then larger exhaust speeds are possible.


Space mirror

Suppose we use mylar film for a space mirror.

Mirror density        =  1390 kg/m^3
Mirror thickness      =  .1 mm
Mirror mass/area      =  .139 kg/m^2
Solar flux            =  1362 Watts/m^2
H2O exhaust speed     =  1.9 km/s
H2O mass/time/area    =  .00075 kg/s/m^2      Mass of propellant per time per area
Mirror acceleration   =  10.3 m/s
The acceleration of a mirror rocket is limited by the strength of the mirror.
Launch cost

If we assume that the kinetic energy of an orbiting object comes from electricity then

Orbital speed                            =  7.8 km/s
Energy of a 1 kg object at orbital speed = 30.4 MJoules
Cost of electricity                      = 36.0 MJoules/$
Cost of a 1 kg object at orbital speed   =  .84 $
For a typical hydrogen+oxygen rocket, the mass fractions are:
Payload                      =  1 kg
Superstructure               =  2 kg
Hyddrogen mass               =  3 kg
Oxygen mass                  = 24 kg
Total mass                   = 30 kg
Oxygen mass / Hydrogen mass  =  8
Cost of liquid hydrogen      =  .70 $/kg
Cost of liquid oxygen        =  .16 $/kg
Cost of liquid hydrogen      = 2.1  $
Cost of liquid oxygen        = 3.8  $
Typical launch cost for 1 kg = 2500 $
The superstructure is everything except the payload and the fuel.
Most of the launch cost is in the superstructure, not the fuel.
If the kinetic energy of the 1 kg payload comes purely from electricity, the cost of the electricity is tiny.
Orbit speed                   =   7.8 km/s
Energy of 1 kg at orbit speed =  30.4 MJoule
Cost of electricity           =  .015 $/MJoule
Electricity cost of the energy=  .46  $

Space habitats

Bigelow BA-330 habitat
Bigelow Genesis habitat

The Bigelow BA-330 has as much room as the bridge of the Enterprise and the Bigelow Genesis has as much room as a Humvee. Bigelow habitats are lighter than NASA habitats and they have thicker walls. Thicker walls are helpful for defending against micrometeorites and radiation.

                 Volume    Mass   Wall thickness
                  (m3)    (tons)       (m)

Bigelow Genesis    11.5     1.36    .15
NASA Orion         19.6     8.91
Bigelow BA-330    330      23       .46
Space Station     837     450       .003
International Space Station
NASA Orion

Artificial gravity

If artificial gravity is generated by spinning a spaceship, then according to en.wikipedia.org/wiki/Artificial_gravity, the spin period has to be at least 30 seconds for the inhabitants to not get dizzy. If we assume a spin period of 30 seconds and a gravity of 1 g,

Spin period    =  T  =  2 π R / V   =  30 seconds
Spin radius    =  R  =  T2 A / (2π)2 =  228 meters
Velocity       =  V  =  2 π R / T   =  48 meters/second
Acceleration   =  A  =  V2 / R      =  10 meters/second2

Tether

Suppose we use a tether to connect a spinning spaceship. Zylon is the material with the best tensile strength to density ratio.

Tether density           =  D  =  1520 kg/m3 for Zylon
Tether tensile strength  =  P  =  F / Ar  =  5.8 GPa
Mass of spaceship        =  M
Radius of tether         =  R  =  T2 A / 4 pi2  =  V2 / A
Tether cross-section     =  Ar
Mass of tether           =  m  =  2 R Ar D
Centripetal acceleration =  A  =  10 meters/second2
Tether tension force     =  F  =  M A
Spaceship spin period    =  T  =  2 π R / V  =  30 seconds
The mass ratio of the tether to the spaceship is
m/M  =  2 T2 D A2 / P / (4 π2)  =  1.33e-6 T2  = .00119
To be safe, the tether can be given a mass 10 times larger then this. Even so, the tether weighs much less than the habitable module, and so the mass of the tether is not a factor in the spaceship design.

If the spaceship mass is M=1000 tons, the tether mass is m=12 tons. Such a tether can easily be launched from the Earth.

For extremely large tethers you can use iron from the moon.


Radiation in space

The Earth's atmosphere is thick enough to block cosmic rays from space and Mars' atmosphere isn't. The walls of spaceships are too thin to protect against cosmic rays.

            Atmosphere thicknes
                (tons/m2)

Venus              1000
Titan                73
Earth, sea level     10
Earth, 12 km high     4.9
Mars                   .16


                                       mSieverts     Shielding thickness
                                         /year       (tons/m2)
Terrestrial radiation                    2.02        n/a
Average medical radiation                 .60        n/a
Earth surface, cosmic rays only           .39         10
Earth surface, all radiation             3.5          10
Earth 2 km altitude, cosmic rays only     .9           8
Earth 3 km altitude, cosmic rays only    1.7           7
Earth 4 km altitude, cosmic rays only    3.3           6
Earth, 12 km altitude, equator          20             2.5
Earth, 12 km altitude, poles           100             2.5
Space station, 420 km altitude         150              .01   1/8 inch aluminum walls
Space                                  600              .01
Space, 4 tons/m2 shield                  2.5           4
Mars surface                           220              .16

"Space" refers to interplanetary space between Earth and Mars.

At the space station, the Earth's magnetic field blocks 3/4 of the radiation from space.

The sun's magnetic field stops cosmic ray particles below 1 GeV.

The Earth's magnetic field deflects all but the highest-energy cosmic rays.

5 hour airplane flight incurs ~ .03 millisieverts.
A dose of 4800 millisieverts has a 50% risk of death.


Radiation shielding

The above data suggests that to shield against cosmic rays, you need at least 3 tons/meter2 of shielding.

Suppose a spherical spaceship is shielded with ice.

Radius of spaceship   =  r  =  3 meters
Radius of ice shield  =  R  =  6 meters
Density of ice shield =  D  =  1000 kg/meter3
Mass of ice shield    =  (4 &pi / 3) (R3-r3)  =  792 tons
The only way to get this much ice is from the moon. This is the source of the mass for the tether calculation.
Life support

Aeroponic plants
Space station life support

The space station life support system:
1 kWatt/person/day
1 liter of water/person/day
1 kg of food/person/day


         Mass fraction
         in human body

Oxygen     .65
Carbon     .18
Hydrogen   .10
Nitrogen   .03
Calcium    .014
Phosphorus .011
Potassium  .0025
Sulfur     .0025
Sodium     .0015
Chlorine   .0015
Magnesium  .0005
Iron       .00006
Air and water are the biggest challenges for life support in space. Water can be obtained from the moon and electrolysized to produce oxygen for air. Air also requires nitrogen, which cannot be found on the moon but is abundant in Mars' atmosphere.

The principal components of fertilizer are nitrogen, phosphorus, and potassium, with nitrogen being the heaviest component. Nitrogen can be obtained from Mars's atmosphere. If you want to grow crops on Mars you will have to bring phosphorus and potassium.

The most efficient way to grow plants in space is with aeroponics, where the roots are grown in open air. http://en.wikipedia.org/wiki/Aeroponics


Space stations

Good locations for space bases are:
Earth orbit
The moon
Moon orbit
The L2 Lagrange point
Mars orbit

Ice can be shipped from the moon to the other stations.

The L2 point stays tethered to the Earth as the Earth orbits.

The L2 point is ideal for telescopes because from there you can shield the sun, the Earth, and the moon all at the same time. The Webb telescope will go there. If we had a manned space station at L2 then we could assemble telescopes on-site and build colossal telescopes.


Radiation shielding

Cosmic rays consist mostly of high-energy protons with energies > 1000 MeV. When a proton passes through matter it loses energy from collisions with electrons and with nuclei. Electron collisions subtract a small amount of energy from the proton and nuclear collisions subtract most of the energy. This is because collisions with electrons are mediated by the electromagnetic force and collisions with nuclei are mediated by the strong force.

E = Proton energy
L = Distance the proton travels through matter (meters)
D = Density of the matter (kg/meter3)
V = Proton velocity
C = Speed of light
Proton kinetic energy is measured in MeV. 1 MeV = 1.6e-13 Joules. The rest energy of a proton is 1000 MeV.

Proton energy loss is governed by the "Bethe-Bloch" formula. For cosmic ray protons with E > 1000 MeV, the formula may be approximated as

EnergyLoss  =  200 L (D/1000) MeV
If the proton is traveling through water with a mass density of 1000 kg/meter3, the energy loss rate is 200 MeV/meter. The amount of matter required to stop a proton with E = 1000 MeV is 5 meters.

Spacecraft walls are thick enough to stop low-energy protons from the solar wind but they are of no help in stopping cosmic rays. Mars' atmosphere isn't thick enough either.

When a high-energy proton collides with a nucleus, most of the energy is lost in the collision, hence the transmission of protons through matter can be modeled as an exponential.

T  =  Initial intensity of protons
t  =  Transmitted intensity of protons passing through a distance L of matter
L  =  Distance the proton has traveled through the matter
S  =  Characteristic stopping-length of the matter
D  =  Density of the matter in kg/meter3
A  =  Atomic number of the nuclei in the matter
   =  1 for protons
   =  8 for oxygen

t = T exp(-L/S)

S  =  .35 A1/3 1000/D  meters
For oxygen, A = 8 and D=1000, hence the characteristic stopping length of protons in water is S = 0.2 meters.

Suppose you want to stop 99% of the protons.

t = .01 T

L = 4.6 S
If water is used to stop cosmic ray protons, the formula predicts you need least 1 meter of it. This translates to a column density of 1 ton/meter2.

This is an underestimate of the shielding required because when a high-energy proton hits a nucleus it creates a shower of secondary particles which must then be shielded. In practice, 4 meters of water are required. Muons are the biggest nuisance because they don't feel the strong force. Most of the cosmic radiation at the Earth's surface is from muons.


Manned Mars mission

Mars Institute base in Northern Canada

Hohmann trajectory

A Hohmann trajectory takes you from one circular orbit to another, such as from Earth's orbit to Mars' orbit.

The spacecraft starts on the cyan circular orbit.

At point "2", the spacecraft fires its rockets and increases its speed. From there, it coasts along the yellow trajectory to point "3".

When the spacecraft arrives at point "3", it fires its rockets to decrease its speed, placing it on the circular red trajectory.

In a trip from the Earth to Mars, the Earth is at point "2" and Mars is at point "3".

Departure velocity from the Earth     =  2.95 km/s
Arrival velocity with respect to Mars =  2.65 km/s
Travel time from Earth to Mars        =  8.5  months
Wait time on Mars for Hohmann window  = 14.9  months
Travel time from Mars to Earth        =  8.5  months
Total mission time                    = 31.9  months
The total change in velocity that the rocket has to generate is 5.60 km/s. This is within the reach of a hydrogen+oxygen rocket, which has an exhaust speed of 4.4 km/s. This is the minimalist trajectory. If more rocket power is available then the travel time decreases.

Calculation of the Earth-Mars Hohmann orbit


Oberth maneuver

The Oberth maneuver allows one to use a planet to magnify the impulse from a rocket.

For a Hohmann trajectory, the travel time from Earth to Mars is 289 days, which is a long time to be in zero gravity and in the radiation of space. The Oberth effect can speed up the trip.

Suppose a spacecraft is on a highly elliptical orbit, with a perigee just larger than the Earth's radius and an apogee much larger than the Earth's radius. Such an orbit would look like the Kuiper belt object "Sedna" pictured above.

An Oberth maneuver procedes as:

1)  Start far from the planet at apogee
2)  Coast toward the planet on a trajectory where the perigee is just above the
    surface of the planet.
3)  At perigee, fire the rockets at maximum power
4)  Coast away from the planet.  The rocket escapes the planet with a speed that is
    enhanced by the Oberth effect

Planet escape speed             =  Vescape  =  11.2 km/s for the Earth
Speed change from the rocket    =  Vrocket  =  10   km/s for a mighty rocket
Departure speed from the planet =  Vdepart  =  18   km/s

V2depart  =  V2rocket + 2 Vrocket Vescape           Derivation

The Oberth effect can speed up the travel time to Mars to 3 months.

The Oberth effect can greatly magnify a small rocket boost. For example, if Vrocket = 1 km/s then Vdepart = 4.8 km/s. This allows one to transport large payloads between planets, if speed isn't important.


Solar system pinball

The Oberth effect can be used on any planet or moon. The larger the mass of the object the more extreme the effect.


        Vescape (km/s)

Moon        2.38
Mars        5.03
Earth      11.2
Saturn     35.5
Jupiter    59.5
Sun       618
Any mission to the outer solar system first passes by Jupiter, both for the gravity assist and for an Oberth boost. Jupiter is the hub of the solar system.

If you have a nonzero approach speed for a planet then the Oberth maneuver gives a departure speed of

Approach speed from deep space  =  Vapproach

V2depart  =  V2approach + V2rocket + 2 Vrocket Vescape           Derivation

Mars Mission

A mission to Mars might use the following strategy:

Mine ice on the moon.

Launch the ice from the moon into space.

Use solar energy to convert ice into hydrogen and oxygen and then liquify it. This is now rocket fuel.

Use this fuel to send supplies to Mars. The supplies will go to Mars with a slow trajectory and the astronauts will go later using a faster trajectory. Using the Oberth effect, it's possible to move a heavy spacecraft to Mars using two light nudges from the rockets, but the travel time is long.

Launch a rocket from the Earth and place it in an Oberth-style elliptical orbit. Fuel the rocket with ice from the moon. This is the rocket that will go to Mars.

Launch the astronauts from Earth and place them on board the Mars rocket.

When the fully-fueled Mars rocket is at perigee, fire the rocket so that it escapes the Earth and heads for Mars. This is the "Oberth maneuver".

Upon arriving at Mars, use an inverse Oberth maneuver to place the rocket into an elliptical orbit around Mars.

The spacecraft must now fire its rockets again to go from an elliptical orbit to a circular low-Mars orbit. It can use fuel that was sent ahead of time from the Earth for this maneuver.

Once the spacecraft is in low-Mars orbit, the astronauts can drop to the surface of Mars using the atmosphere for breaking.

On Mars, ice is used to fuel the rocket that will lift the astronauts into low-Mars orbit.

Once exploration is complete, the astronauts return to the spacecraft.

Using fuel sent ahead of time from the Earth, the spacecraft goes from a low-Mars orbit to an elliptical orbit.

The spacecraft refuels again and uses an Oberth maneuver to depart Mars. Upon reaching Earth, an inverse Oberth maneuver is used to place the spacecraft in an Earth elliptical orbit.

With this mission plan, the manned rocket uses fuel only during the Oberth and inverse Oberth maneuvers. This minimizes the travel time.


Radiation shielding
                                       mSieverts     Shielding
                                         /year       (tons/m^2)
Earth surface, all radiation             3.5          10
Earth surface,       cosmic rays only     .39         10
Earth 2 km altitude, cosmic rays only     .9           8
Earth 3 km altitude, cosmic rays only    1.7           7
Earth 4 km altitude, cosmic rays only    3.3           6
Earth, 12 km altitude, equator          20             2.5
Earth, 12 km altitude, poles           100             2.5
Space station, 420 km altitude         150              .01  1/8 inch aluminum walls
Space                                600                .01  Beyond the Earth's field
Space, 4 tons/m^2 shield               2.5             4
Mars surface                         220                .16
Mars Hellas Basin, 7 km deep           ?                .29
To shield against cosmic rays, you need around 4 tons/meter^2 of material. This means the ship will weigh at least 500 tons, which emphasizes the need to get material from the moon. The liquid hydrogen & oxygen fuel can be used for radiation shielding.

The mass of food and water needed for the journey from Earth to Mars is much less than the mass of the radiation shielding, so you don't need to skimp on food quality.


Getting around the solar system

Lagrange points
Earth Lagrange points
Interplanetary transport network

Using Lagrange points and gravity slingshots, objects can be moved around the solar system with minimal propulsion.

Gravity assists can change a trajectory by of order the escape speed. You can use a sequence of gravity assists like a billiards-style trick shot to move objects around the solar system, requiring only nudges between assists. This is the "interplanetary transport network".

          Mass    Escape  Orbit
         (Earth   speed   speed
         masses)  (km/s)  (km/s)
Sun      333000    618.
Mercury    .0553     4.3    47.9
Venus      .8150    10.46   35.0
Earth     1.0000    11.2    29.8
Mars       .1074     5.03   24.1
Vesta      .000045    .36   19.3
Ceres      .00016     .51   17.9
Pallas     .0000359   .32   17.6
Jupiter 317.83      59.5    13.1
Saturn   95.16      35.5     9.64
Uranus   14.50      21.3     6.81
Neptune  17.20      23.5     5.43
Pluto      .00220    1.23    4.74
Moon       .0123     2.38    1.02
Charon     .000271            .23

Gravity assists

Galileo (Jupiter)
Cassini (Saturn)
Messenger (Mercury)

Voyager 1 & 2
New Horizons (Pluto)
Dawn (Vesta & Ceres)

Each of these missions is powered by chemical rockets except for Dawn, which is powered by an ion drive. Ion drives require fewer gravitational assists than chemical rockets.


Atmospheres
Earth
Titan
Veuns
Mars
      Gravity  Temp  Pressure  Density    N2       O2        CH4      CO2       Ar     Xe     H2S
      (m/s2)   (K)    (Bar)    (kg/m3)   (kg/m3)   (kg/m3)

Earth   9.80   287    1         1.22      .94       .209       0       .00048   .0011  0      0
Mars    3.71   210     .0063     .020     .00054   0           0       .020
Titan   1.35    94    1.46      5.3      5.22      0            .074  ?
Moon    1.62   220    0         0        0         0           0      0        0
Pandora 7.8    290    1.20      1.46     1.2        .30        ?       .26     ?        .080   .015
For humans, xenon is an anaesthetic and H2S is toxic.
Gear for exploring Titan

Arctic scuba gear
Wingsuit (human-powered flight is easy on Titan)
Strontium-90 radioactive power source
Device for extracting nitrogen from the air.
Device for electrolyzing ice to produce oxygen.
You don't need a pressure suit because the pressure is 1.5 times Earth pressure.


Appendix

Calculation of the Hohmann trajectory

Suppose a spaceship is on an elliptical orbit, such as the yellow orbit above.

G  = Gravitational constant
M  = Mass of sun
m  = Mass of spacecraft
R  = Distance of spacecraft from sun
R1 = Radius of perigee (point of closest approach to the star)
R2 = Radius of apogee  (point on orbit furthest from star)
V  = Velocity of spacecraft
V1 = Velocity of spacecraft at perigee
V2 = Velocity of spacecraft at apogee
A  = Semi-major axis of the orbital ellipse
   = .5 (R1 + R2)
E  = Energy
   = Kinetic energy  +  Gravitational potential energy
   = .5 m V^2        -  G M m / R
Angular momentum is conserved. Equating angular momentum at apogee and perigee,
V1 R1 = V2 R2
Equating energy at apogee and perigee,
E  =  .5 m V1^2 - GMm/R1  =  .5 m V2^2 - GMm/R2
Algebra gives
E = - G M m / (2A)
The energy at radius R is
E  =  .5 m V^2 - GMm/R  =  - GMm/(2A)
For a circular orbit, V^2 = GM/R, and
E  = - .5 m V^2
   = - .5 GMm/R
Make dimensionless:
G  = 1
M  = 1
m  = 1
R1 = 1
R2 = r

Solve for V1 and V2

.5 V1^2 - 1   = - 1/(1+r)        ->       V1^2 = 2r / (1+r)

.5 V2^2 - 1/r = - 1/(1+r)        ->       V2^2 = 2/r - 2/(1+r)

For the Earth-Mars system,
r  =  R2/R1
   =  1.524

V1 = 1.09891
V2 =  .72107

U1 =  Velocity of a circular orbit at radius R1
   =  1.
U2 =  Velocity of a circular orbit at radius R2
   =  1/sqrt(r)
   =  .81004
D1 =  Departure velocity from the Earth
   =  V1 - U1
   =  .09891
D2 =  Arrival velocity at Mars
   =  U2 - V2
   =  .08897
To restore the units, multiply the dimensionless velocities by the true value of U1.
U1 = 29.8 km/s   = Velocity of Earth in its orbit
U2 = 24.1 km/s   = Velocity of Mars  in its orbit

D1  -->  D1 * U1  =  .09891 * 29.8  =  2.948 km/s
D2  -->  D2 * U2  =  .08897 * 29.8  =  2.651 km/s
The departure velocity from the Earth is
D1 = 2.95 km/s
Upon arriving Mars, the velocity with respect to Mars is
D1 = 2.65 km/s
The total change in velocity that the spacecraft must generate is
D1 + D2  =  5.60 km/s
A hydrogen+oxygen rocket has an exhaust speed of 4.4 km/s and is capable of generating this change in velocity.
Oberth effect

For a Hohmann trajectory, the travel time from Earth to Mars is 289 days, which is a long time to be in zero gravity and in the radiation of space. The Oberth effect can speed up the trip.

Suppose a spacecraft is on a highly elliptical orbit, with a perigee just larger than the Earth's radius and an apogee much larger than the Earth's radius. Such an orbit would look like the Kuiper belt object "Sedna" pictured above.

G  = Gravitational constant
M  = Mass of Earth
R1 = Perigee radius
   ~ Radius of Earth
R2 = Apogee radius
     >> Radius of the Earth
V1 = Velocity of spacecraft at perigee
V2 = Velocity of spacecraft at apogee
   ~ 0
U1 = Velocity of a spacecraft on a circular orbit at radius R1
U1 = G M / R1
   = 7.2 km/s for the Earth
U2 = Velocity of a spacecraft on a circular orbit at radius R2
   ~ 0
Ve = Escape velocity from the Earth
When the spacecraft is at apogee, the energy is
E  =  Kinetic energy  +  Gravitational energy
   =         0        +         0
When the spacecraft is at perigee, the energy is
E  =  Kinetic energy  +  Gravitational energy
   =    1/2 m V1^2    -  G M m / R1

V1^2 =  2 G M / R1
     =  2 U1^2
     =  Ve^2
V1 is equal to the "Escape velocity". If a spacecraft starts from the surface of the Earth and is launched directly away from the Earth, it must have a velocity of at least Ve to escape the Earth.

The escape velocity can also be obtained from the gravitational potential energy.

1/2 m Ve^2 = G M m / R1     -->     Ve^2 = 2 G M / R1
Suppose the spacecraft fires its rockets at perigee and increases its speed by D1.

The energy is now

E  =  1/2 m (V1 + D1)^2  -  G M m / R1
   =  1/2 m (Ve + D1)^2  -  1/2 m Ve^2
   =  1/2 m (D1^2 + 2 D1 Ve)
The spacecraft is now on a hyperbolic orbit and will escape the Earth, As it recedes from the Earth, it will approach a constant velocity Q1. When it is far from the Earth, the energy is
E  =  1/2 m (D1^2 + 2 D1 Ve)
   =  1/2 m Q1^2

Q1 =  SquareRoot(D1^2 + 2 D1 Ve)

Q1 > D1
If the spacecraft starts in an elliptical orbit and changes its speed by D1 at perigee, it departs the Earth at speed Q1, which is larger than D1. This is the "Oberth effect".

If a rocket changes its velocity by 5 km/s at perigee, it departs the Earth with a velocity of

Q1  =  SquareRoot(5^2 + 2 * 5 * 11.2)
    =  11.7 km/s
This gets you to Mars in about 4 months.

X:  Change in velocity at perigee
Y:  Departure velocity from the planet
Ve: Escape velocity for the planet
This is a plot of
Y^2 = X^2 + 2 X Ve

           Ve (km/s)
Moon           2.38
Mars           5.03
Earth         11.2
Saturn        35.5
Jupiter       59.5
Sun          618.

Bicycle

A typical set of parameters for a racing bike is

Velocity        =  V          =   20 m/s       (World record=22.9 m/s)
Power           =  P          = 2560 Watts     (Typical power required to move at 20 m/s, measured experimentally)
Force on ground =  F  =  P/V  =  128 Newtons

We assume a high gear, with 53 teeth on the front gear and 11 teeth on the rear gear.

Number of links in the front gear      =  Nf  =  53
Number of links in the rear gear       =  Nr  =  11
Length of one link of a bicycle chain  =  L          =  .0127 m =  .5 inches
Radius of the front gear               =  Rf  =  Nf L / (2 π)   =  .107  m
Radius of the rear gear                =  Rr  =  Nr L / (2 π)   =  .0222 m

Torque balance:
Ground force * Wheel radius  =  Chain force * Rear gear radius
Pedal force  * Pedal radius  =  Chain force * Front gear radius

Chain force  =  Ground force * Wheel radius / Rear gear radius
             =  128 * .311 / .0222
             =  1793 Newtons

Pedal force  =  Ground force * Wheel radius / Pedal radius * Front gear radius / Rear gear radius
             =  Ground force * Wheel radius / Pedal radius * Front gear teeth  / Rear gear teeth
             =  128 * .311 / .17 * 53 / 11
             =  1128 Newtons

            Radius  Force  Torque  Gear
              (m)    (N)    (Nm)   teeth
Pedal crank  .170   1128   191.9     -
Front gear   .107   1793   191.9    53
Rear gear    .0222  1793    39.8    11
Rear wheel   .311    128    39.8     -

Wheel frequency =  Velocity / (Radius * 2Pi)
                =  20 / (.311 * 2π)
                =  10.2 Hertz
Pedal frequency =  Wheel frequency * Rear gear teeth / Front gear teeth
                =  10.2 * 53 / 11
                =  2.12 Hertz
                =  127 revolutions per minute
Humans can pedal effectively in the range from 60 rpm to 120 rpm. Gears allow one to choose the pedal frequency. There is also a maximum pedal force of around 1200 Newtons.

When going fast the goal of gears is to slow down the pedals.

When one is climbing a hill the goal of gears is to speed up the pedals so that you don't have to use as much force on the pedals.

Pedal period                   * Rear gear teeth   =  Wheel period                   * Front gear teeth
Pedal radius / Pedal velocity  * Front gear teeth  =  Wheel radius / Wheel velocity  * Front gear teeth

Pedal force  =  Power / Pedal velocity
             =  Power / Wheel velocity * Wheel radius / Pedal radius * Front gear teeth / Rear gear teeth
             =  Power / Wheel velocity * .311 / .17 * Front gear teeth / Rear gear teeth
             =  Power / Wheel velocity * 1.83 * (Front gear teeth / Rear gear teeth)
             =  Power / Wheel velocity * 1.83 * Gear ratio

Gear ratio   =  Front gear teeth / Rear gear teeth
For a given power and wheel velocity, the pedal force can be adjusted by adjusting the gear ratio.

Suppose a bike is going uphill at large power and low velocity.

Power            =  1000 Watts
Velocity         =  3 m/s
Front gear teeth =  34              (Typical for the lowest gear)
Rear gear teeth  =  24              (Typical for the lowest gear)

Pedal force  =  Power / Wheel velocity * 1.83 * Front gear teeth / Rear gear teeth
             =  1000 / 3 * 1.83 * 34 / 24
             =  864 Newtons
             =  88 kg equivalent force
This is a practical force. If you used the high gear,
Pedal force  =  Power / Wheel velocity * 1.83 * Front gear teeth / Rear gear teeth
             =  1000 / 3 * 1.83 * 53 / 11
             =  2939 Newtons
             =  300 kg equivalent force
This force is impractically high.

History of metallurgy

Stone age
Copper age, 5000 BCE
Bronze age, 3200 BCE
Iron age, 1200 BCE
Carbon age, 1987

The carbon age began in 1987 when Jimmy Connors switched from a steel to a carbon racquet.

      Discovery   Yield    Density
       (year)    Strength  (g/cm3)
                  (GPa)

Gold     Ancient   .20     19.3
Silver   Ancient   .10     10.5
Copper   -5000     .12      9.0
Bronze   -3200     .20      9      Copper + Tin.   Stronger than copper
Brass    -2000     .20      9      Copper + Zinc.  Stronger than copper
Iron     -1200     .25      7.9    In the form of steel. Stronger than bronze and brass
Carbon    1963    1.4       1.75   Royal Aircraft develops the first commercial carbon fiber

Discovery of elements

The ancient metals such as iron, copper, tin, and zinc are obtained by carbon smelting minerals. Cobalt was the first metal discovered since iron and it's discovery inspired people to smelt every known mineral in the hope of yielding a new metal. By 1800 nearly all of the carbon smeltable metals had been discovered.

Some elements can't be carbon smelted and require electrolysis to isolate. Electrochemistry began in 1800 with the invention of the battery and most of the remaining metals were discovered soon after. Sodium and potassium were isolated by electrolysis in 1807 and these were used to smelt metals that couldn't be smelted with carbon.

         Discovery   Method of         Abundance in
          (year)     discovery         crust (ppm)

Carbon     Ancient   Naturally occuring      400       Coal, diamond
Gold       Ancient   Naturally occuring         .0031
Silver     Ancient   Naturally occuring         .08
Sulfur     Ancient   Naturally occuring      420
Lead         -6500   Smelt with carbon        10
Copper       -5000   Smelt with carbon        68
Bronze (As)  -4200                                     Copper + Arsenic
Bronze (Sn)  -3200                                     Copper + Tin
Tin          -3200   Smelt with carbon         2.2    
Brass        -2000                                     Copper + Zinc
Mercury      -2000   Heat the oxide             .067
Iron         -1200   Smelt with carbon     63000       In the form of steel
Zinc          1300   Smelt with carbon        79       Date when first produced in pure form
Antimony      1540   Smelt with iron            .2
Arsenic       1649   Heat the oxide            2.1
Phosphorus    1669   Heat the oxide        10000
Cobalt        1735   Smelt with carbon        30       First metal discovered since iron
Platinum      1735   Naturally occuring         .0037
Nickel        1751   Smelt with carbon        90
Hydrogen      1766   Hot iron + steam       1500
Oxygen        1771   Heat HgO             460000
Nitrogen      1772   From air                 20
Manganese     1774   Smelt with carbon      1120
Molybdenum    1781   Smelt with carbon         1.1
Tungsten      1783   Smelt with carbon      1100
Chromium      1797   Smelt with carbon       140
Palladium     1802   Chemistry                  .0063
Osmium        1803                              .0018
Iridium       1803                              .004
Rhodium       1804   Smelt with zinc            .0007  Smelt Na3RhCl6 with zinc
Sodium        1807   Electrolysis          23000
Potassium     1807   Electrolysis          15000
Magnesium     1808   Electrolysis          29000
Cadmium       1817   Smelt with carbon          .15
Lithium       1821   Electrolysis             17
Zirconium     1824   Smelt with potassium
Aluminum      1827   Smelt with potassium  82000
Silicon       1823   Smelt with potassium 270000
Beryllium     1828   Smelt with potassium      1.9
Thorium       1929   Smelt with potassium
Vanadium      1831                           190
Uranium       1841   Smelt with potassium      1.8
Ruthenium     1844   Smelt with carbon          .001
Tantalum      1864   Smelt with hydrogen       1.7
Niobium       1864   Smelt with hydrogen      17
Fluorine      1886   Electrolysis            540
Helium        1895   From uranium ore
Titanium      1910   Smelt with sodium     66000
Hafnium       1924
Rhenium       1928   From molybdenite           .0026
Scandium      1937                            26

Gold was the densest element known until the discovery of platinun in 1735. It was useful as an uncounterfeitable currency until the discovery of tungsten in 1783, which has the same density as gold.

Wood fires are 200 Celsius short of the copper smelting temperature. Coal has to be used.

Titanium can't be smelted with carbon because it produces titanium carbide (TiC).


Discovery of the strong metals

The usefulness of a metal as a sword depends on its strength/density ratio. The table below shows all the metals with a ratio of at least 5 MJoules/kg. For these metals, strength tends to be proportional to density and the strength/density ratio has a characteristic value of 10 MJoules/kg. Beryllium is the sole outlier with a superlatively large value of 71 MJoules/kg.

The low density metals are ones up to vanadium on the periodic table. None are carbon smeltable and all require electrochemistry to isolate. The first low density metal to be produced was magnesium in 1808.

        Protons  Strength  Density  Strength/Density   Carbon    Discovery
                  (GPa)    (g/cm3)   (MJoule/kg)      smeltable    year

Beryllium    4      132      1.85      71.4          no      1828
Magnesium   12       17      1.74       9.8          no      1808
Aluminum    13       26      2.70       9.6          no      1827
Scandium    21       29      3.0        9.7          no      1937
Titanium    22       44      4.5        9.8          no      1910
Vanadium    23       47      6.0        7.8          no      1831
Chromium    24      115      7.2       16.0          yes     1797
Manganese   25       75      7.2       10.4          yes     1774
Iron        26       82      7.9       10.4          yes    -1200
Cobalt      27       75      8.9        8.4          yes     1735
Nickel      28       76      8.9        8.5          yes     1751
Copper      29       48      9.0        5.3          yes    -5000
Zinc        30       43      7.1        6.0          yes     1746
Molybdenum  42      120     10.3       11.7          yes     1781
Ruthenium   44      173     12.4       14.0          yes     1844
Rhodium     45      150     12.4       12.1          yes     1804
Tungsten    74      161     19.2        8.4          yes     1783
Rhenium     75      178     21.0        8.5          yes     1928
Osmium      76      222     22.6        9.8          yes     1803
Iridium     77      210     22.6        9.3          yes     1803
Uranium     92      111     19.1        5.8          no      1841

Strength:          Shear modulus            (GPascals)
Density:           Density                  (grams/cm3)
Strength/Density:  Shear modulus / Density  (MJoules/kg)

Metals known since antiquity

For a metal, the stiffness is characterized by the "shear strength" and the sword worthiness is characterized by the shear strength over the density (the "strength to weight ratio"). For example for iron,

Shear modulus    =  S         =   82 GJoules/meter3
Density          =  D         = 7900 kg/meter3
Sword worthiness =  Q  = S/D  = 10.4 MJoules/kg

Metals

This plot includes all metals with a strength/density at least as large as lead, plus mercury. Beryllium is beyond the top of the plot.


Wootz steel

-600  Wootz steel developed in India and is renowned as the finest steel in the world.
1700  The technique for making Wootz steel is lost.
1790  Wootz steel begins to be studied by the British Royal Society.
1838  Anosov replicates Wootz steel.
Wootz steel is a mix of two phases: martensite (crystalline iron with .5% carbon), and cementite (iron carbide, Fe, 6.7% carbon).

Iron meteorites

In prehistoric times iron meteorites were the only source of metallic iron. They consist of 90% iron and 10% nickel.


Alloys

Copper
Orichalcum (gold + copper)
Gold

Alloy of gold, silver, and copper


Strongest alloys
                   Yield    Density   Yield/Density
                  strength  (g/cm3)    (GJoule/kg)
                   (GPa)
Magnesium  + Li        .14     1.43      .098
Magnesium  + Y2O3      .31     1.76      .177
Aluminum   + Be        .41     2.27      .181
LiMgAlScTi            1.97     2.67      .738
Titanium   + AlVCrMo  1.20     4.6       .261
AlCrFeCoNiTi          2.26     6.5       .377
AlCrFeCoNiMo          2.76     7.1       .394
Steel      + Co Ni    2.07     8.6       .241
VNbMoTaW              1.22    12.3       .099
Molybdenum + W Hf     1.8     14.3       .126

Sapphire               .4      3.98      .101
Diamond               1.6      3.5       .457
Magnesium              .10     1.74      .057
Beryllium              .34     1.85
Aluminum               .020    2.70
Titanium               .22     4.51
Chromium               .14     7.15
Iron                   .10     7.87
Cobalt                 .48     8.90
Nickel                 .19     8.91
Copper                 .12     8.96
Molybdenum             .25    10.28
Tin                    .014    7.26
Tungsten               .947   19.25
Rhenium                .290   21.02
Osmium                        22.59
Iridium                       22.56
Alloys can be vastly stronger than their constituent elements. Alloys such as "TiScAlLiMg" are "high entropy alloys", which are a mix of elements in approximately equal proportions.
For comparison, the table includes pure metals, diamond, and sapphire. Large synthetic sapphires and small synthetic diamonds can be constructed. The recently developed LiMgAlScTi alloy is the first metal to outpeform diamond.
Alloy types
Beryllium + Li           →  Doesn't exist. The atoms don't mix
Beryllium + Al           →  Improves strength
Magnesium + Li           →  Weaker and lighter than pure Mg. Lightest existing alloy
Magnesium + Be           →  Only tiny amounts of beryllium can be added to magnesium
Magnesium + Carbon tubes →  Improves strength, with an optimal tube fraction of 1%
Aluminum  + Li,Mg,Be,Sc  →  Stronger and lighter than aluminum
Titanium  + Li,Mg,Sc     →  Stronger and lighter than titanium
Steel     + Cr,Mo        →  Stronger and more uncorrodable than steel. "Chromoly"
Copper    + Be           →  Stronger than beryllium and cannot spark

Column buckling

If too much weight is placed on a column it buckles. Suppose a column is constructed with constant mass and varying density. The lower the density the wider and stronger the column.

Radius            =  R
Length            =  L
Density           =  D
Mass              =  M  =  π D L R2
Buckling constant =  C
Tensile modulus   =  K
Force             =  F  =  C K R4 L-2      Force requird to buckle the column
Quality           =  Q  =  F / M  =  K M D-2 L-4
The figure of merit for a material for columns is K D2. Balsa wood has a density of .16 g/cm3 and outperforms the strongest alloys.
                   Yield    Density  Yield/Density  Yield/Density2
                  strength  (g/cm3)    (GJoule m3/kg2)
                   (GPa)
Balsa                  .006     .16      .037    .234
Bamboo                 .0079    .35      .023    .064
Magnesium  + Li        .14     1.43      .098    .068
Magnesium  + Y2O3      .31     1.76      .177    .100
Aluminum   + Be        .41     2.27      .181    .080
LiMgAlScTi            1.97     2.67      .738    .276
Titanium   + AlVCrMo  1.20     4.6       .261    .057
AlCrFeCoNiTi          2.26     6.5       .377    .053
AlCrFeCoNiMo          2.76     7.1       .394    .055

High-temperature metals (refractory metals)
          Melting point (Celsius)

Tungsten    3422
Rhenium     3186
Osmium      3033
Tantalum    3017
Molybdenum  2623
Niobium     2477
Iridium     2446
Ruthenium   2334
Hafnium     2233
Technetium  2157
Rhodium     1964
Vanadium    1910
Chromium    1907

High-temperature superalloys

Most alloys weaken with increasing temperature except for a small subset called "superalloys" that strengthen with temperature, such as Ni3Al and Co3Al. This is called the "yield strength anomaly".

Nickel alloys in jet engines have a surface temperature of 1150 Celsius and a bulk temperature of 980 Celsius. This is the limiting element for jet engine performance. Half the mass of a jet engine is superalloy.

Current engines use Nickel superalloys and Cobalt superalloys are under development that will perform even better.

Yield strength in GPa as a function of Celsius temperature.

                   20   600   800  900  1000  1100 1200  1400  1600 1800  1900  Celsius

VNbMoTaW          1.22         .84        .82       .75  .66   .48   .4
AlMohNbTahTiZr    2.0   1.87  1.60  1.2   .74  .7   .25
Nickel superalloy 1.05        1.20   .90  .60  .38  .15
Tungsten           .95   .42   .39        .34  .31  .28  .25   .10   .08  .04
Below 1100 Celsius AlMohNbTahTiZr has the best strength-to-mass ratio and above this VNbMoTaW has the best ratio. Both alloys supercede nickel superalloy and both outperform tungsten, the metal with the highest melting point. Data:   
Entropy, nickel superalloy
Copper alloys
                  Yield strength (GPa)

Copper                  .27
Brass                   .41     30% zinc
Bronze                  .30     5% tin
Phosphor bronze         .69     10% tin, .25% phosphorus
Copper + beryllium     1.2      2% beryllium, .3% cobalt
Copper + nickel + zinc  .48     18% nickel, 17% zinc
Copper + nickel         .40     10% nickel, 1.25% iron, .4% manganese
Copper + aluminum       .17     8% aluminum

Bells and cymbals

Bells and cymbals are made from bell bronze, 4 parts copper and 1 part tin.


Mohs hardness

Carbide

Carbides are the hardest metallic materials.

10     Diamond
 9.5   BN, B4C, B, TiB2, ReB2
 9.25  TiC, SiC
 9.0   Corundum, WC, TiN
 8.5   Cr, TaC, Si3N4
 8     Topaz, Cubic zirconia
 7.5   Hardened steel, tungsten, emerald, spinel
 7     Osmium, Rhenium, Vanadium, Quartz

Full list of alloys
Primary  Added   Yield  Break  Stiff  Strain  Poi-  Density Vick  Elong  Yield/   Melt
metal    metals  (GPa)  (GPa)  (Gpa)          sson  (g/cm3)              density  (C)

Magnesium  Li              .16    45                 1.43             .098
Magnesium  Y2O3     .312   .318                      1.76             .177
Magnesium  Tube     .295   .39    49                 1.83        .05  .161
Beryllium           .345   .448  287  .0016  .032    1.85             .186
Aluminum   Be40     .41    .46   185                 2.27        .07  .181
Aluminum   Mg Li    .21    .35    75  .0047          2.51
Aluminum   Cu Li    .48    .53                       2.59             .185
Aluminum   Mg Sc    .433   .503                      2.64        .105 .164
LiMgAlScTi         1.97                              2.67  5.8        .738
Titanium   Be Al                                     3.91
Titanium   Al6V4    .89   1.03   114         .33     4.43   .34  .14          1660
Titanium   VCrMoAl 1.20   1.30                       4.6         .08  .261
Vit 1              1.9                               6.1   5.7
AlCoCrFeNiTih      2.26   3.14                       6.5         .23  .377
Zirconium  Liquid  1.52   1.52    93                 6.57   .56  .018 .231
AlCoCrFeNiMo       2.76                              7.1              .394
AlMohNbTahTiZr     2.0    2.37                       7.4              .270
Inconel 718                                          8.19
Copper     Be      1.2    1.48   130         .30     8.25             .145     866
CrFeNiV.5W                2.24                      ~8.5
Iron       Co Ni   2.07   2.38                       8.6         .11  .241
Iron       Cr Mo                                     9      .32
Nickel     Cr      1.2    2.3    245         .32     8.65  6.6
TiZrNbHfTa          .93                              9.94  3.83  .5
TiVNbMoTaW                                          11.70  4.95
VNbMoTaW                                            12.36
NbMoTaW                                             13.75
Molybdenum W45Hf1 ~1.8    2.14                     ~14.3   3.6   .126
Tungsten   MoNiFe   .62    .90   365                17.7         .10  .035

Yield:     Yield modulus
Break:     Tensile strength (breaking point)
Stiffness: Young's modulus
Strain:    Fractional strain at the breaking point
Poisson:   Poisson ratio
Many properties of alloys are approximately equal to a linear sum of the properties of its constituent elements. This applies for density, stiffness modulus, and Poisson's ratio.

Many properties of alloys can be dramatically different from those of its constiuent elements. This applies for the yield modulus, the tensile breaking modulus, and the hardness.

For aluminum alloys, density = 2.71 - .01 Mg - .079 Li.

Magnesium strengthens when alloyed with aluminum, nickel, copper, and neodymium.

Data:    TiVNbMoTaW    AlTiNbMo½Ta½Zr    Mg    Be+Al    Aluminum+Mg+Li    Table    Al+Be    Mg + Li    Mg alloys    Elasticity    Ti alloy    Ti alloy    Ti alloy textbook    Liquidmetal    Mg + tubes    Elasticity table    Al Cu Li    Al + tubes    Mg + tubes    W + Mo    Al Mg Sc    Fe + Co + Ni    Li2MgSc2Ti3Al2    Entropy survey    Entropy survey    Entropy survey *    CrFeNiV½W    Entropy rev 2014    Nickel Chromium    Copper textbook    TiZrNbHfTa


Vickers hardness
                       Min   Max

Valence compounds     1000  4000     carbides, borides, silicides
Intermetallic          650  1300
BCC lattice            300   700
FCC lattice            100   300

Metal smelting

Prehistoric-style smelter

Most metals are in oxidized form. The only metals that can be found in pure form are gold, silver, copper, platinum, palladium, osmium, and iridium.

Smelting is a process for removing the oxygen to produce pure metal. The ore is heated in a coal furnace and the carbon seizes the oxygen from the metal. For copper,

Cu2O + C  →  2 Cu + CO
At low temperature copper stays in the form of Cu2O and at high temperature it gives the oxygen to carbon and becomes pure copper.

For iron, the oxidation state is reduced in 3 stages until the pure iron is left behind.

3 Fe2O3 + C  →  2 Fe3O4 + CO
Fe3O4   + C  →  3 FeO   + CO
FeO     + C  →    Fe   + CO
Oxidation state  =  Number of electrons each iron atom gives to oxygen

       Oxidation state
CuO          2
Cu2O         1
Cu           0
Fe2O3        3
Fe3O4       8/3
FeO          2
Fe           0

Smelting temperature

The following table gives the temperature required to smelt each element with carbon.

        Smelt  Method  Year  Abundance
         (C)                   (ppm)

Gold        <0   *   Ancient      .0031
Silver      <0   *   Ancient      .08
Platinum    <0   *    1735        .0037
Mercury     <0  heat -2000        .067
Palladium   <0  chem  1802        .0063
Copper      80   C   -5000      68
Sulfur     200   *   Ancient   420
Lead       350   C   -6500      10
Nickel     500   C    1751      90
Cadmium    500   C    1817        .15
Cobalt     525   ?    1735      30
Tin        725   C   -3200       2.2
Iron       750   C   -1000   63000
Phosphorus 750  heat  1669   10000
Tungsten   850   C    1783    1100
Potassium  850   e-   1807   15000
Zinc       975   C    1746      79
Sodium    1000   e-   1807   23000
Chromium  1250   C    1797     140
Niobium   1300   H    1864      17
Manganese 1450   C    1774    1120
Vanadium  1550   ?    1831     190
Silicon   1575   K    1823  270000
Titanium  1650   Na   1910   66000
Magnesium 1875   e-   1808   29000
Lithium   1900   e-   1821      17
Aluminum  2000   K    1827   82000
Uranium   2000   K    1841       1.8
Beryllium 2350   K    1828       1.9

Smelt:      Temperature required to smelt with carbon
Method:     Method used to purify the metal when it was first discovered
            *:  The element occurs in its pure form naturally
            C:  Smelt with carbon
            K:  Smelt with potassium
            Na: Smelt with sodium
            H:  Smelt with hydrogen
            e-: Electrolysis
            heat:  Heat causes the oxide to decompose into pure metal. No carbon required.
            chem:  Chemical separation
Discovery:  Year the element was first obtained in pure form
Abundance:  Abundance in the Earth's crust in parts per million
Elements with a low carbon smelting temperature were discovered in ancient times unless the element was rare. Cobalt was discovered in 1735, the first new metal since antiquity, and this inspired scientists to smelt every known mineral in the hope that it would yield a new metal. By 1800 all the rare elements that were carbon smeltable were discovered.

The farther to the right on the periodic table, the lower the smelting temperature, a consequence of "electronegativity".

The battery was invented in 1800, launching the field of electrochemistry and enabling the the isolation of non-carbon-smeltable elements. Davy used electrolysis in 1807 to isolate sodium and potassium and then he used these metals to smelt other metals. To smelt beryllium with potassium, BeO + 2 K ↔ Be + K2O.

Titanium can't be carbon smelted because it forms the carbide Ti3C.

Data


Thermite

Thermite is smelting with aluminum. For example, to smelt iron with aluminum,

Fe2O3 + 2 Al  →  2 Fe + Al2O3

Smelting reactions

The following table shows reactions that change the oxidation state of a metal. "M" stands for an arbitrary metal and the magnitudes are scaled to one mole of O2. The last two columns give the oxidation state of the metal on the left and right side of the reaction. An oxidation state of "0" is the pure metal and "M2O" has an oxidation state of "1".

                            Oxidation state   Oxidation state
                                at left          at right
 2  M2O   ↔  4  M     + O2        1                0
 4  MO    ↔  2  M2O   + O2        2                1
 2  M3O4  ↔  6  MO    + O2       8/3               2
 6  M2O3  ↔  4  M3O4  + O2        3               8/3
 2  M2O3  ↔  4  MO    + O2        3                2
 2  MO    ↔  2  M     + O2        2                0
2/3 M2O3  ↔ 4/3 M     + O2        3                0
 1  MO2   ↔  1  M     + O2        4                0
 2  MO2   ↔  2  MO    + O2        4                2

Gibbs energy

Let MO be a metal oxide for which the Gibbs energy of CO is larger than MO and the oxygen binds to the metal preferentially over carbon.

The entropies of most metal oxides are similar and so changing the temperature has little effect on their relative Gibbs energies. CO is special because it is a gas and hence has a larger entropy than the solid metal oxides. As temperature increases the Gibbs energy of CO decreases faster than that of MO and at the critical smelting temperature they are equal. Above this temperature the oxygen unbinds to the metal and binds to carbon.

For the smelting of cobalt,

Standard temperature                 =  T0  =  298 Kelvin  =  25 Celsius
Smelting temperature                 =  Tsmelt
Temperature change                   =  t   =  Tsmelt - T0
Gibbs energy at standard temperature =  G
Entropy at standard temperature      =  S
Gibbs energy at temperature Tsmelt    =  g   =  G - t S
CoO Gibbs energy per mole O2         =  GCoO =  -428.4   kJoule/mole
CO  Gibbs energy per mole O2         =  GCO  =  -274.4   kJoule/mole
CoO entropy per mole O2              =  SCoO =      .12  kJoule/mole
CO  entropy per mole O2              =  SCO  =      .396 kJoule/mole
At the smelting temperature, the Gibbs energies of CoO and CO are equal and the reaction is in equilibrium. Below this temperature oxygen binds to cobalt and above this temperature it binds to carbon. The calculation is approximate because it assumes entropy is a constant as a function of temperature. To calculate the smelting temperature,
    gCoO      =      gCO
GCoO - t SCoO  =  GCO - t SCO

t  =  (GCoO - GCO) / (SCoO - SCO)
   =  558 Celsius

Tsmelt  =  583 Celsius  =  t + 25 Celsius      (The actual smelting temperature is 525 Celsius)

Smelting thermodynamics
       Gibbs       Gibbs            Entropy       Entropy
    kJoule/mole  kJoule/mole(O2)  kJoule/mole  kJoule/mole(O2)

Li2O     -561.9    -1123.8
Na2O     -377       -754
K2O      -322.2     -644.4
Cu2O     -146.0     -292.0
Ag2O      -11.2      -22.4

BeO      -579.1    -1158.2
CO       -137.2     -274.4     .198      .396
MgO      -596.3    -1192.6     .0269     .0538
CaO      -533.0    -1066.0     .0398     .0769
VO       -404.2     -808.4
MnO      -362.9     -725.8     .0597     .1194
CoO      -214.2     -428.4
NiO      -211.7     -423.4
CuO      -129.7     -259.4     .0426     .0852
ZnO      -318.2     -636.4
CdO      -228.4     -456.8
HgO       -58.5     -117.0

Fe3O4   -1014       -507       .0146     .0073
Co3O4    -795.0     -397.5

B2O3    -1184       -789
Al2O3   -1582.3    -1054.9     .0509     .0339
Ti2O3   -1448       -965.3
V2O3    -1139.3     -759.5
Cr2O3   -1053.1     -702.1     .0812     .0541
Fe2O3    -741.0     -494.0     .0874     .0583

CO2      -394.4     -394.4     .214      .214
SO2                            .2481     .2481
SiO2     -856.4     -856.4     .0418     .0418
TiO2     -852.7     -852.7
MnO2     -465.2     -465.2     .0530     .0530
MoO2     -533.0     -533.0
WO2      -533.9     -533.9
PbO2     -219.0     -219.0

MoO3     -668.0     -445.3
WO3      -764.1     -509.4

V2O4    -1318.4     -659.2

Cu          0
C (gas)   672.8

Minerals

Spodumene: LiAl(SiO3)2
Beryl: Be3Al2(SiO3)6
Periclase: MgO
Magnesite: MgCO3
Dolomite: CaMg(CO3)2
Bauxite: Al(OH)3 and AlO(OH)

Quartz: SiO2
Rutile: TiO2
Vanadinite: Pb5(VO4)3Cl
Chromite: FeCr2O4
Pyrolusite: MnO2
Hematite: Fe2O3

Hematite: Fe2O3
Pyrite: FeS2
Iron meteorite
Cobaltite: CoAsS
Millerite: NiS
Chalcocite: Cu2S

Chalcopyrite: CuFeS2
Sphalerite: ZnS
Germanite: Cu26Fe4Ge4S32
Zircon: ZrSiO4
Molybdenite: MoS2
Acanthite: Ag2S
Cassiterite: SnO2
Wolframite: FeWO4
Cinnabar: HgS
Platinum nugget
Gold nugget
Galena: PbS

Fluorite: CaF2
Volcanic sulfur
Alumstone: KAl3(SO4)2(OH)6


Aliens

Timeline of the universe

An alien planet could conceivably have formed as early as 1 billion years after the big bang, meaning that there are likely aliens with a head start on us by billions of years.

An alien civilization could easily build a fission or fusion rocket that travels at 1/10 the speed of light, which would take 1 million years to cross the galaxy. The aliens have plenty of time to get here.

                Millions of years ago

Big bang             13700
First planets formed 13000
Earth formed          4500
Photosynthesis        3000
Oxygen atmosphere      600
Multicellular life     600
Vertebrates            480
Tetrapod vertebrates   400       Mammals, birds, and reptiles are all tetrapods
Mammals                170
Dinosaur extinction     66
Cats                    25
Cheetahs                 6       Fastest land animal
Tigers                   1.8
Humans                   1
Lions                     .9
Agriculture               .01
Civilization              .005
Calculus                  .0004
Smartphones               .00001

Starship

We presently possess the technology to build a fission and fusion rocket, each of which can reach a speed of .1 times the speed of light, and such a rocket can cross the Milky Way galaxy in a time that is a small fraction of the age of the universe. If aliens had built such a rocket they could easily have already colonized the galaxy.

Speed of light                       =  C
Speed of a fission or fusion rocket  =  V  =  .1 C
Diameter of the Milky Way            =  X           =     .1 million light years
Time to cross the galaxy             =  T  =  X/V   =      1 million years
Age of the universe                                 =  13800 million light years

Divinity

The divinity hypothesis becomes persuasive if there is a physical mechanism allowing it to happen, a mechanism that obeys the known laws of physics. Such a mechanism exists. An advanced alien civilization is equivalent to a diety.


Stone Churches

Ulm Minster
Saint Peter's Basilica
Basilica of Saint Paul

Washington National Cathedral
Liverpool Cathedral
Washington National Cathedral, state funeral for Ronald Reagan

                               Volume    Area   Nave  Tower  Year
                               (k m3)   (k m2)  (m)   (m)

Saint Peter's Basilica            5000   15.2   46    136.6  1626   Vatican
Basilica of Our Lady of Aparecida 1200   12     40    100.0  1980   Brazil   Aparecida
Saint Joseph's Oratory             660    6.8                1967   Canada   Montreal*
Seville Cathedral                  500   11.5   42    105    1528   Spain    Seville
Cathedral of St. John the Divine   480   11.2   37.8   70.7  1941   USA      Manhattan
Liverpool Cathedral                450    9.7         101.0  1978   UK       Liverpool
Milan Cathedral                    440   10.2   45.0  108.5  1965   Italy    Milan
Abbey of Santa Giustina                   9.7          75    1606   Italy    Padua
Cologne Cathedral                  407    7.9   43.4 *157.4  1880   Germany  Cologne
Basilica of Our Lady of Lichen     300    9.2   45    141.5  2004   Poland   Lichen Stary
San Petrino Basilica               270    7.9   45           1479   Italy    Bologna
Hagia Sophia                       256    8.0                 537   Turkey   Istanbul*
Amiens Cathedral                   200    7.7   42.3  112.7  1270   France   Amiens
Ulm Minster                        190    8.3   41   *161.5  1890   Germany  Ulm
Saint Mary's Church                190    5.0                1502   Poland   Gdansk*
Frauenkirche                       190    4.2                1525   Germany  Munich*
Palma Cathedral                    160    6.7                1346   Spain    Palma*
Holy Trinity Cathedral             137    5.0                2004   Georgia  Tbilisi*
Church of the Most Holy Trinity    130    8.7                2007   Portugal Fatima
Basilica St. Paul Outside the Walls       8.5   29.7   73    1823   Italy    Rome
Basilica Cat. Lady of the Pillar          8.3                1872   Spain    Zaragoza
Florence Cathedral                        8.3   45    114.5  1436   Italy    Florence
Basilica of the Sacred Heart              8.0          90.0  1970   Belgium  Brussels
Basilica of Our Lady of Guatalupe         8.2   34     42    1976   Mexico   Mexico City
Cathedral of Our Lady                     8.0                1521   Belgium  Antwerp
Basilica of Our Lady of Peace             8.0         158.0  1989   Ivory C. Yamoussoukro
Saint Paul's Cathedral                    7.9   37.5  111.3  1708   UK       London
Washington National Cathedral             7.7          91.7  1990   USA      DC
Basilica of the National Shrine           7.1                1961   USA      DC*
Cathedral of La Plata                     7.0                1932   Argen.   La Plata*
Mexico City Metropolican Cathedral        6.7                1813   Mexico   Mexico City*
Reims Cathedral                           6.7                1275   France   Reims*
Strasbourg Cathedral                      6.0        *142.0  1439   France   Strasbourg
Cathedral of Our Lady of the Angels       6.0                2002   USA      LA*
De Hoeksteen, Barneveld             43    6.0                       Neth.    Barneveld*
Padre Pio Pilgrimage Church               6.0                2004   Italy    San Giovani Rotondo*
Bourges Cathedral                         5.9                1230   France   Bourges*
Esztergom Basilica                        5.6                1869   Hungary  Esztergom*
Notre Dame de Paris                       5.5                1345   France   Paris*
Sagrida Familia                           5.4   45    170    2026   Spain    Barcelona
Primate Cathedral of Bogota               5.3                1823   Colombia Bogota*
Cathedral of Christ the Savior            5.2                1883   Russia   Moscow*
Chartres Cathedral                        5.2                1220   France   Chartres*
New Cathedral, Linz                       5.2         134.8  1924   Austria  Linz*
Westminster Cathedral                     5.0                1910   UK       London*
Winchester Cathedral                      5.0                1525   UK       Winchester*
Dresden Cathedal                          4.8                1755   Germany  Dresden*
Basilica of St. Therese, Lisieux          4.5                1954   France   Lisieux*
Basilica de San Martin de Tours           4.3                1878   Philip.  Taal*
Ely Cathedral, Cambridgeshire             4.3                1375   UK       Ely*
Cathedral Basilica of St. Louis           4.1                1914   USA      St. Louis*
Westminster Abbey                         3.0                1018   UK       London*
*: Held the status of world's tallest building.

Among these types of buildings the cathedral is probably the best known, to the extent that the word "cathedral" is sometimes mistakenly applied as a generic term for any very large and imposing church. In fact, a cathedral does not have to be large or imposing, although many cathedrals are. The cathedral takes its name from the word "cathedra", or "biship's throne".

Saint Peter's Basilica was designed principally by Bramante, Michelangelo, Maderno, and Bernini. It is the most renowned work of Renaissance architecture and is the largest stone church in the world.


Church towers
Ulm Minster
Cologne Cathedral
                       Height (m)  Year

Ulm Minster                161.5   1890 *
Lincoln Cathedral          159.7   1311 * Collapsed in 1549
Our Lady of Peace Basilica 158.0   1989
Cologne Cathedral          157.4   1880 *
Beauvais Cathedral         153.0   1569   Collapsed in 1573
Saint Mary's Church        151.0   1478 * Collapsed in 1647
Rouen Cathedral            151.0   1876 *
Old St. Paul's Cathedral   150.0   1240 * Collapsed in 1561
Saint Nikolai, Hamburg     147.3   1874 *
Strasbourg Cathedral       142.0   1439 *
Basilica Lady of Lichen    141.5   2000
Saint Peter's Basilica     136.6   1626
Saint Stephen's Cathedral  136.4   1433
New Cathedral, Linz        134.8   1924
Notre Dame et St. Lambert  134.5   1433  Destroyed in 1794
Saint Peter's Church       132.2   1878
Saint Michaelis Church     132.1   1786
Malmesbury Abbey           131.3   1180  Collapsed around the year 1500
*: Held the status of world's tallest building.
Arches


Catenary

Sir Robert Hooke
Blue: catenary      Red: parabola
Sir Christopher Wren

The catenary arch was discovered by Sir Robert Hooke.

Hooke: As hangs a flexible cable so, inverted, stand the touching pieces of an arch.


Parabola

A suspension bridge under zero load hangs as a catenary and under infinite load it hangs as a parabola.


Cathedral naves

Lincoln Cathedral
Saint Paul's Cathedral
Salisbury Cathedral

Bristol Cathedral
Salisbury Cathedral
Laon Cathedral


Flying buttresses

A flying buttress delivers the compression force from the arch to the ground.


Organs

Sydney Town Hall
LDS Conference Center


Organ frequency
Pipe length    =  L              =  8.5 meters
Wavelength     =  W  =  2L       = 17.0 meters
Sound speed    =  V              =  340 meters/second
Pipe frequency =  F  =  ½ V / L  =   20 Hertz    (Lower limit of human sensitivity)

Stained glass

NaO2       Colorless
FeO        Green                  Beer bottles
S          Amber
S + B2O3   Blue
S + Ca     Yellow
MnO2       Purple
Co         Blue
CuO        Turquoise
Ni         Blue or violet or black
Cr         Dark green or black
Au         Red
Cu         Dark red
Se         Pink
AgNO3      Orange
Cd         Yellow
U          Yellow
SnO        White


Glass composition
                Egypt     Rome   Europe    Modern
                1500 BC   0 AD   1300 AD

Silica    SiO2    65      68       53       73
Soda      NaO2    20      16        3       16
Potash    K2O      2        .5     17         .5
Lime      CaO      4       8       12        5
Magnesia  MgO      4        .5      7        3
Numbers are percentages.
Tallest structure in the world
                     Height   Year
                      (m)
Gobekli Tepe, Turkey   15   -11500
Pyramid of Djoser      62    -2650
Meidum Pyramid         93.5  -2610
Bent Pyramid          101.1  -2605
Red Pyramid           105    -2600
Great Pyramid of Giza 146    -2560
Lincoln Cathedral     160     1311-1549   Collapsed
St. Mary's Church     151     1549-1647   Collapsed
Strasbourg Cathedral  142     1647-1874
St. Nikolai           147     1874-1876
Rouen Cathedral       151     1876-1880
Cologne Cathedral     157     1880-1884
Washington Monument   169     1884-1889
Eiffel Tower          300     1889-1930
Chrysler Building     319     1930-1931
Empire State Building 381     1931-1967
During the years 1311-1884 the tallest structure in the world was always a church.
Marble

Marble is metamorphic limestone (CaCO3).

Marble Canyon, Colorado River

Masonic Temple, Washington DC
George Washington
Marble Arch, London

Tomb of the Unknown Soldier
Vietnam War Memorial, Illinois


Cement
CaO             Calcium oxide          Lime
Al2O3           Aluminum oxide
SiO2            Silicon dioxide        Rock, quartz
CaCO3           Calcium carbonate      Limestone
CO2             Carbon dioxide
Al2O3⋅SiO2       Aluminum silicate      Volcanic ash, andalusite, kyanite, sillimanite
CaSO4⋅2(H2O)                            Gypsum
Fe2O3           Iron(III) oxide        Rust
Non-hydraulic cement hardens upon contact with CO2 in the air and cannot be set in the presence of water.

Hydraulic cement hardens upon contact with water.

Aluminum silicates are compounds derived from aluminum oxide (Al2O3) and silicon dioxide (SiO2) and can be found in volcanic ash.

The Ancient Romans made hydraulic cement from volcanic ash and lime.

Non-hydraulic cement:

CaCO3          →  CaO + CO2           Heat for 10 hours at above 825 Celcius
CaO + H2O      →  Ca(OH)2             Add water and then let dry
Ca(OH)2 + CO2  →  CaCO3 + H2O          Hardens in the presence of CO2
Hydraulic cement is made from a mixture of
2(CaO)⋅SiO2         Belite
3(CaO)⋅SiO2         Alite
3(CaO)⋅Al2O3        Tricalcium aluminate
4(CaO)⋅Al2O3⋅Fe2O3   Brownmillerite
Portland cement is a hydraulic cement.

History of cement:

Ancient Babylon            Bitumen (asphalt). Viscous petroleum
Ancient Egypt              Sand and burnt gypsum
Ancient Greece and Rome    Hydraulic cement
1780                       James Parker advances the art of hydraulic cement
1840                       Aspdin develops "Portland cement"
Portland cement:
CaO      .63
SiO2     .219
Al2O3    .069
Fe2O3    .03
MgO      .025
SO3      .017
Portland cement is the most widely used modern cement. It is made by heating limestone (calcium carbonate) with other materials (such as clay) to 1450 Celsius. The resulting hard substance is ground into powder and mixed with gypsum and water, after which it hardens.

Concrete is cement mixed with gravel and sand.


Industrial revolution
-3000  Coal is used to smelt copper in Britain
 1700  5/6 of the world's coal is mined in Britain
 1800  Volta invents the battery
 1830  Faraday develops the generator for converting mechanical to electric energy
 1856  Bessemer process developed to transform pig iron to steel
 1882  First commercial electricity plants, using hydro or coal power
 1884  Sir Parsons develops the steam turbine, which today is used for 80% of electrical power

World efficiency for converting heat to electricity          = .36
World efficiency for delivering electricity to consumers     = .90   (Transmission efficiency)
World efficiency for converting heat to consumer electricity = .33

Obelisks

San Jacinto Monument
Washington Monument
Lincoln Tomb

Bunker Hill Monument
Perry's Memorial
Wellington Monument

                        Height  Base   Year

San Jacinto Monument    172.92  15     1939   Texas. Topped by a 220 ton star
Washington Monument     169.05  16.80  1884   Washington DC
Perry's Memorial        107            1915   Perry's Vicry and Int. Peace Memorial
Jefferson Davis Mon.    107.0          1924   Kentucky
Capas National Shrine    70            2003   Philippines
High Point Monument      67            1930   New Jersey
Bunker Hill Monument     67            1843   Massachusetts
Wellington Monument      62            1861   Dublin
Wellington Monument      53.34  24     1854   Somerset
Nelson's Column          51.6          1843   London
Lateran Obelisk          45.7         -1500   Rome
Vendome Column           42            1810   Paris
Flaminio Obelisk         36.5         -1300   Rome
Lincoln Tomb             36            1874   Illinois
Trajan's Column          35.1    3.7    113   Rome
Obelisk of Montecitorio  34.0          -592   Rome
Solare Obelisk           33.97         -592   Rome
Veteran's Memorial       33.5          1876   Pennsylvania
Vatican Obelisk          25.5           -29   Rome
Pompey's Pillar          20.46   2.71   297   Egypt
Nelson's column
Lateran Obelisk
Trajan's Column
Montecitorio Obelisk

Trinity nuclear test site
Pompei's Column
Raising the obelisk


Wind

For a solid granite obelisk with a square cross section, we estimate the wind force required to topple the obelisk.

Height            =  H
Base side length  =  L
Pillar density    =  D     =  2650  kg/meter3  (Granite)
Air density       =  d     =  1.22  kg/meter3
Pillar mass       =  M     =  D H L2
Wind speed        =  V     =    80  meters/second   (Extreme hurricane)
Gravity consant   =  g     =   9.8  meters/second2
Gravity force     =  Fgrav  =  M g
Wind force        =  Fwind  =  ½ d L H V2
Gravity torque    =  Τgrav  =  ½ L Fgrav
Wind torque       =  Τgrav  =  ½ H Fwind

Gravity torque  =  Wind torque
  ½ D H L3 g    =  ¼ L H2 d V2

H  =  2 g L2 D / (d V2)
L2 =  ½ H V2 d / (D g)
If H = 169 meters (Washington Monument), then L = 25.4 meters.
Density
               grams/cm3    Tensile    Compression
                            strength    strength
                             (MPa)       (MPa)
Air               .00122
Water            1.0
Brick, light                    .28         7
Brick, hard                    2.8         80
Sediment         2.0  ± .3
Sandstone        2.3  ± .3     2.1         60
Slate                          3.5         95
Shale            2.35 ± .35
Limestone        2.65 ± .15    2.1         60
Granite          2.65 ± .15    4.8        130
Metamorphic      2.8  ± .2
Basalt           2.9  ± .2
Portland cement  3.15          3.5         21
Portland concrete              2.8         14

Beryllium        1.85          448
Magnesium        1.74          232
Silicon          2.33
Aluminum         2.70           50
Titanium         4.51          370
Zinc             7.14
Chromium         7.15          282
Tin              7.26          200
Iron             7.86          350
Cobalt           8.90          760
Nickel           8.91          195
Copper           8.96          210
Molybdenum      10.28          324
Silver          10.49          170
Lead            11.34           12
Tungsten        19.25         1510
Gold            19.3           127
Platinum        21.45          165
Iridium         22.56         2000
Osmium          22.59         1000

Eiffel Tower


The Pyramids


Ruby

Ruby in a green laser
Synthetic rubies

Emerald

Sapphire

Synthetic sapphire

Diamond

Raw diamond
Raw diamond
Synthetic diamond
Synthetic diamonds

Topaz

Quartz


Crystals
Crystal, polycrystal, and amorphous

Diamond
Diamond
Diamond
Diamond
Diamond

Diamond and graphite
Carbon phase diagram
Corundum (Al2O3)
Corundum
Corundum unit cell

Corundum
Tungsten Carbide
Metal lattice

Alpha quartz (SiO2)
Beta quartz
Glass (SiO2)
Ice
Salt (NaCl)

Corundum is a crystalline form of aluminium oxide (Al2O3). It is transparent in its pure orm and can have different colors when metal impurities are present. Specimens are called rubies if red, padparadscha if pink-orange, and all other colors are called sapphire, e.g., "green sapphire" for a green specimen.

Metal impurity   Color

Chromium         Red
Iron             Blue
Titanium         Yellow
Copper           Orange
Magnesium        Green

Price
1 Carat                     =  .2 grams
Price of a 1 Carat diamond  =  C  ≈  1000 $     (This varies according to quality)
Mass of diamond in Carats   =  M
Price of diamond in dollars =  C M2
Pure gold                   =  24 Karats
3/4 pure gold               =  18 Karats


1837  Gaudin produces the first synthetic ruby.
1905  Bridgman invents the diamond anvil, which reached a pressure of 10 GPa.
      He was awarded the Nobel prize for this in 1946.
1910  Synthetic ruby begins to be mass produced.
1928  Sir Parsons produces the first synthetic diamonds.
1954  Hall produces the first commercially successful synthetic diamonds.
1970  First gem-quality synthetic diamonds produced.
2015  Synthetic diamonds reach 10 carats in size.

Polymers

Zylon
Vectran
Aramid (Kevlar)
Polyethylene

Aramid
Nylon
Hydrogen bonds in Nylon

Spider silk
Lignin

Lignin comprises 30 percent of wood and it is the principal structural element.


Rope

               Year   Young  Tensile  Strain  Density   Common
                      (GPa)  strength         (g/cm3)   name
                              (GPa)
Gut           Ancient           .2
Cotton        Ancient                   .1       1.5
Hemp          Ancient   10      .3      .023
Duct tape                       .015
Gorilla tape                    .030
Polyamide      1939      5     1.0      .2       1.14    Nylon, Perlon
Polyethylene   1939    117                       1.4     Dacron
Polyester      1941     15     1.0      .067     1.38
Polypropylene  1957                               .91
Carbon fiber   1968            3.0               1.75
Aramid         1973    135     3.0      .022     1.43    Kevlar
HMPE           1975    100     2.4      .024      .97    Dyneema, Spectra
PBO            1985    280     5.8      .021     1.52    Zylon
LCAP           1990     65     3.8      .058     1.4     Vectran
Vectran HT              75     3.2      .043     1.41    Vectran
Vectran NT              52     1.1      .021     1.41    Vectran
Vectran UM             103     3.0      .029     1.41    Vectran
Nanorope             ~1000     3.6      .0036    1.3
Nanotube              1000    63        .063     1.34
Graphene              1050   160        .152     1.0


Strain  =  Strength / Young
Carbon fiber is not useful as a rope.

A string ideally has both large strength and large strain, which favors Vectran.

Suppose Batman has a rope made out of Zylon, the strongest known polymer.

Batman mass            =  M         =    100 kg               (includes suit and gear)
Gravity constant       =  g         =     10 meters/second2
Batman weight          =  F         =   1000 Newtons
Zylon density          =  D         =   1520 kg/meter3
Zylon tensile strength =  Pz        = 5.8⋅109 Newtons/meter2
Rope load              =  P         = 1.0⋅109 Newtons/meter2   (safety margin)
Rope length            =  L              100 meters
Rope cross section     =  A  = F/P  =1.0⋅10-6 meters2
Rope radius            =  R  =(A/π)½=     .56 mm
Rope mass              =  Mr = DAL  =     .15 kg

Wood

         Density   Tensile   Young  Crush    Compress    Compress
                   strength                  with grain  against
         (g/cm^3)  (Gpa)     (Gpa)  (Gpa)                grain

Balsa         .12    .020      3.7     .012
Corkwood      .21
Cedar         .32    .046      5.7                               Northern white
Poplar        .33    .048      7.2                               Balsam
Cedar         .34    .054      8.2                               Western red
Pine          .37    .063      9.0                               Eastern white
Buckeye       .38    .054      8.3                               Yellow
Butternut     .40    .057      8.3
Basswood      .40    .061     10.3
Alder, red    .41                              5820         9800
Spruce, red   .41    .072     10.7
Aspen         .41    .064     10.0
Fir, silver   .42    .067     10.8
Hemlock       .43    .061      8.5                               Eastern
Redwood       .44    .076      9.6             1500  650   1553
Ash, black    .53    .090     11.3
Birch, gray   .55    .069      8.0
Walnut, black .56    .104     11.8
Ash, green    .61    .100     11.7
Ash, white    .64    .110     12.5
Oak, red      .66    .100     12.7
Elm, rock     .66    .106     10.9
Beech         .66    .102     11.8
Birch, yellow .67    .119                      1200  715   1668
Mahogany      .67    .124     10.8                                 West Africa
Locust        .71    .136     14.5                                 Black or Yellow
Persimmon     .78    .127     14.4
Oak, swamp    .79    .124     14.5                                 Swamp white
Gum, blue     .80    .118     16.8
Hickory       .81    .144     15.2                                 Shagbark
Eucalyptus    .83    .122     18.8
Bamboo        .85    .169     20.0     .093
Oak, live     .98    .130     13.8
Ironwood     1.1     .181     21.0
Lignum Vitae 1.26    .127     14.1
Fir, Douglas                                   1700   625   1668
Data #1     Data #2
Fission

A neutron triggers the fission of Uranium-235 and plutonium-239, releasing energy and more neutrons.


Chain reaction

Fizzle

Fission releases neutrons that trigger more fission.

Chain reaction simulation

Critical mass

Two pieces of uranium, each with less than a critical mass, are brought together in a cannon barrel.

If the uranium is brought together too slowly, the bomb fizzles.


Plutonium fission

Plutonium is more difficult to detonate than uranium. Plutonium detonation requires a spherical implosion.


Nuclear isotopes relevant to fission energy

Abundance of elements in the sun, indicated by dot size

Blue elements are unstable with a half life much less than the age of the solar system.

The only elements heavier than Bismuth that can be found on the Earth are Thorium and Uranium, and these are the only elements that can be tapped for fission energy.

Natural Thorium is 100% Thorium-232
Natural Uranium is .72% Uranium-235 and 99.3% Uranium-238.
Plutonium doesn't exist in nature.


           Protons  Neutrons  Halflife   Critical   Isotope
                              (10^6 yr)  mass (kg)  fraction

Thorium-232    90    142      14000          -       1.00     Absorbs neutron -> U-233
Uranium-233    92    141           .160     16        -       Fission chain reaction
Uranium-235    92    143        700         52        .0072   Fission chain reaction
Uranium-238    92    146       4500          -        .9927   Absorbs neutron -> Pu-239
Plutonium-238  94    144           .000088   -        -       Produces power from radioactive heat
Plutonium-239  94    145           .020     10        -       Fission chain reaction
The elements that can be used for fission energy are the ones with a critical mass. These are Uranium-233, Uranium-235, and Plutonium-239. Uranium-233 and Plutonium-239 can be created in a breeder reactor.
Thorium-232  +  Neutron  ->  Uranium-233
Uranium-238  +  Neutron  ->  Plutonium-239
The "Fission" simulation at phet.colorado.edu illustrates the concept of a chain reaction.

Natural uranium is composed of .7% Uranium-235 and the rest is Uranium-238. Uranium-235 can be separated from U-238 using centrifuges, calutrons, or gas diffusion chambers. Uranium-235 is easy to detonate. A cannon and gunpowder gets it done.

Plutonium-239 is difficult to detonate, requiring a perfect spherical implosion. This technology is beyond the reach of most rogue states.

Uranium-233 cannot be used for a bomb and is hence not a proliferation risk.

Plutonium-238 emits alpha particles, which can power a radioisotope thermoelectric generator (RTG). RTGs based on Plutonium-238 generate 540 Watts/kg and are used to power spacecraft.

Teaching simulation for nuclear isotopes

Generating fission fuel in a breeder reactor

Creating Plutonium-239 and Uranium-233:

Uranium-238 + Neutron  ->  Plutonium-239
Thorium-232 + Neutron  ->  Uranium-233

Detail:

Uranium-238 + Neutron  ->  Uranium-239
Uranium-239            ->  Neptunium-239 + Electron + Antineutrino    Halflife = 23 mins
Neptunium-239          ->  Plutonium-239 + Electron + Antineutrino    Halflife = 2.4 days

Thorium-232 + Neutron  ->  Thorium-233
Thorium-233            ->  Protactinium-233 + Electron + Antineutrino   Halflife = 22 mins
Protactinium-233       ->  Uranium-233      + Electron + Antineutrino   Halflife =

Nuclear fusion bombs

A nuclear fusion bomb contains deuterium and lithium-6 and the reaction is catalyzed by a neutron.

N + Li6  ->  He4 + T +  4.87 MeV
T + D    ->  He4 + N + 17.56 MeV

Total energy released  =  22.43 MeV
Nucleons               = 8
Energy / Nucleon       = 22.434 / 8  =  2.80

Energy
1 ton of TNT                  4*10^9  Joules
1 ton of gasoline             4*10^10 Joules
North Korea fission device    0.5 kilotons TNT
10 kg uranium fission bomb    10  kilotons TNT
10 kg hydrogen fusion bomb    10  megatons TNT
Tunguska asteroid strike      15  megatons TNT        50 meter asteroid
Chixulub dinosaur extinction  100 trillion tons TNT   10 km asteroid

History of nuclear physics
1885       Rontgen discovers X-rays
1899       Rutherford discovers alpha and beta rays
1903       Rutherford discovers gamma rays
1905       E=mc^2. Matter is equivalent to energy
1909       Nucleus discovered by the Rutherford scattering experiment
1932       Neutron discovered
1933       Nuclear fission chain reaction envisioned by Szilard
1934       Fermi bombards uranium with neutrons and creates Plutonium. First
           successful example of alchemy
1938       Fission discovered by Hahn and Meitner
1938       Bohr delivers news of fission to Princeton and Columbia
1939       Fermi constructs the first nuclear reactor in the basement of Columbia
1939       Szilard and Einstein write a letter to President Roosevelt advising
           him to consider nuclear fission
1942       Manhattan project started
1942-1945  German nuclear bomb project goes nowhere
1945       Two nuclear devices deployed by the United States

History of nuclear devices
           Fission Fusion

U.S.A.       1945  1954
Germany                  Attempted fission in 1944 & failed
Russia       1949  1953
Britain      1952  1957
France       1960  1968
China        1964  1967
India        1974        Uranium
Israel       1979   ?    Undeclared. Has both fission and fusion weapons
South Africa 1980        Dismantled in 1991
Iran         1981        Osirak reactor to create Plutonium. Reactor destroyed by Israel
Pakistan     1990        Centrifuge enrichment of Uranium. Tested in 1998
                         Built centrifuges from stolen designs
Iraq         1993        Magnetic enrichment of Uranium. Dismantled after Gulf War 1
Iraq         2003        Alleged by the United States. Proved to be untrue.
North Korea  2006        Obtained plutonium from a nuclear reactor. Detonation test fizzled
                         Also acquired centrifuges from Pakistan
                         Also attempting to purify Uranium with centrifuges
Syria        2007        Nuclear reactor destroyed by Israel
Iran         2009?       Attempting centrifuge enrichment of Uranium.
Libya         --         Attempted centrifuge enrichment of Uranium.  Dismantled before completion.
                         Cooperated in the investigation that identified
                         Pakistan as the proliferator of Centrifuge designs.
Libya        2010        Squabbling over nuclear material
Libya        2011        Civil war

Fusion power

A tokamak fusion reactor uses magnetic fields to confine a hot plasma so that fusion can occur in the plasma.

Deuterium + Tritium fusion

The fusion reaction that occurs at the lowest temperature and has the highest reaction rate is

Deuterium  +  Tritium  ->  Helium-4  +  Neutron  +  17.590 MeV
but the neutrons it produces are a nuisance to the reactor.

A potential fix is to have "liquid walls" absorb the neutrons (imagine a waterfall of neutron-absorbing liquid lithium cascading down the walls of the reactor).


Energy

The unit of energy used for atoms, nuclei, and particle is the "electron Volt", which is the energy gained by an electron upon descending a potential of 1 Volt.

Electron Volt (eV)  =  1  eV  =  1.602e-19 Joules
Kilo electron Volt  =  1 keV  =  103 eV
Mega electron Volt  =  1 MeV  =  106 eV
Giga electron Vlt   =  1 GeV  =  109 eV

Fusion

Fusion of hydrogen into helium in the sun

Proton + Proton  ->  Deuterium + Positron + Neutrino
Hydrogen fusion requires a temperature of at least 4 million Kelvin, which requires an object with at least 0.08 solar masses. This is the minimum mass to be a star. The reactions in the fusion of hydrogen to helium are:
P    + P    -->  D    +  Positron + Neutrino +   .42 MeV
P    + D    -->  He3  +  Photon              +  5.49 MeV
He3  + He3  -->  He4  +  P   +  P            + 12.86 MeV

Helium fusion

As the core of a star star runs out of hydrogen it contracts and heats, and helium fusion begins when the temperature reaches 10 million Kelvin.


Heavy element fusion

A heavy star continues to fuse elements until it reaches Iron-56. Beyond this, fusion absorbs energy rather than releasing it, triggering a runaway core collapse that fuses elements up to Uranium. If the star explodes as a supernova then these elements are ejected into interstellar space.


Instruments

Range of instruments

Green dots indicate the frequencies of open strings.

An orchestral bass and a bass guitar have the same string tunings.

The range of organs is variable and typically extends beyond the piano in both the high and low direction.


Stringed instruments

A violin, viola, cello, and double bass
String quartet
Orchestra


Violin and viola
Cello
Bass
Guitar
Electric guitar


Strings on a violin


Strings on a viola or cello


Violin fingering
Strings on a guitar


Wind and brass instruments

Flute
Oboe
Clarinet
Bassoon

Trumpet
French horn
Trombone
Tuba

In a reed instrument, a puff of air enters the pipe, which closes the reed because of the Bernoulli effect. A pressure pulse travels to the other and and back and when it returns it opens the reed, allowing another puff of air to enter the pipe and repeat the cycle.


Orchestra
              String   Baroque    Classical  Modern
              quartet  orchestra  orchestra  orchestra

First violin    1        6          12         16
Second violin   1        4          10         14
Viola           1        4           8         12
Cello           1        4           8         12
Bass                     2           6          8
Flute                    2           2          4
Oboe                     2           2          4
Clarinet                             2          4
Bassoon                  2           2          4
Trumpet                  2           2          4
French Horn              2           2          4
Trombone                                        4
Tuba                                            2
Harpsichord              1           1
Timpani                  1           1          1

Tuning

Violins, violas, and cellos are tuned in fifths. String basses, guitars, and bass guitars are tuned in fourths. Pianos are tuned with equal tuning.

             Hertz
Violin E      660      =  440*1.5
Violin A      440
Violin D      293      =  440/1.5
Violin G      196      =  440/1.52

Viola  A      440      Same as a violin A
Viola  D      293
Viola  G      196
Viola  C      130

Cello  A      220      One octave below a viola A
Cello  D      147
Cello  G       98
Cello  C       65

String bass G  98      =  55 * 1.52
String bass D  73      =  55 * 1.5
String bass A  55      3 octaves below a violin A
String bass E  41      =  55 / 1.5

Guitar E      326
Guitar B      244
Guitar G      196
Guitar D      147
Guitar A      110      2 octaves below a violin A
Guitar E       82
When an orchestra tunes, the concertmaster plays an A and then everyone tunes their A strings. Then the other strings are tuned in fifths starting from the A.

A bass guitar is tuned like a string bass.

The viola is the largest instrument for which one can comfortably play an octave, for example by playing a D on the C-string with the first finger and a D on the G-string with the fourth finger. Cellists have to shift to reach the D on the G-string.

According to legend Bach used a supersized viola, the "Viola Pomposa"


Low note

Singers typically have a range of 2 octaves. The low note for each instrument is:

    Strings   Winds      Brass      Voice

D             Piccolo
C             Flute                 Soprano
Bb            Oboe
A
G   Violin
F#                       Trumpet    Alto
E   Guitar    Clarinet
D
C   Viola                           Tenor
Bb
A
G                                   Baritone
F#                       Horn
E                        Trombone   Bass
D
C   Cello
Bb            Bassoon
A
G
F
E   Bass
D                      Tuba

Treble clef:  Violin, flute, oboe, clarinet, saxophone, trumpet, French horn, guitar,
              soprano voice, alto voice, tenor voice.
Alto clef:    Viola
Base clef:    Cello, bass, bass guitar, bassoon, trombone, tuba, timpani,
              baritone voice, bass voice
String basses and bass guitars have the same string tuning.

For guitars, tenors, basses, and bass guitars, the tuning is an octave lower than written.


Piano


Viola d'amore

The viola d'amore has 7 playing strings and 6 resonance strings.


Instruments of Indian classical music

Sitar

A sitar has 6 or 7 playing strings and 11 or more sympathetic strings.

There is no standard tuning for sitar strings. An example tuning is to set the playing strings to {C, C, G, C, G, C, F} and the sympathetic strings to {C, B, A, G, F, E, E, D, C, B, C}

The fret positions can be tuned.

The bridge is curved so that the contact point between the string and the bridge is not sharp, which has the effect of transferring energy between the string modes.

Sarod and Sitar
Sarod
Surbahar

Tanpura
Tanpura
Bansuri

Shehnai
Sarangi
Sarangi
Santoor

Pakhawaj
Tabla

The surbahar is typically tuned 2 to 5 whole steps below the sitar.

The tanpura does not play melody but rather supports and sustains the melody of another instrument or singer by providing a continuous harmonic drone.


Electric sitar


Guitar frets

Guitars frets are set by equal tuning.


L   =  Length of an open A-string
    =  .65 meters
T   =  Wave period
F   =  Frequency of the A-string
    =  220 Hertz
V   =  Speed of a wave on the A-string
    =  2 L F
    =  2 * .65 * 220
    =  286 meters/second
I   =  Index of a fret
    =  1 for B flat
    =  2 for B
    =  3 for C, etc.
f   =  Frequency of note I
    =  F * 2^(I/12)
X   =  Distance from the bridge to fret I
    =  V / (2 f)
    =  V / (2 F) * 2^(-I/12)
    =  L * 2^(-I/12)

 I  Note   X     L-X

 0   A    .650   .0
 1   Bb   .614   .036
 2   B    .579   .071
 3   C    .547   .103
 4   C#   .516   .134
 5   D    .487   .163
 6   Eb   .460   .190
 7   E    .434   .216
 8   F    .409   .241
 9   F#   .386   .264
10   G    .365   .285
11   Ab   .344   .306
12   A    .325   .325

Flexibility of just tuning

The frequency of a note depends on context. Suppose a set of viola strings is tuned in fifths so that the frequencies are

G  =  1
D  = 3/2
A  = 9/4
The G-string has been normalized to have a frequency of 1. There are several possibilities for assigning the pitch of the "E" on the D-string.

If the note "E" is chosen to resonate with the G-string its frequency is

E  =  5/3  =  1.6666
If the note "E" is chosen to resonate with the "A-string" then it is placed a perfect fourth below the A.
E  =  (9/4) / (4/3)  =  27/16  =  1.688
If the note "E" is played with equal tuning with the G-string as the tonic,
E  =  2^(9/12)  =  1.682
All three values for the E are different. Musicians have to develop a sensitivity for this.

Indian tuning

Red:    Equal tuning
Green:  Just tuning
Orange: Pythagorean tuning
Indian music has two separate tones for each half step, one from just tuning and the other from Pythagorean tuning. For the tonic and the fifth these tones are the same for both tunings. There are 22 tones in total.


Lagavulin
Ardbeg
Caol Ila

Bruichladdich
Bowmore
Bunnahabhain

Kilchoman
Port Askaig
Lagavulin


Alcohol

A typical bottle of beer has a volume of 12 ounces, is 5% alcohol, and contains
.6 ounces of alcohol. We use this amount as a reference unit and define
.6 ounces of alcohol to be one "Bond".

Volume of the drink           =  V
Fraction of alcohol           =  F
Volume of alcohol             =  Valc   =  F V
Volume of one beer            =  Vbeer  =  12 ounces
Fraction of alcohol in beer   =  Fbeer  =  .05
Volume of alcohol in one beer =  VBond  =  .6 ounces
One ounce                     =  29.6 mL
One "Bond" of alcohol         =    .6 ounces
One wine or Scotch bottle     =  25.4 ounces  =  750 ml


              Alcohol   Volume  Alcohol  Alcohol   $    $/Bond
              fraction   (oz)    (oz)    (Bonds)

Beer (12 oz)      .05     12       .6       1       .67   .67   Budweiser
Wine glass        .13      4.6     .6       1      8     8.0    Napa Valley
Scotch shot       .40      1.5     .6       1      8     8.0    Laphroaig
Beer pitcher      .05     64      3.2       5.3   16     3.0    Budweiser
Beer keg          .05   1984     99.2     165.3  100      .60   Budweiser
Wine bottle       .13     25.4    3.3       5.5    3      .55   Charles Shaw
Vodka bottle      .40     25.4   10.1      16.9   15      .89   Smirnoff
Scotch bottle     .40     25.4   10.1      16.9   50     3.0    Laphroaig
Distilled ethanol .95     25.4   24.1      40.2   15      .37   Everclear

Wine regions


Spices

Turmeric: curcumin
Cumin: cuminaldehyde
Chili: capsaicin
Mustard: allyl isotyiolcyanate

Bay: myrcene
Garlic and onion: allicin
Clove: eugenol

Raspberry ketone
Tangerine: tangeritin
Lemon: citral
Lemon peel: limonene

Chocolate: theobromine
Smoke: guaiacol
Cardamom: terpineol
Wintergreen: methyl salicylate

Hydrogen   White
Carbon     Black
Nitrogen   Blue
Oxygen     Red
Sulfur     Yellow
        Scoville scale (relative capsaicin content)

Ghost pepper    1000000
Trinidad        1000000      Trinidad moruga scorpion
Naga Morich     1000000
Habanero         250000
Cayenne           40000
Tabasco           40000
Jalapeno           6000
Pimento             400


Molecule        Relative hotness

Rresiniferatoxin   16000
Tinyatoxin          5300
Capsaicin             16         Chili pepper
Nonivamide             9.2       Chili pepper
Shogaol                 .16      Ginger
Piperine                .1       Black pepper
Gingerol                .06      Ginger
Capsiate                .016     Chili pepper
Caraway: carvone
Black tea: theaflavin
Cinnamon: cinnamaldehyde
Citrus: hesperidin
Fruit: quercetin

Mint: menthol
Juniper: pinene
Saffron: picrocrocin
Saffron: safranal
Wine: tannic acid

Black pepper: piperine
Oregano: carvacrol
Sesame: sesamol
Curry leaf: girinimbine
Aloe emodin
Whiskey lactone


Signalling molecules

Alcohol
Caffeine
Tetrahydrocannabinol
Nicotine

Adrenaline
Noadrenaline
Dopamine
Seratonin

Aspirin
Ibuprofen
Hydrocodone
Morphone

Vitamin A (beta carotene)
Vitamin A (retinol)
Vitamin C (ascorbic acid
Vitamin D (cholecalciferol)


Time zones


History

220 BCE

210 CE

450

475

480

526

998

1092

1345

1370

1470

1547

Spanish colonies

1648

1812

British Empire, 1921

British colonies

1929-1938

Japanese Empire, 1942

1943-1945

American Empire

2008


Languages


Monarchies and republics

1815
1914
1930

1950
2015


Gross Domestic Product


Miscellaneous

Speed limit

Airports


Balls

Ball sizes are in scale with each other and court sizes are in scale with each other.
Ball sizes are magnified by 10 with respect to court sizes.
The distance from the back of the court to the ball is the characteristic distance the ball travels before losing half its speed to air drag.

             Ball    Ball   Court   Court    Ball
           diameter  Mass   length  width   density
             (mm)    (g)     (m)     (m)    (g/cm3)

Ping pong      40      2.7    2.74    1.525   .081
Squash         40     24      9.75    6.4     .716
Golf           43     46                     1.10
Badminton      54      5.1   13.4     5.18    .062
Racquetball    57     40     12.22    6.10    .413
Billiards      59    163      2.84    1.42   1.52
Tennis         67     58     23.77    8.23    .368
Baseball       74.5  146                      .675  Pitcher-batter dist. = 19.4 m
Hockey puck    76    163     61      26      1.44   25 mm thick
Whiffle        76     45                      .196
Football      178    420     91.44   48.76    .142
Rugby         191    435    100      70       .119
Bowling       217   7260     18.29    1.05   1.36
Soccer        220    432    105      68       .078
Basketball    239    624     28      15       .087
Cannonball    220  14000                     7.9    For an iron cannonball

Sports

Rugby
Rugby scrum
Football

    Forwards                     Backs

 1  Loose-head prop           8  Number 8
 2  Hooker                    9  Scrum half
 3  Tight-head prop          10  Fly half
 4  Lock                     11  Left wing
 5  Lock                     12  Inside center
 6  Blind-side flanker       13  Outside center
 7  Open-side flanker        14  Right wing
                             15  Fullback
American football

Football

The following data is a five-year average of results from the NFL Combine, from 2008-2013.

                Weight   Reps  40 yard   20 yard   Broad   Vertical
                                dash     shuttle   jump      jump
               (pounds)       (seconds) (seconds) (inches) (inches)

Wide receiver      202.3  15.4   4.55    4.25     120      35
Cornerback         193.2  15.5   4.55    4.17     121      35
Running back       213.3  20.5   4.59    4.28     117      34.5
Safety             208.9  18.1   4.62    4.24     114      34.5
Outside linebacker 238.1  22.7   4.74    4.34     117      33.5
Tight end          251.6  21.5   4.77    4.37     116      33.5
Fullback           242.6  24.1   4.80    4.39     120      33.5
Inside linebacker  241.5  22.7   4.80    4.31     115      33
Quarterback        223.1  17.8   4.87    4.34     110      31
Defensive end      266.3  25.6   4.88    4.46     113      32.5
Defensive tackle   304.8  28.3   5.13    4.66     105      29
Offensive center   303.1  27.3   5.30    4.66     100      27
Offensive tackle   314.7  25.3   5.32    4.80     101      27
Offensive guard    314    26.2   5.36    4.85      99      27

Reps:      Bench press repetitions at 225 pounds.
40:        Time for the 40 yard dash.  Reaction time is not counted.
20:        Time for the 20 yard shuttle.  Reaction time is not counted.
           5 yards to the right, 10 yards to the left, and 5 yards to the right.

Sports records
                 Men      Women
   50 meter      5.47      5.96  seconds
   60 meter      6.31      6.92  seconds
  100 meter      9.58     10.49  seconds
  200 meter     19.19     21.34  seconds
  400 meter     43.18     47.60  seconds
  800 meter    100.91    113.28  seconds
 1000 meter    131.96    148.98  seconds
 1500 meter    206.00    230.07  seconds
 5000 meter  12:37.35  14:11.15  seconds
10000 meter  26:17.53  29:31.78  seconds
Marathon      2:02:57   2:17:42  seconds
40 km         1:56:29            seconds
90 km         5:20:49            seconds
4x100 meter     36.84     40.82  seconds
4x200 meter     78.63     87.46  seconds
4x400 meter    174.29    195.17  seconds
High jump        2.45      2.09  meters
Pole vault       6.16      5.06  meters
Long jump        8.95      7.52  meters
Shot put        23.12     22.63  meters    Men=7.26 kg.  Women=4.0 kg
Discus          74.08            meters
Hammer          86.74            meters
Javelin         98.48     72.28  meters
Broad jump       3.73            meters
Baseball pitch  46.98     30.85  meters/second

Sports leagues and playoff systems

Sports teams tend to organize themselves into "power leagues" to exclude competition. We could all the members of the power leagues "oligarchs" and the nonmembers "plebes". Leagues also tend to chose poor tournament structures that limit a team's individual choices about who they play and they prevent interesting matchups from happening. Two especially weak formats are single elimination and pools.
Power leagues
                   Number of
                   power leagues

College basketball       8   Big 10, ACC, Big East, SEC, Big 12, Pac 10, MAC, Mountain West
College football         5   Big 10, SEC, Pac 10, Big 12, ACC
College rugby            2   Varsity Club Championship,  Division 1-A rugby
World rugby              2   South Hemisphere 4-Nations, North Hemisphere 6-Nations
College hockey           2   Big 10, National Collegiate Hockey Conference
College wrestling        2   Big 10, Big 12
English club soccer      1   Premiere League
American pro basketball  1
American pro football    1
American pro baseball    1
Professional hockey      1
Formula-1                1
NASCAR                   1
Pro Golf Association     1

Tournament format
                         Regular  Teams  Post-   Postseason format
                         season          season
                         games           teams
National Football League   16      32     12     single elimination
Professional Basketball    82      30     16     single elimination
National Hockey League     82      30     16     single elimination
Major League Baseball     162      31     10     single elimination
College football           12    >100      2     single elimination
Big Ten football            8      12      2     single elimination
College Basketball         27    >300     64     single elimination
Men's college hockey       35      59     16     single elimination
English Premier League     38      20      -     none
Soccer World Cup            -       -     32     pools, single elimination
England Soccer FA Cup       -       -    763     single elimination
The Rugby Championship      6       4      -     none
6 Nations Rugby             5       6      -     none
Rugby Pacific Nations Cup   3       4      -     none
Rugby World Cup             -       -     20     pools, single elimination
Professional Sumo          90      42      0     matches determined each round by judges
Mongolian Wrestling                    >1000     single elimination
NASCAR                     36      59     16     chase
Formula-1                  20      24      -     none
The Rugby Championship is a double round robin between New Zealand, Australia, S. Africa, and Argentina.

6 Nations Rugby is a round robin between England, France, Wales, Scotland, Ireland, and Italy.

The Rugby Pacific Nations Cup is a round robin between Fiji, Japan, Samoa, and Tonga.

The three rugby leagues are rare examples where there is no postseason.


Pools

The world cup is the most prominent example of pools.


World Cup 2014 qualification
                     Teams   Teams   Teams   Games   Quality  Max
                     in 1st  in 2nd  that    played  games    Losses
                     stage   stage   qualify

Europe      UEFA        -     48     13       10       1       3
S. America  Conmebol    -     10      4.5     18       9       5
Africa      CAF        40     10      5        8       0       2
Asia        AFC        20     10      4.5     14       3       5
N. America  Concacaf   12      6      3.5     16       5       7
Pacific     OFC         8      4       .5      9       0       2
Brazil, a member of Conmebol, automatically qualifies, giving Conmebol effectively 5.5 spots.

Two finalists are determined by playoffs. One is determined by a playoff between 5th place Conmebol team and the 5th place AFC team, and another between the 4th place Concacaf team and the 1st place OFC team.

"Games played" is the number of qualifying games played by a team that qualifies as a finalist.

If a team qualifies for the World Cup. The number of "quality games" is the average number of games played against teams that also qualified.

"Max Losses" is the average number of losses incurred by a team that barely qualifies.

Among all 32 finalists:

Average number of games played against finalists during qualifying  ~  2.6
Average number of games played during qualifying                    ~ 12
Average number of games played during finals                        ~  4.0
Average number of losses by a team barely qualifying for the finals ~  4
In Conmebol qualifying, teams get lots of experience playing worthy competition, and a team can afford to lose several times and still qualify.

For CAF, qualification is largely luck and teams don't get a chance to play other finalists.


World Cup history
           Win  Final  Semifinal

Brazil      5     7     11
Germany     4     8     13
Italy       4     6      8
Argentina   2     5      5
Uruguay     2     2      5
France      1     2      5
England     1     1      2
Spain       1     1      2
Netherlands 0     3      5
Czech.      0     2      2
Hungary     0     2      2
Sweden      0     1      3
Teams that usually qualify:
Europe       Germany, Italy,    Spain,  Netherlands,  France,  England,  Portugal
Conmebol     Brazil,  Argentina
Concacaf     U.S.,    Mexico
Asia         Japan,   South Korea
Germany and Brazil are the most sucessful teams and they have only played one World Cup game.
Conmebol 2014 qualifying
           Pts  W  D  L 
Argentina   32  9  5  2
Colombia    30  9  3  4
Chile       28  9  1  6
Ecuador     25  7  4  5
Uruguay     25  7  4  5    Defeated Jordan in a playoff
Venezuela   20  5  5  6
Peru        15  4  3  9
Bolivia     12  2  6  8
Paraguay    12  3  3 10
Brazil plus the top 4 teams qualify. The 5th-place team plays a playoff game against a team from another federation to qualify.
CONCACAF qualifying
           Points  Win  Draw  Loss
U.S.         22     7    1     2
Costa Rica   18     5    3     2
Honduras     15     4    3     3
Mexico       11     2    5     3
Panama        8     1    5     4
Jamaica       5     0    5     5

UEFA 2014 qualifying
Group    Pot 1        Pot 2         Pot 3        Pot 4        Pot 5       Pot 6
  A    + Croatia      Serbia      * Belgium      Scotland     Macedon     Wales
  B    * Italy        Denmark       Czech.       Bulgaria     Armenia     Malta
  C    * Germany    + Sweden        Ireland      Austria      Faroe.      Kazakhstan
  D    * Nether.      Turkey        Hungary    + Romania      Estonia     Andorra
  E      Norway       Slovenia    * Switzer.     Albania      Cyp       + Iceland
  F    + Portugal   * Russia        Israel       N.Ireland    Azerbai.    Luxembourg
  G    + Greece       Slovakia    * B+H          Lithuan.     Latvia      Leich
  H    * Eng land     Monteneg.   + Ukraine      Poland       Moldova     San Marino
  I    * Spain      + France        Belarus      Georgia      Finland

Pot 1 consists of the top 9 seeds, pot 2 consists of the next 9 seeds, etc.
"*" = pool winner
"+" = pool 2nd place
Each pool is a double round robin.

The winner of each pool qualifies for the World Cup. The best 8 2nd-place teams play head-to-head matchups, with the winner of each matchup qualifying.

          Agg
Portugal  4-2   Sweeden
France    3-2   Ukraine
Greece    4-2   Romania
Croatia   2-0   Iceland

Pools

In the World Cup, teams that advance from the pool round to the round of 16

Pools tend to feature many games that have no outcome on qualifying. The Conmebol pool, consisting of 10 teams of which 5.5 advance, features the largest fraction of games that impact qualifying.

A UEFA pool consists of 6 teams, of which 1.5 advance. This pool features the greatest potential for a premier team to not qualify. For example, one of the 2014 pools consists of both Spain and France.


NCAA Basketball
Division  Conferences  Teams
   1          32        351
   2          23
   3          45
Power conferences: Big Ten, ACC, Big East, Pac 10, Big 12, SEC, MAC, Mountain West

College Wrestling
            Championships

Oklahoma State  34
Iowa            23
Iowa State       8
Oklahoma         7
Penn State       5
Minnesota        3
Ohio State       1
Arizona State    1
Michigan State   1
Northern Iowa    1
Cornell College  1
Indiana          1


Big Twelve      49
Big Ten         33
Other            3
Few schools outside of the Midwest have won championships.

NCAA hockey

Big Ten
ECAC Hockey

USA Hockey
Hockey East

Men's hockey:

            Championships   Frozen Four

Michigan             9       24
North Dakota         7       21
Denver               7       14
Wisconsin            6       12
Boston College       5       24
Boston University    5       22
Minnesota            5       21
Lake Superior State  3        4
Michigan State       3       11
Michigan Tech        3       10
Cornell              2        8
Maine                2       11
Colorado College     2       10
RPI                  2        5
Minnesota-Duluth     1        4
Harvard              1       12
Providence           1        4
Bowling Green        1
Northern Michigan    1
Union                1
Yale                 1
St Lawrence                   9
Clarkson                      7
New Hampshire                 7
Dartmouth                     4
Women's hockey:
               Championships   Final

Minnesota-Duluth     5          6
Minnesota            5          7
Wisconsin            4          6
Clarkson             1          1
Harvard                         4

NCAA rugby

Varsity Club Championship:
Air Force
Arizona State
Arkansas State
BYU
California
Central Washington
Clemson
Dartmouth
Navy
Notre Dame
Oklahoma
Texas
UCLA
UTAH


Division 1-A Rugby:
East:        Army,  Delaware,  Kutztown,  Penn State,  Wheeling Jesuit
             St. Bonaventure,  Iona,  University at Buffalo
Mid-South:   Lindenwood,  Life University,  Davenport University
California:  Cal Poly,  Saint Mary's,  UC Davis,  San Diego State,
             UC Santa Barbara,  Santa Clara,  Sacramento State,  Stanford
West:        Colorado,  Colorado State,  Wyoming  Northern Colerado,  New Mexico


            Championships

California       25
BYU               4
Air Force         3
Harvard           1
San Diego State   1
The Varsity Club conference was formed in 2013.

http://en.wikipedia.org/wiki/Varsity_Cup_Championship
http://en.wikipedia.org/wiki/Division_1-A_Rugby


American football playoffs

In American football, the regular season matters. Each regular season win tends to increase the playoff seed by 1, and the playoff seed matters because it determines home field advantage and wildcard byes.

The following table gives the average number of regular season wins for each seed, using data from 2007-2016. The regular season has 16 games.

Seed   Wins   Privilege

  1    13.3   Home field advantage for the semifinal and quarterfinal. Wildcard game bye.
  2    12.1   Home field advantage for the quarterfinal. Wildcard game bye.
  3    10.9   Home field advantage for the wildcard game.
  4     9.2   Home field advantage for the wildcard game.
  5    11.0
  6    10.0
Seeds 1-4 go to division winners and seeds 5-6 go to the best records among the remaining teams.

The frozen tundra of Lambeau Field, Green Bay, Wisconsin


National Basketball Association playoffs

In the NBA, the regular season has little impact on the playoffs. For example, suppose the Milwaukee Bucks are the best team and they win the conference and gain the #1 seed. In the conference playoffs they can expect to face the 8th, 4th, and 2nd seeds. If the Bucks decide to slack and place 8th, they can expect to face the 1st, 2nd, and 3rd seeds in the playoffs. But since the Bucks are the best team the 1st seed is the 2nd best team, the 2nd seed is the 3rd best team, etc.

                                                      Rank of teams played in playoffs

If the Bucks play hard and place 1st in the conference:   2nd, 4th, and 8th
If the Bucks slack and place 8th in the conference:       2nd, 4rd, and 3rd
No matter what the Bucks will face the 2nd and 4th ranked temas. The only difference between playing hard and slacking is that they have to play the 3rd ranked team instead of the 8th ranked team.

The following table gives the average number of games won by each seed, using data from 2009-2016. The regular season has 82 games. The difference between the 1st and 8th seeds is typically 20 games. That's a lot of slack.

Seed   Wins

  1    60.8
  2    55.9
  3    52.2
  4    49.9
  5    47.6
  6    45.2
  7    43.7
  8    41.9

World cup history

6: Win    5: Final    4: Semifinal    3: Quarterfinal    2: Round 16    1: Round 32
W: Wins
F: Finals
S: Semiinals
Q: Quarterfinals
Z: Sum of the columns from 1950 to 2014

          3 3 3 5 5 5 6 6 7 7 7 8 8 9 9 9 0 0 1 1   W  F  S  Q    Z
          0 4 8 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4

Brazil    3 2 4 5 3 6 6 2 6 4 4 3 2 3 6 5 6 3 3 4   5  2  4  5   80
Germany     4 2   6 4 3 5 4 6 3 5 5 6 3 3 5 4 4 6   4  4  5  4   78
Italy       6 6 3 2   2 2 5 2 4 6 2 4 5 3 2 6 1 1   4  2  2  2   62
Argentina 5 2       2 2     3 6 3 6 5 2 3 1 3 3 5   2  3     5   51
France    3 2 3 3 2 4   2     2 4 4     6 1 5 1 3   1  1  3  4   45
England         3 3 2 3 6 3     3 3 4   2 3 3 2 1   1     1  8   41
Uruguay   6     6 4   2 3 4 2     2 2     1   4 2   2     3  1   38
Spain       3   4     2 2     2 3 3 2 3 1 3 2 6 1   1     1  5   37
Neth.       2 2             5 5     2 3 4   2 5 4      3  2  1   34
Sweden      3 4 4   5     2 3 2     1 4   2 2          1  3  2   32
Mexico    3     3 2 2 2 2 3   2   3   2 2 2 2   2            4   32
Hungary     3 5   5 2 3 3     2 2 1                    2     3   31
Serbia    4     3 3 3 4     3   2   3   2   1 1           2  5   29
Czech       5 3   2 2 5   2     2   3       1          2     2   29
Belgium   3 2 2   2       2     3 4 2 2 1 2     3         1  3   28
Switz.      3 3 3 3   2 2             2     2 1 2            4   27
USA       4 2   3                   1 2 1 3 1 2 2         1  2   24
Austria     4 2   4 2         3 3   1   1                 2  2   24
Poland        2           3 4 3 4 2       1 1             1  2   23
Chile     3     3     4 2   2   2       2     2 2         1  2   22
Russia              3 3 4 3     3 2 1 1   1     1         1  4   22
Paraguay  3     3   2             2     2 2 1 3              3   18
Bulgaria              2 2 2 2     2   4 1                 1      15
S. Korea          2               1 1 1 1 4 1 2 1         1      14
Scotland          2 2       2 2 2 1 1   1                        13
Portugal                4         1       1 4 2 1         2      13
Cameroon                        2   3 1 1 1   1 1            1   11
Denmark                           2     3 2   1              1    9
Japan                                   1 2 1 2 1        1        8
Costa Rica                          2     1 1   3            1    8
Croatia                                 4 1 1   1                 7


UFC

               Mir  Mioc  Werd  Over  Lesn  Vela  Sant  Hunt  Arlo  Silv  Cout  Carw  Nogu

Frank Mir                        -     +-          -     -     -     +           +     ++
Stipe Miocic               +     +                 -     +     +
F. Werdum            -           +-          +     -     +     -     +                 +-
A. Overeem     +     -     +-          +           +     +     +     -                 --
Brock Lesnar   +-                -           -                             +     +
Cain Velasquez             -           +           ++-               ++                +
J. dos Santos  +     +     +     -           +--         +                       +
Mark Hunt      +     -     -     -                 -                 +o
A. Arlovski    +     -     +     -                                   ++
Antonio Silva  -           -     +           --          -     +-
Randy Couture                          -                                               -
Shane Carwin   +                       -           -
A. Nogueira    --          +-                -                             +

+: Victory
-: Defeat
o: Draw
For example, Miocic defeated Werdum


               Holm  Tate  Rousey  Zing  Nunes  Penn  Pena  McMann  Eye  Kaufman  Shev  Davis
Holly Holm            -      +                   +                                 -     f
Miesha Tate      +           -      -      -     -            +      +     -
Ronda Rousey     -    +             +      -                  +            +             +
Cat Zingano           +      -             +     +     -
Amanda Nunes          +      +      -                         +                    +     -
Raq. Pennington  -    +             -
Julianna Pena                       +                                +             -
Sarah McMann          -      -             -                         +                   +
Jessica Eye           -                                -      -                          -
Sarah Kaufman         +      -                                                     -     +-
Val. Shevchenko  +                         -           +                   +
Alexis Davis                 -             +                  -      +     +-
Ger. de Randamie +                         -

Vehicles

Harrier
Harrier
Osprey

Hovercraft

For submarines,

Test depth                  Maximum depth under peacetime conditions,
                               typically 4/7 of the crush depth
Maximum operating depth     Maximum depth under battle conditions
Crush depth                 Depth at which a submarine crushes


Maximum recreational scuba diving depth   =    30 meters
Max scuba depth to avoid N2 narcosis      =    30 meters
Max scuba depth using standard air        =    66 meters   (Oxygen toxicity)
German U boat crush depth                 =   250 meters
American nuclear submarine crush depth    =   730 meters
Average ocean depth                       =  3688 meters
Marana Trench, deepest point in the ocean = 10994 meters

Knots

Windsor
Trinity
Cross
Four-in-hand
Eldredge

Square
Granny (use a square instead)
Grief (use a square instead)

Sheet bend, for binding a thick and thin rope
Double sheet bend
Surgeon's knot
Surgeon's knot

Slipped sheet bend
Becket hitch
Knife lanyard

Standard bowline and Cowboy bowline
Bowline
Bowline on a bight
Spanish bowline
Portuguese bowline
Double knot

Clove hitch
Miller's
Strangle
Transom

Constrictor
Icicle hitch
Mastworp slipknot
Slippery hitch

Miller's
Slipped constrictor

Square lashing
Tripod lashing

Cleat knot
Bale sling hitch
Barrel hitch

Ashley's stopper
Ashley's stopper
Figure eight
Knife lanyard

Trefoil
Savoy

Alpine coil
Versatackle
Water knot


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