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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
Dr. Jay Maron


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   Cartridge
                 diam    mass                                       rate    mass
                  mm      kg      m/s    kJoule   meters     kg     Hertz   tons

Swiss Mini Gun      2.3    .00013  122       .00097 .0018      .020
Chiappa 17          4.4    .0010   560       .16    .121                            .17 PMC/Aguila
Chiappa 17          4.4    .0010   640       .20    .121                            .17 HM2
SPP-1               4.5    .0128   245       .38               .95                  4.5x40mmR
Heckler Koch MP7    4.6    .0020   735       .54    .180      1.9                   HK 4.6x30mm
Walther PPK         5.6    .0020   530       .281   .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
Luger 9mm           9.0    .0081   354              .102
Winchester 9x23     9.0    .0081   442
Colt 45            11.4    .0100   262              .127
Magnum 44          11.2    .0156   448              .165
Smith Wesson 460   11.5    .019    630      3.77    .213                            .460 SW Magnum
Magnum DesertEagle 12.7    .019    470              .254      1.996                 .50 Action Express
Smith Wesson 50mag 12.7    .026    550              .267      2.26
Smith Wesson 50mag 12.7    .029    520      3.92    .267      2.26
Smith Wesson 50mag 12.7    .032    434      3.01    .267      2.26                  .500 SW Magnum
MagnumResearch BFR 12.7    .026    550      3.93    .254      2.40                  .50 Beowulf

Ruger 96            4.4    .0013   720       .34    .47       2.38                  .17 HMR
Ruger M77           5.2    .0026  1200      1.83    .61       3.74                  .204 Ruger
CMMG MK47 Mutant    5.6    .0036   975                                              Remmington 22
Remmington 9mm      9.0    .0091   975
M4 Carbine          5.56   .0041   936      1.80    .370      2.88    15.8
FN SCAR-H Rifle     7.62   .011    790      3.51    .400      3.58    10.4          20 round magazine
Barrett M82        13.0    .045    908     18.9     .74      14.0                   10 round magazine
Hannibal           14.9    .049    750     13.8
CZ-550             15.2    .065    914     27.2                                     .600 Overkill
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  Rocket  Shock  Density  Boil
                 /kg     km/s   km/s   g/cm3  Kelvin 

Beryllium+ O2    23.2   5.3
Aluminum + O2    15.5
Magnesium+ O2    14.8
Hydrogen + O2    13.2   4.56             .07    20
Kerosene + O3    12.9 
Octanitrocubane  11.2          10.6     1.95
Methane  + O2    11.1   3.80             .42   112  CH4
Octane   + O2    10.4                    .70   399  C8H18
Kerosene + O2    10.3   3.52             .80   410  C12H26
Dinitrodiazeno.   9.2          10.0     1.98
C6H6N12O12        9.1                   1.96        China Lake compound
Kerosene + H2O2   8.1   3.2
Kerosene + N2O4   8.0   2.62
HMX (Octogen)     8.0   3.05    9.1     1.86
RDX (Hexagen)     7.5   2.5     8.7     1.78
Al + NH4NO3       6.9
Nitroglycerine    7.2           8.1     1.59        Unstable
PLX               6.5                   1.14        95% CH3NO2 + 5% C2H4(NH2)2
Composition 4     6.3           8.04    1.59        91% RDX. "Plastic explosive"
Kerosene + N2O    6.18
Dynamite          5.9           7.2     1.48        75% Nitroglycerine + stabilizer
PETN              5.8           8.35    1.77
Smokeless powder  5.2           6.4     1.4         Used after 1884. Nitrocellulose
TNT               4.7           6.9     1.65        Trinitrotoluene
Al + Fe2O3        4.0                               Thermite
H2O2              2.7   1.59            1.45   423  Hydrogen peroxide
Black powder      2.6            .6     1.65        Used before 1884
Al + NH4ClO4            2.6
NH4ClO4                 2.5
N2O               1.86  1.76
N2H4              1.6   2.2             1.02   387  Hydrazine
NH4NO3            1.4   2.0     2.55    1.12        Ammonium nitrate
Bombardier beetle  .4                               Hydroquinone + H2O2 + protein catalyst
N2O4               .10                  1.45   294

Rocket: Rocket exhaust speed
Shock:  Shock speed
Nitrocellulose
TNT
RDX
HMX
PETN
Octanitrocubane

Nitrocellulose
TNT
RDX
HMX
PETN
Octanitrocubane

Dinitrodiazenofuroxan
Nitromethane


High explosives

High explosives have a large shock velocity.


                MJoules   Shock  Density
                  /kg     km/s    g/cm3

Octanitrocubane    11.2   10.6     1.95
Dinitrodiazeno.     9.2   10.0     1.98
C6H6N12O12          9.1            1.96    China Lake compound
HMX (Octogen)       8.0    9.1     1.86
RDX (Hexagen)       7.5    8.7     1.78
PLX                 6.5            1.14    95% CH3NO2 + 5% C2H4(NH2)2
Composition 4       6.3    8.04    1.59    91% RDX. "Plastic explosive"
Dynamite            5.9    7.2     1.48    75% Nitroglycerine + stabilizer
PETN                5.8    8.35    1.77

Liquid oxygen

The best oxidizer is liquid oxygen, and the exhaust speed for various fuels when burned with oxygen is:

                Exhaust  Energy   Density of fuel + oxidizer
                 speed   /mass
                 km/s    MJ/kg      g/cm3

Hydrogen   H2      4.46   13.2    .32
Methane    CH4     3.80   11.1    .83
Ethane     C2H6    3.58   10.5    .9
Kerosene   C12H26  3.52   10.3   1.03
Hydrazine  N2H4    3.46          1.07
Liquid hydrogen is usually not used for the ground stage of rockets because of its low density.
Oxidizer

We use kerosene as a standard fuel and show the rocket speed for various oxidizers. Some of the oxidizers can be used by themselves as monopropellants.

    Energy/Mass       Energy/Mass        Rocket           Rocket         Boil    Density
   with kerosene   as monopropellant  with kerosene  as monopropellant  Kelvin   g/cm3
       MJoule/kg         MJoule/kg          km/s             km/s

O3        12.9           2.97                                              161
O2        10.3           0                  3.52             0             110     1.14
H2O2       8.1           2.7                3.2              1.6           423     1.45
N2O4       8.00           .10               2.62                           294     1.44
N2O        6.18          1.86                                1.76          185
N2H4       -             1.58                                2.2           387     1.02

Solid rocket fuel
               MJoules  Rocket   Density
                 /kg     km/s    g/cm3

C6H6N12O12        9.1             1.96        China Lake compound
HMX (Octogen)     8.0   3.05      1.86
RDX (Hexagen)     7.5   2.5       1.78
Al + NH4ClO4            2.6
NH4ClO4                 2.5
NH3OHNO3                2.5       1.84        Hydrxyammonium nitrate
Al + NH4NO3       6.9
NH4NO3            1.4   2.0       1.12        Ammonium nitrate

History
~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

Black powder           =  .75 KNO3  +  .19 Carbon  +  .06 Sulfur

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


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

Hydrogen + Oxygen     13.16
Gasoline + Oxygen     10.4


        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            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

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

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.

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

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
Drag force           =  F  =  ½ C A D V2
Drag power           =  P  =  ½ C A D V3  =  F V
Drag parameter       =  K  =  C A
"Terminal velocity" occurs when the drag force equals the gravitational force.
M g  =  ½ C D A V2
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

Rolling drag

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

Typical car tires have a rolling drag coefficient of .01 and specialized tires can achieve lower values.
                             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
Best car tires                .0075
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

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

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

Wing lift

An wing generates lift at the cost of drag. Lift exceeds drag.

Wing drag force        =  F
Wing lift force        =  F
Wing lift-to-drag coef.=  Qw =  F / F

Wing aspect ratio

The lift-to-drag coefficient Qw is proportional to wing length divided by wing width.

Wing length            =  L
Wing width             =  W
Wing lift-to-drag coef.=  Qw ~  L/W    =  Wing aspect ratio.

Wing lift-to-drag coefficient

Wing width varies along the length of the wing. We define an effective width as

Width = ½ Area / Length

"Area" is the total for both wings, and "Length" is for one wing.

Aspect ratio is Length/Width.

               Qw     Aspect   Wing    Wing   Wing
                      ratio    length  width  area
                               meter   meter  meter2

U-2             23     10.6                            High-altitude spy plane
Albatros        20               1.7                   Largest bird
Gossamer        20     10.4     14.6     1.4    41.3   Gossamer albatross, human-powered aircraft
Hang glider     15
Tern            12
Herring Gull    10
Airbus A380      7.5    7.5     36.3    11.6   845
Concorde         7.1     .7     11.4    15.7   358.2
Boeing 747       7      7.9     23.3    11.3   525
Cessna 150       7      2.6      4.5     1.7    15
Sparrow          4
Human wingsuit   2.5    1        1.0     1.0     2
Flying lemur     ?                                      Most capable gliding mammal.  2 kg max
Flying squirrel  2.0

Wing angle of attack

Changing wing angle changes lift and drag. There is an optimum angle that maximizes the lift-to-drag coefficient.

If the angle is larger than the optimal angle, you gain lift at the expense of drag. If you make the angle of attack too large, lift ceases and the plane stalls.


Air drag

The air drag force is

Air density            =  D  =  1.22 kg/meter2
Velocity               =  V
Cross-sectional area   =  A
Drag coefficient       =  C
Drag force             =  F  =  ½ C D A V2

Parachute at terminal velocity
Human mass             =  M        =  80  kg
Gravity                =  g        =  10  meter/second2
Gravity force          =  F       = 800  Newton
Chute drag coefficient =  C        =   1  Dimensionless
Air density            =  D        =1.22  kg/meter2
Parachute area         =  A        = 100  meter2
Drag force             =  F = ½ C D A V2 = F
Terminal velocity      =  V        = 3.6  meter/second

Maximum speed

Drag force             =  F  =  ½ C D A V2
Drag power             =  P  =  F V  =  ½ C D A V3

     Drag coef    Drag area   Power   Max speed
   dimensionless   meter2     Watt   meter/second


Bike     1            .5        400      11
Car       .4         3       300000      74

Wing drag coefficient

             Cw

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

Gliding

A glider is unpowered. The more efficient the glider, the smaller the glide angle. The minimum glide angle is determined by the wing lift/drag coefficient.

Drag force             =  F
Lift force             =  F  =  Fgrav
Wing lift/drag ratio   =  Qw =  F / F
Horizontal speed       =  V
Vertical descent speed =  V
Glide ratio            =  G  =  V / V
Gravitational force    =  Fgrav
Drag power             =  Pdrag  =  F   V
Power from gravity     =  Pgrav  =  Fgrav V
If the glider descends at constant velocity,
Pdrag  =  Pgrav
The goal of a glider is to maximize the glide ratio
V / V  =  (Pdrag / F)  /  (Pgrav / Fgrav)
         =  Fgrav / F
         =  Qw
The glide ratio is equal to the lift coefficient. Qw = G

Level flight

Air density           =  D
Wing area             =  A
Wing drag coefficient =  Cw
Wing drag             =  F  =  ½ Cw D A V2
Wing lift             =  F
Wing lift/drag ratio  =  Qw  =  F / F
Aircraft speed        =  V
Aircraft mass         =  M
Gravity               =  g   =  9.8 meters/second2
Gravity force         =  Fgrav=  M g
Engine force          =  Feng =  V F
Drag power            =  P  =  F V  =  ½ Cw D A V3
Agility (Power/mass)  =  p   =  P / M  =  V g / Qw
For flight at constant velocity,
Feng = F         Horizontal force balance

F   = Fgrav      Vertical force balance

F   = F Qw      Definition of the wing lift/drag coefficient

Fgrav= Fdrag Qw   →   M g = Qw ½ Cw D A V2

Cruising speed       =  V  =  M½ g½ Qw (½ Cw D A)   ~  M1/6

Agility (Power/mass) =  p  =  M½ g3/2 Qw-3/2 (½ Cw D A)  ~  M1/6

Aircraft energy/mass =  e                              ~  M0

Flight time          =  T  =  e/p                      ~  M-1/6

Range                =  X  =  V T                      ~  M0

For the mass scalings, we assume that wing area scales as M2/3.


Wingtip vortex

A wingtip creates a vortex as it moves. Wingtips are often equipped with a vertical element to damp the vortex. The vertical element increases the effective wing length and improves the lift-to-drag coefficient. coefficient.

Birds fly in a "V" formation to use the updraft from their neighbor's wingtip vortices.


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
F-14 Tomcat      2.34  19.8   33.7     15.2    268    2960         712   1974
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.

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  Range   Year
                   mach    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         17000   2 Test flights
USA      X-41          8                 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.
Flight time

A electric propeller-driven aircraft can hover for more than an hour. The hovering time is determined by the battery energy per mass and by the rotor radius. Example values:

Drone mass         =  M          =  1.0  kg
Battery mass       =  m          =   .5  kg
Battery energy/mass=  e  =  E/m  =   .8  MJoules/kg
Battery energy     =  E          =   .4  MJoules
Hover power/mass   =  p  =  P/M  =   94  Watts/kg     (Hover power for a 1 kg drone with a 1/4 meter radius rotor)
Hover power        =  P  =  p M  =   94  Watts
Flight time        =  T  =  E/P  = 3990  seconds  =  66 minutes
The flight time is
T  =  (e/p)⋅(m/M)

Hovering power per mass

The power per mass required to hover is determined by the physics of rotors. For a 1 kg vehicle with a 1/4 meter radius rotor,

Mass               =  M  =  1    kg
Gravity constant   =  g  =  9.8  meters/second
Rotor radius       =  R  =   .25 meters
Rotor quality      =  q  =  1.3
Hover power/mass   =  p  =  M½ g3/2 q-1  R-1  =  94 Watts
The rotor radius scales as M1/3 and the hover power/mass scales as M1/6. If we scale the above vehicle from 1 kg up to 300 kg (the mass of a 1-person vehicle) the hovering power/mass is 240 Watts/kg and the total power is 73 kWatts, or 98 horsepower.
Capacitors
Voltage          =  V             Volts
Capacitance      =  C             Farads
Total energy     =  E  =  ½ C V2  Joules
Effective        =  Ee =  ¼ C V2  Joules
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.

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  Bombs * #
                           speed          mass         alt                   Built
                            kph    ton    ton    ton   km    kWatt     km                   MTon

UK       Avro Lancaster        454  16.6   32.7  10.0   6.5   4x 954   4073   7377  1942    74
USA      B-17 Flying Fortress  462  16.4   29.7   3.6  10.5   4x 895   3219  12731  1938    46
USA      B-29 Superfortress    574  33.8   60.6   9.0   9.7   4x1640   5230   3970  1944    36
UK       Handley Page Halifax  454  17.7   24.7   5.9   7.3   4x1205   3000   6176  1940    36.4
Soviet   Petlyakov Pe-2        580   5.9    8.9   1.6   8.8   2x 903   1160  11427  1941    18
UK       Short Stirling        454  21.3   31.8   6.4   5.0   4x1025   3750   2371  1939    15.2
Soviet   Tupolev Tu-2          528   7.6   11.8   3.8   9.0   2x1380   2020   2257  1942     8.6
Germany  Heinkel He 177        565  16.8   32.0   7.2   8.0   2x2133   1540   1169  1942     8.4
Germany  Fokke-Wulf Condor     360  17.0   24.5   5.4   6.0   4x 895   3560    276  1937     1.5
Japan    Mitsubishi Ki-67      537   8.6   13.8   1.6   9.5   2x1417   3800    767  1942     1.2
Japan    Yokosuka P1Y Ginga    547   7.3   13.5   1.0   9.4   2x1361   5370   1102  1944     1.1
Japan    Mitsubishi G4M        428   6.7   12.9   1.0   8.5   2x1141   2852   2435  1941     2.4

UK       Total                                                                             126
USA      Total                                                                              82
Soviet   Total                                                                              27
Germany  Total                                                                              10
Japan    Total                                                                               5

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
Netherlands       4                                10*
Norway            6                                 4*
Mexico                              1                                    TC
Brazil            5        1                                             TC

*:  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.
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

Rockets

Rocket types

Chemical rocket
Fission thermal hydrogen rocket
Solar thermal hydrogen rocket
Ion rocket
Fusion rocket
Fission afterburner
Alpha rocket

The rockets that are possible with current technology are:

Chemical             React chemicals, for example hydrogen + oxygen
Thermal hydrogen     Hydrogen exhaust is heated with nuclear or solar power
Ion rocket           Accelerate ions, powered by electricity from nuclear or solar
Fusion bomb rocket   Can move large objects like asteroids
Fission afterburner  Neutrons trigger fission in the exhaust
Alpha rocket         A radioisotope emits alpha particles as exhaust

Exhaust speed


The possible energy sources are:

Chemical        H2+O2, Kerosene+O2, or solid fuel (Al+NH4NO3)
Fission         Uranium-235, Plutonium-239, Americium-242m, Beryllium-7
Radioactivity   Plutonium-238, Cobalt-60, Lead-210
Solar           Photovoltaic cells or solar mirror heat
Fusion          A fusion bomb using Deuterium + Lithium-6

The maximum exhaust speed is given by the energy/mass of the energy source.

Exhaust speed   =  V
Energy/Mass     =  e  =  ½ V2

This is the "perfect" exhaust speed. In practice the exhaust speed is less.

Energy                      Practical   Perfect   Energy/Mass
source                       exhaust    exhaust
                              speed      speed
                              km/s       km/s      MJoule/kg

Fusion       Deuterium+Li6    6900      23000   270000000
Fission      Uranium-235      4000      12000    74000000
Fission      Plutonium-239    4000      12000    76000000
Radioactive  Plutonium-238      50       2100     2300000
Chemical     Hydrogen+O2         4.4        5.1        13.2
Chemical     Methane +O2         3.7        4.7        11.1
Chemical     Kerosene+O2         3.3        4.5        10.3
Chemical     Al+NH4NO3           2.7        3.7         6.9

Rocket properties

The relationship between rocket quantities is:

Exhaust speed       =  V
Exhaust energy/mass =  e  =  ½V2
Exhaust power/mass  =  p  =  ½V2/T  =  e/T
Burn time           =  T
Acceleration        =  A  =  2p/V  =  V/T  

"Burn time" is the time for a rocket to expel its own mass of propellant. The shorter the burn time the better.

For ion drives the exhaust speed is customizable. The power/mass is fixed and there is a tradeoff between exhaust speed and acceleration. In the table below we choose a burn time of 107 seconds (3 months) for ion rockets.

It is difficult to make a rocket with both large power/mass and large exhaust speed. Fusion bombs qualify but they can only be used to move large objects like asteroids.

Rocket type               Exhaust  Power/Mass  Acceleration   Burn time
                           km/s     Watts/kg    meters/s/s     seconds

Chemical, Hydrogen+O2           4.4   1700000    770                5.6
Chemical, Methane +O2           3.7   2500000   1350                2.7
Chemical, Kerosene+O2           3.3   5000000   3030                1.1
Chemical, Al+NH4NO3             2.5   9000000   7200                 .3   Solid fuel

Fission thermal H2              9      160000     36              253     A fission reactor heats H2
Fission thermal H              13       70000     11             1200     A fission reactor heats atomic hydrogen

Ion drive                      45         100       .0045    10000000

Fission afterburner, Am-242m   93         200       .0043    22000000
Fission afterburner, Pu-239    30         200       .013      2200000
Fission afterburner, Be-7     134         200       .0030    45000000
Fission afterburner, He-3      40         200       .010      4000000
Fission afterburner, Li-6      31         200       .013      2400000
Fission afterburner, B-10      35         200       .011      3100000

Alpha, Po-210                  77         171       .0016    12000000     Half life   .38   years
Alpha, Ca-248                  71          60       .002     28700000     Half life   .91   years
Alpha, Ca-252                  68          19       .00019   82000000     Half life  2.6    years
Alpha, Pb-210                  77           2.9     .000075 730000000     Half life 22.3    years
Alpha, U-230                  200        7000                             Half life   .0554 years
Alpha, Th-228                 180         170       .0019    62000000     Half life  1.91   years
Alpha, Ra-228                 180          57       .00063  190000000     Half life  5.75   years
Alpha, Ac-227                 180          15       .00017  710000000     Half life 21.8    years
Alpha, U-232                  200           5.7                           Half life 68.9    years

Fission bomb                Large       Large   Large           Short
Fusion bomb                 Large       Large   Large           Short       Deuterium + Lithium-6

Solar sail                 300000      228000       .00152

Chemical rockets
Hydrogen
Methane
Dodecane (kerosene)
Oxygen
NH4NO3 (solid fuel)

The chemical rocket with the fastest exhaust is hydrogen+oxygen (HOX). It is the most expensive chemical rocket because hydrogen has to be liquefied at 20 Kelvin. Because the liquid hydrogen has low density, HOX can't be used for the first stage of a ground rocket, but it can be used for upper stages. HOX is easy in space.

Kerosene is often used because it's liquid at room temperature. Methane is slightly better than kerosene but it must be stored as a liquid at 112 Kelvin.

Solid fuel is cheap, simple, and reliable, and is often used for the first stage.

Fuel     Exhaust    Fuel    Fuel boiling
          speed    density     point
         (km/s)   (g/cm3)      (K)

Hydrogen    4.4     .07       20     Complex because of the low boiling point of hydrogen
Methane     3.7     .42      112     New technology pioneered by SpaceX
Kerosene    3.3     .80      410     Simple because kerosene is a liquid at room temperature
Solid fuel  2.7    1.2         -     Simple and cheap.  Often Aluminum + NH4NO3

Air launch

Stratolaunch
Pegasus
Pegasus
The Kingfisher, the first ramjet, built in 1951
SR-71 Blackbird ramjet
Scramjet

In air launch, an aircraft launches a rocket at high altitude. Air launch has advantages over ground launch, such as:

*) The aircraft speed adds to the rocket speed.

*) Less air drag. Air at 15 km has 1/4 the density of air at sea level.

*) The rocket can be launched at the equator so that the Earth's equatorial speed adds to the rocket speed.

*) Ramjet launch makes it possible to reach orbit with cheap solid rockets.

                              Speed    Speed
                              (km/s)   (Mach)

Earth rotation at equator        .46   1.6
Turbofan aircraft                .27     .9
Ramjet aircraft                 1.5     5
Scramjet aircraft              >1.5    >5
Solid rocket exhaust            2.7     9
Kerosene rocket exhaust         3.3    11
Hydrogen+Oxygen rocket exhaust  4.4    15
low Earth orbit                 7.8    26.4

At present, turbofan aircraft are being built for launching rockets, and in the near future ramjet aircraft will be used. For ramjet launch:

Stage   Type           Speed increase   End speed
                            km/s           km/s

  0     Equator motion        .45          .45
  1     Ramjet               1.55         2.0
  2     Solid rocket         4.0          6.0
  3     HOX                  2.0          8.0       Orbit speed


Fission thermal hydrogen rocket

A fission thermal hydrogen rocket uses fission to heat hydrogen propellant. It has a higher exhaust speed than chemical rockets (13 km/s vs. 4 km/s).

In a chemical rocket the energy comes from a chemical reaction and this puts an upper limit on the exhaust speed. Reacting hydrogen and oxygen gives an exhaust speed of 4.4 km/s. Nuclear power can reach higher exhaust speed.

Rocket type               Exhaust speed (km/s)

Nuclear    Fission thermal H         13
Nuclear    Fission thermal H2         9
Chemical   Hydrogen + Oxygen          4.4
Chemical   Methane  + Oxygen          3.7
Chemical   Kerosene + Oxygen          3.3
Chemical   Solid fuel                 2.7

Hydrogen is used for its low mass. The lower the mass of the propellant molecule, the higher the exhaust speed.

Exhaust speed is determined by temperature. For a temperature of 2750 Kelvin, the speed of monatomic hydrogen propellant is 13 km/s.

Monatomic hydrogen mass=  M  =1.66⋅10-27 kg
Temperature            =  T  =     2750 Kelvin
Exhaust speed          =  V  =       13 km/s
Boltzmann constant     =  k  =1.38⋅10-23 Joules/Kelvin
Exhaust constant       =  K  =  ½ M V2 / (1.5 k T)  =  2.46

At 2750 Kelvin,

Molecule   Mass  Exhaust speed
           AMU       km/s

H            1        13.0
H2           2         9.2
H2O         18         3.1

H2 dissociates into atomic H at high temperature and low density. A thermal rocket has two modes: use H2, which has high pressure and high power/mass, or use H, which has low pressure and low power/mass. The low-pressure mode still has enough power/mass to propel humans to the outer planets.

In a mirror rocket, a space mirror focuses sunlight onto hydrogen propellant.


High-temperature fission reactor

The fission material with the highest melting point is uranium oxide, which melts at 3138 Kelvin. Achieving a higher temperature requires liquid fission fuel that's embedded in a material with a higher melting point. The materials with the highest melting points are:

                         Melt    Boil
                        Kelvin  Kelvin

  HfCN                   4400
  Ta4HfC5                4263
  Hafnium carbide HfC    4201
  Tantalum carbide TaC   4150
  Niobium carbide NbC    3881
  Zirconium carbide ZrC  3805    5370
  Carbon (graphite)      3800
  Tungsten               3695    6203

Solar mirror thermal rocket

A thin film mirror can focus sunlight and heat hydrogen. The power/mass of the rocket is limited by the power/mass of the mirror. JPL designed a mirror with the goal of minimizing mass/area.

Solar flux        =  f         =  1365  Watts/meter2
Mirror mass/area  =  d         =  .006  kg/meter2
Mirror power/mass =  p  = f/d  =228000  Watts/kg

The mirror consists of mylar coated with aluminum.

Mylar density      =  1.39 g/cm3
Aluminum density   =  2.70 g/cm3
Mylar thickness    =  .025 mm
Aluminum thickness =  .010 mm

Ion drive

An ion drive uses electric power to accelerate ions. Electric power can come from nuclear fission, radioactivity, or solar cells. The ion speed is customizable. It should be at least as large as 20 km/s otherwise you might as well use a chemical or thermal rocket. The upper limit is given by the mission duration. We calculate the ion speed assuming a Plutonium-238 heat source and a burn time of 107 seconds.

Heat source Power/Mass =  P           =   567    Watts/kg     Plutonium-238
Electrical efficiency  =  D           =      .05              Convert heat to electricity
Drive efficiency       =  d           =      .6               Convert electricity to ion energy
Electric Power/Mass    =  p  = edP    =    28    Watts/kg
Burn time              =  T           =    10    Million seconds  =  3 months
Ion speed              =  V  = (2pT)½ =    24    km/s
Ion Energy/Mass        =  e  = pT     =   280    MJoules/kg

Power in space

Power in space is usually limited by cooling. For an electric generator that is closed-cycle and doesn't expel mass, cooling is limited by blackbody radiation and the maximum power/mass is determined by temperature. An example of an open-cycle system is a thermal hydrogen rocket, where the expelled mass serves as coolant and the power/mass can be much higher than for a closed-cycle system.


Solar cells

Solar panels on the space station

The power/mass of solar cells depends on distance from the sun. At the Earth, solar cells have similar power/mass as nuclear, and beyond the Earth, solar cells are worse than nuclear.

Location   Power/Mass   Distance from sun
            Watts/kg          AU

Earth         100           1
Mars           43           1.52
Ceres          13           2.77
Jupiter         3.7         5.2

Generators

Thermoelectric generator
Stirling engine
Stirling engine
Nuclear reactor

Heat can be converted to electricity with a Stirling engine, a thermoelectric generator, or photovoltaics. Usually the generator and cooling system have more mass than the heat source. Heat comes from fission or radioactivity.

Examples of existing space generators:

Heat source    Generator   Electricity  Heat source   Electric     Fuel      Total
                            Watts/kg     Watts/kg    efficiency  fraction  efficiency

Uranium-235    Brayton         195                    .25                          SAFE-400 reactor
Uranium-235    Stirling         44        991         .23        .193     .044     Kilopower reactor
Plutonium-238  Thermoelectric    5.3      567         .068       .137     .0093    GPHS-RTG. Galileo and Cassini
Plutonium-238  Thermoelectric    4.2      567         .062       .119     .0074    Voyager 1 and Voyager 2
Plutonium-238  Thermoelectric    2.8      567                             .049     MMRTG. Curiosity rover

Features of the Kilopower reactor:

*) Fuel cast as Uranium-Molybdenum alloy. Molybdenum has a melting point of 2896 Kelvin.
*) Beryllium oxide moderator, which has a melting point of 2780 Kelvin.
*) Boron carbide control rods, with a melting point of 3036 Kelvin.
*) Passive cooling with liquid sodium coolant, which melts at 371 Kelvin and boils at 1156 Kelvin.
*) Cannot melt down because the reaction rate decreases with increasing temperature.
*) Stirling engine to convert heat to electricity.


Radioactivity

Plutonium-238
Plutonium-238

The most important radioactive isotopes for power are:

Plutonium-241     Good balance of power/mass and half life.  Easy to produce.
Curium-244        Good balance of power/mass and half life.
Strontium-90      Abundant because it's present in burnt fission fuel.
Polonium-210      Alpha rocket
Lead-210          Alpha rocket
Thorium-228       Alpha rocket
Caesium-137       Abundant because it's present in burnt fission fuel.
Plutonium-238     Outperforms Strontium-90. Has to be bred in a reactor.
Cobalt-60         Larger power/mass than plutonium-238. Easily produced with neutron transmutation.
Scandium-46       Ludicrously high power/mass, and easily produced by neutron transmutation.
Californium-252   Superlatively large power/mass. Capable of powering an Iron Man suit.
Curium-252        Decays by spontaneous fission 3% of the time
Curium-254        Decays primarily by spontaneous fission

                  Half life    Heat      Decay      Decay    Obtainable by
                    year      Watt/kg                MeV     neutron transmutation

 Californium-254       .166 11200000    Fission      207      *
 Scandium-46           .229   485000    β              2.366  *
 Polonium-210          .379   144000    α              5.41   *
 Ruthenium-106        1.023    71200    2β             3.584  *
 Californium-252      2.64     41400    α or Fission  12.33   *   α 96.9% (6.12 Mev). Fission 3.09% (207 MeV)
 Cobalt-60            5.27     19300    β              2.82   *
 Osmium-194           6.02      4313    2β             2.330  *
 Lead-210            22.3       2907    α              9.100  *
 Plutonium-238       87.7        578    α              5.59   *
 Americium-242m     141          725    2α            12.33   *
 Curium-250        8300          170    Fission      148      *   Fission 74%, Alpha 18%, Beta 8%

 Beryllium-7           .146  1822000    EC              .547
 Sodium-22            2.6      68700    β+ or EC       2.842

 Uranium-230           .0554 9280000    6α+2&beta
 Thorium-228          1.912   235000    5α+2&beta         34.784
 Radium-228           5.75     90660    5α+4&beta         40.198
 Polonium-208         2.898             α
 Actinium-227        21.8      21600    5α+3&beta         36.18
 Uranium-232         68.9       7545    6α+2&beta         40.79
 Radium-226        1599          286    5α+4&betz         34.958
 Thorium-229       7917           57.7  5α+3&beta         35.366
 Protactinium-231 32600           16.2  6α+3&beta         41.3

High-temperature radioisotopes

For a heat engine, the higher the temperature, the more efficient. High temperature also means higher radiative cooling power. Isotopes with a high boiling point are:

              Half life     Heat      Decay  Decay energy   Melt    Boil    Obtainable by
                year       Watt/kg               MeV       Kelvin  Kelvin   neutron transmutation

Uranium-230           .0554 9280000    6α                    1405    4404
Tungsten-188          .191   148700    2Beta                 3695    6203    *
Tantalum-182          .313    68866    Beta                  3290    5731    *
Tungsten-181          .332    59200    EC        1.732       3695    6203    *
Iridium-194m          .468    50400    Gamma     2.229       2719    4403    *
Rhodium-102           .557    67000    Beta+                 2237    3968
Ruthenium-106        1.023    71200    2Beta     3.584       2607    4423    *
Hafnium-172          1.87     11700    EC        1.835       2506    4876
Thorium-228          1.912   235000    5α       34.784       2023    5061
Rhodium-101          4.07      9890    EC        1.980       2237    3968
Osmium-194           6.02      4313    2Beta     2.33        3306    5285    *
Actinium-227        21.8      12000    5α       36.18        1500    3500
Uranium-232         68.9       7545    6α       40.79        1405    4404
Thorium-229       7917           57.7  5α+3&beta         35.366     2023   5061
Protactinium-231 32600           16.2  6α+3&beta         41.33      1841   4300

Neutron transmutation

Neutron capture transmutes an isotope one space to the right and beta decay transmutes an isotope one space up.

The most massive nuclei that exist naturally are thorium-232, uranium-235, or uranium-238. All are unstable but have half lives larger than 700 million years. The road starts with these isotopes and then adding neutrons transmutes them according to the orange lines. The road forks at beta isotopes, which can either beta decay or capture a neutron.

The end of the road is fermium. Neutrons can't further increase the proton number because no fermium isotopes on the road beta decay. The road goes as far as fermium-258, which has a half life of .00037 seconds and spontaneously fissions. Producing heavier isotopes requires an accelerator or an extreme neutron flux (such as occurs in a fission bomb).


TRIGA nuclear reactor

A TRIGA 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 elements is easy. The reactor is easy to build and can be operated in space.


Fusion rocket
A fission or fusion bomb delivers both high power/mass and high exhaust speed. For fusion bombs,

Fusion bomb maximum practical yield =     =  e           = 21700  GJoules/kg
Mass fraction converted to energy   =  f  =  e/C2  =.00072
Exhaust speed                       =  V  =  (2e)½ =  6588  km/s

Mass is usually added to the bomb to increase momentum and decrease exhaust speed. Or, the bomb is detonated underground in an asteroid, which adds mass.


Alpha rocket

Alpha particles can be used as exhaust. An alpha emitter is coated on a mylar sail, with the coating thin enough so that most of the alphas escape into space. This rocket is simple and can be made arbitrarily small.

The recoil momentum of the nucleus is absorbed by the sail and the recoil speed of the nucleus is the effective exhaust speed. For example, polonium-208 alpha decays to lead-204 and the recoil speed of lead-204 is:

Alpha mass             =  m           =    4 AMU  =  6.64⋅10-27 kg
lead-204 mass          =  M           =  204 AMU
Alpha energy           =  e  = ½ m v2 = 5.22 MeV       The alpha gets almost all the kinetic energy
Alpha speed            =  v  = (2e/m)½=15871 km/s
lead-204 recoil speed  =  V  = vm/M   =  311 km/s

The speed calculated is for the most fortunate case where the alpha goes directly aft of the spacecraft. Calculating the average recoil speed involves integrating over all emission directions and accounting for absorption by the emitting material. In the ideal case, all particles emitted in the aft hemisphere escape and all particles emitted in the forward hemisphere are absorbed, in which case the effective recoil speed is 1/4 the fortunate speed. For polonium-208 the effective recoil speed is 78 km/s.

To calculate the power/mass of an alpha rocket,

Speed of projectile                =  v
Speed of emitter (recoil)          =  V
Mass of projectile                 =  m =   4 AMU
Mass of emitter                    =  M = 204 AMU
Energy of projectile               =  e = ½mv2
Energy of emitter                  =  E = ½MV2
Momentum of projectile             =  q
Perfect exhaust speed              =  S = q/M
Effective exhaust speed            =  s = S/4
Effective exhaust energy           =  g = ½Ms2
Exhaust energy / Decay energy      =  r = g/e = 16-1 m/M

For a polonium-208 alpha rocket, the power/mass and energy/mass are:

Effective recoil speed  =  V         =  78 km/s
Energy/Mass             =  e  = ½V2  =3042 MJoules/kg
Half life               =  T         =2.90 years
Power/Mass              =  p  = e/t  =  33 Watts/kg

The lighter the isotope the larger the recoil speed. Almost all alpha emitters are actinides, the only exceptions being gadolinium-148, polonium-208, polonium-209, and polonium-210. Alpha rockets using actinides have an average recoil speed of ~75 km/s.

The best alpha-emitting isotopes are:

               Half life  Power/Mass  Decay   Energy   Neutron transmutable
                  year     Watts/kg            MeV

Polonium-210          .379   144000    α       5.41     *
Einsteinium-254       .755   105432    α       6.616    *
Californium-248       .91     59760    α       6.36
Californium-252      2.64     21640    α       6.12     *
Polonium-208         2.898             α
Californium-250    13.1        5779    α       6.02     *
Curium-244         18.1        4014    α       5.80     *
Curium-243         29.1        1885    α
Lead-210           22.3        2907    α       9.100    *
Gadolinium-148     70.9         952    α       3.18
Plutonium-238      87.7         578    α       5.59     *
Americium-242m    141           725    2α      12.33    *

Thorium-227           .0512 9194000    6α+2β   36.14
Uranium-230           .0554 9280000    6α+2β
Thorium-228          1.912   235000    5α+2β   34.784
Radium-228           5.75     90660    5α+4β   40.198
Actinium-227        21.8      21600    5α+3β   36.18
Uranium-232         68.9       7545    6α+2β   40.79
Radium-226        1599          286    5α+4β   34.958
Thorium-229       7917           57.7  5α+3β   35.366
Protactinium-231 32600           16.2  6α+3β   41.33

Stopping length

A charged particle passing through matter loses energy from collisions with electrons, with the energy loss rate given by the Bethe formula. For an alpha particle passing through water,

Particle energy            =  E              =      6  MeV
Particle mass              =  M              =      4  AMU
Particle charge            =  Z              =      2  Proton charges
Material density           =  D              =   1000  g/cm2
Stopping distance          =  X = EZ-2M-1D-1C =   .049  meters      (Inputs in MeV, AMU, and g/cm3. Output in meters)
Material stopping constant =  C              =.000131
Stopping power             =  S              = 778000  MeV meter2
Mass/Area                  =  m = DX         =   .049  kg/meter2

Spaceships

Habitat

Bigelow BA-330 habitat
Bigelow Genesis habitat

Astronauts can expect luxuriously large spaceships. 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 have thicker walls. Thicker walls are helpful for defending against micrometeorites and radiation.

               Mass (tons)   Volume (m3)

Bigelow BA-330     23          330
Bigelow Genesis     3           11

Life support

Aeroponic plants
Space station life support

The space station life support system requires:

Power  =  1  kWatt/person
Water  =  1  kg/person/day
Food   =  1  kg/person/day

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.


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.

Ancient metallurgy

Stone
Copper
Bronze
Iron
Carbon

The earliest metals were gold and silver, the only ones that occur naturally in pure form. Iron can occasionally be found as iron meteorites.

Gold nugget
Silver nugget
Iron meteorite

Copper was discovered around 7000 BCE by smelting copper minerals in a wood fire. Around 3200 BCE it was found that copper is strenghened by tin, and this is called bronze. Around 2000 BCE it was found that copper is also strengthed by zinc, and this is called brass.

The earliest metals were smeltable with a wood fire and they consist of copper, lead, silver, tin, zinc, and mercury. They come from the following minerals:

Gold and silver were known since antiquity, but gold mining didn't start until 6000 BC, and silver smelting didn't start until 4000 BC.

The minerals that were used by ancient civilizations to smelt metal are:

Lead. Galena. PbS
Copper. Chalcocite. Cu2S
Silver. Acanthite. Ag2S
Tin. Cassiterite. SnO2
Zinc. Sphalerite. ZnS
Mercury. Cinnabar. HgS

The next metal to be discovered was iron (c. 1200 BC), which requires a bellows-fed coal fire to smelt.

Iron. Hematite. Fe2O3
Iron. Pyrite. FeS2

No new metals were discovered until cobalt in 1735. Once cobalt was discovered, it was realized that new minerals may have new metals, and the race was on to find new minerals. This yielded nickel, chromium, manganese, molybdenum, and tungsten.

Cobalt. Cobaltite. CoAsS
Nickel. Millerite. NiS
Chromium. Chromite. FeCr2O4
Manganese. Pyrolusite. MnO2
Molybdenum. Molybdenite. MoS2
Tungsten. Wolframite. FeWO4

Chromium is lighter and stronger than steel and it was discovered in 1797. It satisfies the properties of "Valyrian steel" from Game of Thrones. There's no reason chromium couldn't have been discovered earlier.

Coal smelting can't produce the metals lighter than chromium. For these you need electrolysis. The battery was invented in 1799, enabling electrolysis, and the lighter metals were discovered shortly after. These include aluminum, magnesium, titanium, and beryllium.

Aluminum. Bauxite. Al(OH)3 and AlO(OH)
Mangesium. Magnesite. MgCO3
Titanium. Rutile. TiO2
Beryllium. Beryl. Be3Al2(SiO3)6

Carbon fiber eclipses metals. The present age could be called the carbon age. The carbon age became mature in 1987 when Jimmy Connors switched from a wood to a carbon racket.

The plot shows the strength of materials.

Alloys can be much stronger than pure metals.

Wood rivals alloys for strength.


Currency

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. Today, we could use iridium, platinum, or rhenium as an uncounterfeitable currency.


Modern chemistry and the discovery of elements

Prior to 1800, metals were obtained by smelting minerals, and the known metals were gold, silver, copper, iron, tin, zinc, mercury, cobalt, manganese, chromium, molybdenum, and tungsten. Elements to the left of chromium titanium and scandium cant's be obtained by smelting, and neither can aluminum, magnesium, and beryllium. They require electrolysis, which was enabled by Volta's invention of the battery in 1799.

Prior to 1800, few elements were known in pure form. Electrolyis enabled the isolation of most of the rest of the elements. The periodic table then became obvious and was discovered by Mendeleev 1871. The battery launched modern chemistry, and the battery could potentially have been invented much earlier.

Electrolysis enabled the isolation of sodium and potassium in 1807, and these were used to smelt metals that can't be smelted with carbon.

         Discovery   Method of             Source
          (year)     discovery

Carbon     Ancient   Naturally occuring
Gold       Ancient   Naturally occuring
Silver     Ancient   Naturally occuring
Sulfur     Ancient   Naturally occuring
Lead         -6500   Smelt with carbon     Galena       PbS
Copper       -5000   Smelt with carbon     Chalcocite   Cu2S
Bronze (As)  -4200   Copper + Arsenic      Realgar      As4S4
Tin          -3200   Smelt with carbon     Calamine     ZnCO3
Bronze (Sn)  -3200   Copper + Tin
Brass        -2000   Copper + Zinc         Sphalerite   ZnS
Mercury      -2000   Heat the sulfide      Cinnabar     HgS
Iron         -1200   Smelt with carbon     Hematite     Fe2O3
Arsenic       1250   Heat the sulfide      Orpiment     As2S3
Zinc          1300   Smelt with wool       Calamine     ZnCO3 (smithsonite) & Zn4Si2O7(OH)2·H2O (hemimorphite)
Antimony      1540   Smelt with iron       Stibnite     Sb2S3
Phosphorus    1669   Heat NaPO3 Excrement
Cobalt        1735   Smelt with carbon     Cobaltite    CoAsS
Platinum      1735   Naturally occuring
Nickel        1751   Smelt with carbon     Nickeline    NiAs
Bismuth       1753   Isolated from lead
Hydrogen      1766   Hot iron + steam      Water
Oxygen        1771   Heat HgO
Nitrogen      1772   Isolated from air
Manganese     1774   Smelt with carbon     Pyrolusite   MnO2
Molybdenum    1781   Smelt with carbon     Molybdenite  MoS2
Tungsten      1783   Smelt with carbon     Wolframite   (Fe,Mn)WO4
Chromium      1797   Smelt with carbon     Crocoite     PbCrO4
Palladium     1802   Isolated from Pt
Osmium        1803   Isolated from Pt
Iridium       1803   Isolated from Pt
Rhodium       1804   Isolated from Pt
Sodium        1807   Electrolysis
Potassium     1807   Electrolysis
Magnesium     1808   Electrolysis          Magnesia     MgCO3
Cadmium       1817   Isolated from zinc
Lithium       1821   Electrolysis of LiO2  Petalite     LiAlSi4O10
Zirconium     1824   Smelt with potassium  Zircon       ZrSiO4
Aluminum      1827   Smelt with potassium
Silicon       1823   Smelt with potassium
Beryllium     1828   Smelt with potassium  Beryl        Be3Al2Si6O18
Thorium       1929   Smelt with potassium  Gadolinite   (Ce,La,Nd,Y)2FeBe2Si2O10
Vanadium      1831   Smelt VCl2 with H2    Vanadinite   Pb5(VO4)3Cl
Uranium       1841   Smelt with potassium  Uranite      UO2
Ruthenium     1844   Isolated from Pt
Tantalum      1864   Smelt with hydrogen   Tantalite    [(Fe,Mn)Ta2O6]
Niobium       1864   Smelt with hydrogen   Tantalite    [(Fe,Mn)Ta2O6]
Fluorine      1886   Electrolysis
Helium        1895   From uranium ore
Titanium      1910   Smelt with sodium     Ilmenite     FeTiO3
Hafnium       1924   Isolated from zirconium
Rhenium       1928   Isolated from Pt
Scandium      1937   Electrolysis          Gadolinite   FeTiO3

History of mineralogy

 -384  -322   Aristotle. Wrote "Meteorology"
 -370  -285   Theophrastus. Wrote "De Mineralibus"
         77   Pliny the Elder publishes "Natural History"
  973  1050   Al Biruni. Published "Gems"
       1546   Georgius Agricola publishes "On the Nature of Rocks"
       1556   Georgius Agricola publishes "On Metals"
       1609   de Boodt publishes a catalog of minerals
       1669   Brand: Discovery of phosphorus
       1714   John Woodward publishes "Naturalis historia telluris illustrata & aucta", a mineral catalog
       1735   Brandt: Discovery of cobalt
       1777   Lavoisier: Discovery of sulfur
       1778   Lavoisier: Discovery of oxygen and prediction of silicon
       1783   Lavoisier: Discovery of hydrogen
       1784   T. Olof Bergman publishes "Manuel du mineralogiste, ou sciagraphie du regne mineral",
              and founds analytical chemistry
       1778   Lavoisier: Discovery of oxygen
       1801   Rene Just Huay publishes "Traite de Mineralogie", founding crystallography
       1811   Avogadro publishes "Avogadro's law"
       1860   The Karlsruhe Congress publishes a table of atomic weights
       1869   Mendeleev publishes the periodic table

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


Superstrong amorphous alloys

Crystal, polycrystal, amorphous

New alloys have been discovered that are stronger and ligher than diamond. These alloys have an amorphous structure rather than the crystalline structure of conventional alloys. A crystaline alloy tends to be weak at the boundaries between crystals and this limits its strength. Amorphous alloys don't have these weaknesses and can be stronger.

Pure metals and alloys consisting of 2 or 3 different metals tend to be crystaline while alloys with 5 or more metals tend to be amorphous. The new superalloys are mixes of at least 5 different metals.

A material's strength is characterized by the "yield strength" and the quality is the ratio of the yield strength to the density. This is often referred to as the "strength to weight ratio".

Yield strength  =  Y            (Pascals)
Density         =  D            (kg/meter3)
Quality         =  Q  =  Y/D    (Joules/kg)
The strongest allyos are:
       Yield strength   Density   Quality
       (GPa)        (g/cm3)    (MJoule/kg)

Magnesium + Lithium             .14        1.43        98
Magnesium + Y2O3                .31        1.76       177
Aluminum  + Beryllium           .41        2.27       181
Amorphous LiMgAlScTi           1.97        2.67       738
Diamond                        1.6         3.5        457
Titanium  + AlVCrMo            1.20        4.6        261
Amorphous AlCrFeCoNiTi         2.26        6.5        377
Steel     + Cobalt, Nickel     2.07        8.6        241
Amorphous VNbMoTaW             1.22       12.3         99
Molybdenum+ Tungsten, Hafnium  1.8        14.3        126
The strongest pure metals are weaker than the strongest alloys.
       Yield strength   Density   Quality
       (GPa)        (g/cm3)    (MJoule/kg)

Magnesium                        .10       1.74        57
Beryllium                        .34       1.85       184
Aluminum                         .02       2.70         7
Titanium                         .22       4.51        49
Chromium                         .14       7.15        20
Iron                             .10       7.87        13
Cobalt                           .48       8.90        54
Molybdenum                       .25      10.28        24
Tungsten                         .95      19.25        49

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 is unsparkable

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 supersede 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.


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

For an expanded discussion of smelting physics, see jaymaron.com/metallurgy.html.


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

Minerals

These elements are not necessarily on the Science Olympiad list.

We list minerals by element, with the most abundant mineral for each element listed first.

Lithium

Spodumene: LiAl(SiO3)2
Stilbite: LiAlSi2O6
Tourmaline: (Ca,Na,K,)(Li,Mg,Fe+2,Fe+3,Mn+2,Al,Cr+3,V+3)3(Mg,Al,Fe+3,V+3,Cr+3)6((Si,Al,B)6O18)(BO3)3(OH,O)3(OH,F,O)

Beryllium

Beryl: Be3Al2(SiO3)6
Morganite: Be3Al2(SiO3)6
Emerald

Carbon

Diamond: C

Sodium

Halite: NaCl

Magnesium

Periclase: MgO
Magnesite: MgCO3
Dolomite: CaMg(CO3)2
Peridot: (Mg,Fe)2SiO4
Spinel: MgAl2O4
Spinel: MgAl2O4

Aluminum

Bauxite: Al(OH)3 and AlO(OH)
Alumstone: KAl3(SO4)2(OH)6
Muscovite mica: KAl2(AlSi3O10)(F,OH)2 or KF2(Al2O3)3(SiO2)6(H2O)
Corundum: Al2O3
Topaz: Al2SiO4(F,OH)2

Epidote: Ca2(Al2,Fe)(SiO4)(Si2O7)O(OH)
Jadeite: NaAlSi2O6
Albite: NaAlSi3O8
Amazonite: KAlSi3O8
Labradorite: (Na,Ca)(Al,Si)4O8

Silicon

Amethyst: SiO2
Quartz: SiO2
Citrine: SiO2
Opal: SiO2·nH2O
Agate: SiO2

Sulfur

Volcanic sulfur

Calcium

Fluorite: CaF2
Calcite: CaCO3
Satin Spar: CaSO4 · 2H2O
Selenite: CaSO4 · 2H2O
Aragonite: CaCO3
Pearl: CaCO3
Calcite: CaCO3

Titanium, vanadium, chomium, and manganese

Rutile: TiO2
Vanadinite: Pb5(VO4)3Cl
Chromite: FeCr2O4
Pyrolusite: MnO2
Rhodonite: MnSiO3
Rhodochrosite: MnCO3

Iron

Hematite: Fe2O3
Hematite: Fe2O3
Pyrite: FeS2
Iron meteorite
Goethite: FeO(OH)

Cobalt and nickel

Cobaltite: CoAsS
Millerite: NiS

Copper

Chalcocite: Cu2S
Chalcopyrite: CuFeS2
Malachite: Cu2CO3(OH)l2
Azurite: Cu3(CO3)2(OH)2
Bornite: Cu5FeS4
Turquoise: CuAl6(PO4)4(OH)8•4(H2O)

Zinc and germanium

Sphalerite: ZnS
Germanite: Cu26Fe4Ge4S32

Strontium, zirconium, molybdenum

Celestine: SrSO4
Strontianite: SrCO3
Zircon: ZrSiO4
Molybdenite: MoS2

Silver

Argentite: Ag2S
Acanthite: Ag2S
Silver nugget

Tin

Cassiterite: SnO2

Caesium, barium, rare-earths

Pollucite: (Cs,Na)2Al2Si4O12·2H2O
Barite: BaSO4
Monazite: (Ce,La,Nd,Th)PO4

Tungsten

Wolframite: FeWO4
Scheelite: WCaO4
Hubnerite: WMnO4

Platinum, gold, mercury, lead

Sperrylite: PtAs2
Platinum nugget
Gold nugget
Cinnabar: HgS
Galena: PbS
Anglesite: PbSO4
Thorite: (Th,U)SiO4


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.


Largest churches

Saint Peter's Basilica
Saint Joseph's Oratory
Seville Cathedral

Cathedral of Saint John the Divine
Liverpool Cathedral
Milan Cathedral

Basilica of Saint Paul Outside the Walls
Saint Paul's Cathedral
Lincoln Cathedral
Washington National Cathedral

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

Saint Peter's Basilica            5000   15.2   46    136.6  1626   Vatican
Seville Cathedral                  500   11.5   42    105    1528   Spain    Seville
Cathedral of St. John the Divine   480   11.2   39     70.7  1941   USA      Manhattan
Milan Cathedral                    440   10.2   45    108.5  1965   Italy    Milan
Abbey of Santa Giustina                   9.7          75    1606   Italy    Padua
Liverpool Cathedral                450    9.7         101.0  1978   UK       Liverpool
Basilica St. Paul Outside th Walls        8.5   30     73    1823   Italy    Rome
Florence Cathedral                        8.3   45    114.5  1436   Italy    Florence
Ulm Minster                        190    8.3   41   *161.5  1890   Germany  Ulm
Basilica Cat. Lady of the Pillar          8.3                1872   Spain    Zaragoza
Hagia Sophia                       256    8.0                 537   Turkey   Istanbul
Cathedral of Our Lady                     8.0                1521   Belgium  Antwerp
Cologne Cathedral                  407    7.9   43   *157.4  1880   Germany  Cologne
San Petrino Basilica               270    7.9   45           1479   Italy    Bologna
Saint Paul's Cathedral                    7.9   38    111.3  1708   UK       London
Washington National Cathedral             7.7   31     92    1990   USA      DC
Amiens Cathedral                   200    7.7   42    112.7  1270   France   Amiens
Basilica of the National Shrine           7.1                1961   USA      DC
Palma Cathedral                    160    6.7                1346   Spain    Palma
Reims Cathedral                           6.7                1275   France   Reims
Strasbourg Cathedral                      6.0        *142.0  1439   France   Strasbourg
Bourges Cathedral                         5.9                1230   France   Bourges
Notre Dame de Paris                       5.5                1345   France   Paris
Chartres Cathedral                        5.2                1220   France   Chartres
Winchester Cathedral                      5.0                1525   UK       Winchester
Saint Mary's Church                190    5.0                1502   Poland   Gdansk
Westminster Abbey                         3.0                1018   UK       London
Basilica of Saint John Lateran                               1735   Italy    Rome
Oliwa Cathedral                                              1350   Poland   Gdansk
Cluny III                                                    1130   France   Cluny
Canterbury Cathedral                                         1077   UK       Canterbury
York Minster                                                 1472   UK       York
Lincoln Cathedral                                            1311   UK       Lincoln
Peterborough Cathedral                                       1237   UK       Peterborough
Rouen Cathedral                                              1202   France   Rouen

*:  Held the status of world's tallest building.
All churches larger than 7000 meters2 are listed, plus the largest churches constructed before 1500 CE. The
appendix contains an expanded list of churches.

Among these types of buildings the cathedral is probably the best known, to the extent that the word "cathedral" is often mistakenly applied as a generic term for any large and imposing church. In fact, A cathedral doesn't have to be large or imposing, though many are. The cathedral takes its name from the word "cathedra", or "bishop's throne".


Church towers

Ulm Minster
Cologne Cathedral
Lincoln 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.
Saint Peter's Basilica

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.


Arch


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.


Nave

Lincoln Cathedral
Saint Paul's Cathedral
Salisbury Cathedral

Bristol Cathedral
Salisbury Cathedral
Laon Cathedral


Flying buttress

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)

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


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


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    Collapsed
St. Mary's Church     151     1549    Collapsed
Strasbourg Cathedral  142     1647
St. Nikolai           147     1874
Rouen Cathedral       151     1876
Cologne Cathedral     157     1880
Washington Monument   169     1884
Eiffel Tower          300     1889
Chrysler Building     319     1930
Empire State Building 381     1931-1967
During the years 1311-1884 the tallest structure in the world was always a church.
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.
Eiffel Tower


The Pyramids


Gems

Ruby
Diamond
Topaz
Zircon: ZrSiO4
Spinel: MgAl2O4

Sapphire
Sapphire
Sapphire

Emerald
Beryl: Be3Al2(SiO3)6
Morganite

Quartz
Amethyst: SiO2
Amethyst: SiO2
Citrine: SiO2

Garnet: [Mg,Fe,Mn]3Al2(SiO4)3 & Ca3[Cr,Al,Fe]2(SiO4)3
Peridot: (Mg,Fe)2SiO4
Opal: SiO2·nH2O
Jadeite: NaAlSi2O6
Pearl: CaCO3
Amber: Resin

Corundum is a crystalline form of aluminium oxide (Al2O3). It is transparent in its pure form and can have different colors when metal impurities are present.

             Color    Colorant  carat ($)

Painite                          55000  CaZrAl9O15(BO3)
Diamond      Clear                1400  C
Ruby         Red      Chromium   15000  Al2O3
Sapphire     Blue     Iron         650  Al2O3
Sapphire     yellow   Titanium          Al2O3
Sapphire     Orange   Copper            Al2O3
Sapphire     Green    Magnesium         Al2O3
Emerald      Green    Chromium          Be3Al2(SiO3)6
Beryl        Aqua     Iron              Be3Al2(SiO3)6   AKA "aquamarine"
Morganite    Orange   Manganese    300  Be3Al2(SiO3)6
Topaz        Topaz                      Al2SiO4(F,OH)2
Spinel       Red      Red               MgAl2O4
Quartz       Clear                      SiO2
Amethyst     Purple   Iron              SiO2
Citrine      Yellow                     SiO2
Zircon       Red                        ZrSiO4
Garnet       Orange                     [Mg,Fe,Mn]3Al2(SiO4)3 & Ca3[Cr,Al,Fe]2(SiO4)3
Garnet       Blue                 1500  [Mg,Fe,Mn]3Al2(SiO4)3 & Ca3[Cr,Al,Fe]2(SiO4)3
Opal                                    SiO2·nH2O
Opal         Black               11000  SiO2·nH2O
Jet          Black                      Lignite
Peridot      Green                      (Mg,Fe)2SiO4
Pearl        White                      CaCO3
Jade         Green                      NaAlSi2O6
Amber        Orange                     Resin

Crystals
Crystal, polycrystal, and amorphous

Diamond
Carbon phase diagram

Corundum (Al2O3)
Corundum unit cell
Corundum

Metal lattice
Salt (NaCl)
Tungsten Carbide

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


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
                   strength
         (g/cm^3)  (Gpa)     (Gpa)

Balsa         .12    .020      3.7
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
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
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
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
Oak, live     .98    .130     13.8
Ironwood     1.1     .181     21.0
Lignum Vitae 1.26    .127     14.1
Data #1     Data #2
Plastic

           Density   Tensile   Young
                     strength
           (g/cm^3)  (Gpa)     (Gpa)


Polyamide            .11       4.5
Polyimide            .085      2.5
Acrylic              .07       3.2
Polycarbonate        .07       2.6
Acetyl copoly        .06       2.7
ABS                  .04       2.3
Polypropylene  .91   .04       1.9
Polystyrene          .04       3.0
Polyethylene   .95   .015       .8

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

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.


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.


Orchestra
              String    Baroque     Modern
              quartet  orchestra   orchestra

First violin     1        4           16
Second violin    1        4           14
Viola            1        4           12
Cello            1        4           12
Bass                      2            8
Flute                     2            4
Oboe                      2            4
Clarinet                               4
Bassoon                   2            4
Trumpet                   2            4
French Horn               2            4
Trombone                               4
Tuba                                   2
Harpsichord               1
Timpani                   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
Alcohol fraction of beer   =  Fbeer =   .05
Alcohol volume in one beer =  VBond =   .6  ounces
One "Bond" of alcohol              =   .6  ounces
One wine or Scotch bottle          = 25.4  ounces  =  750 ml
One ounce                          = 29.6  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 pepper     40000
Malagueta pepper   40000
Tabasco            40000
Jalapeno            5000
Guajillo pepper     5000
Cubanelle            500
Banana pepper        500
Bell pepper           50
Pimento               50

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)


Maps

American geography

College conferences

Topography

Rain


Rivers


American national parks

Click for larger image.

Yellowstone
Zion
Bryce Canyon
Acadia

Time zones


Languages


Monarchies and republics

1815
1914
1930

1950
2015


Gross Domestic Product


Miscellaneous

Speed limit

Airports


Geography


Rain

Precipitation by month


Sun


Wind


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


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   Racquet  Racquet     Racquet  Fastest   Max       Drag
          diameter Mass  length  width  density  mass   max length  max width  shot   distance  distance
             mm    gram  meter   meter  gram/cm3  gram    cm       cm      m/s     meter     meter

Ping pong      40      2.7   2.74   1.525  .081   70                           31.2                1.8
Squash         40     24     9.75   6.4    .716                                78.22              15.6
Golf           43     46                  1.10                                 94.3   214.2       25.9
Paintball              1.25                                                    85
Snooker        52.5  149     3.658  1.829
Badminton      54      5.1  13.4    5.18   .062   85                          136.9                1.8
Racquetball    57     40    12.22   6.10   .413                                85.4               12.8
Billiards      59    163     2.84   1.42  1.52                                 15.6               48.7
Tennis         67     58    23.77   8.23   .368                                73.2               13.4
Cricket        72    160    80             .82                                        128.6       32.8  Throw   80ii meters from batter to home run boundary
                                                                                      158               Hit
Field hockey   73    160    91.4   55      .78
Baseball       74.5  146   122             .675                                46.9   135.88      27.3  Throw   Pitcher-batter distance = 19.4 m
                                                                               54.14  177               Hit
Pickleball     74     24    13.41   6.10   .151            61.0       21.0
Hockey puck    76    163    61     26     1.44                                 51.0                            25 mm thick
Whiffle        76     45                   .196                                                    8.1
Polo           82    130   274.3  146.3    .45
Croquet        92    454                  1.11
Softball       97.1  188                   .39                                         97.8             Throw
Softball                                                                              175.56            Hit
Football      178    420    91.44  48.76   .142                                26.8    69.5       13.8  Throw
                                                                               35.8   ~65               Placekick
                                                                                       90.5             Punt
Rhythmic gymn 190    400    12     12      .111
Rugby         191    435   100     70      .119                                21.46              12.4  Throw
Volleyball    210    270    18      9      .056
Bowling       217   7260    18.29   1.05  1.36                                                   160
Soccer        220    432   105     68      .078                                35.84               9.3  Placekick
                                                                                       75.35            Punt
                                                                                       59.82            Throw-in
                                                                                       61.26            Throw
Basketball    239    624    28     15      .087                                                   11.4
Disc ultimate 273           64     37                                                                            27.3 mm thick    18 meter end zonesDisc golf     300    200
Beach ball    610    120                   .0011
Javelin              800                                   270                         98.48
Discus        219   2000                                                               74.08                     44 mm thick
Hammer        102   7260                                   121.3                       86.74
Shot put      125   7260                  7.10                                         23.56
Cannonball    220  14000                  7.9                                                    945
Sumo                         4.55

"Fastest shots" are world records.


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

Cycling .25 km      17.005     18.282 seconds     standing start
Cycling  .2 km       9.1       10.154 seconds     flying start
Cycling  .5 km      24.758     28.970 seconds     flying start
Cycling 1   km      56.303            seconds     flying start
Cycling 3   km                196.937 seconds     flying start
Cycling 4   km     241.934            seconds     flying start
Cycling  10 mi   16:35      18:36     seconds
Cycling  25 mi   44:04      49.28     seconds
Cycling  50 mi 1:34:43    1:42:20     seconds
Cycling 100 mi 3:13:37    3:42:03     seconds
Cycling 1 hour      55.089            km
Cycling 1 hour                 48.007 km
Cycling 12 hour    321.44     290.07  km
Cycling 24 hour    544.32     478.42  km

Cycling




College football conference alignment

Colleges wage feudal war over TV markets, and the Big Ten and SEC won. For football, they are far stronger than other conferences.

Stadiums

The Big Ten and SEC have the biggest stadiums. Outside the Big Ten and SEC, the biggest stadiums are Clemson, Florida State, and Notre Dame.

TV markets

The Big Ten and SEC have TV supremacy. The Big Ten now owns the Los Angeles market, and the SEC owns a chunk of the Texas and Oklahoma markets. Big Ten schools will get $70 million of TV revenue per year, and Pac 12 schools will get much less. TV revenue:

          Team  Conference  Teams/Conference
           M$       M$

SEC         40     560         14
Big 10      31.4   440         14
Big 12      20     200         10
Pac 12      21     252         12
ACC         17     240         14
Notre Dame  15      15          1
MW           4      48         12
MAC           .6     8         12
Sun Belt      .5     7         14
CUSA          .45    5         11

The playoffs generate $470 million and total TV revenue is $2250 million.

Oligarchs and plebes

Big Ten and SEC teams are oligarchs and other teams are plebes. The plebes need to adjust. Some might declare independence. We might see an alliance of independents, and independents might form their own version of a conference championship. They could keep an open week on the schedule for spontaneous games.

Texas Christian University achieved upward mobility by winning the Rose Bowl in the 2010 season, and in 2012 they moved from the Mountain West to the Big 12. They made the national championship bracket in 2022.

European soccer leagues have promotion and relegation. College football conferences have promotion but not relegation.

The Big Ten and the SEC don't need the NCAA. They could secede from the NCAA and go pro. Other conferences will be forced to follow.

Promotion and relegation

Conference realignment saw 5 teams relegated from power conferences and 6 teams promoted to power conferences. The promoted teams are all in the Big 12 and these are TCU, BYU, Utah, Cincinnati, Houston, and UCF. The relegated teams were from the Big East and are Connecticut, Temple, and South Florida.

In 2004, there were 6 power football conferences, one of which was the Big East. The Big East has 12 schools, 9 of which defected to other power conferences. The 3 that didn't were relegated and are now minnows. We went from the Power6 to the Power5.

The Pac 12 once had 12 teams. 10 left for other power conferences, leaving 2 behind. Relegation. We are now down to the Power 4.

The Big Ten and SEC deserve their own tier. We could call them "superpowers".

Flexible schedule

In the 2020 covid season, during conference championship week, the Big Ten had all teams play a game. The idea wasn't continued the next season.

In the 2020 season, the game between Liberty and Coastal Carolina was canceled, and Liberty scheduled a game with BYU on 3 day's notice.

The Big Ten has 18 teams and the SEC has 16, forcing them to adjust their scheduling systems. Divisions will be scrapped, and the new systems will favor local games and traditional rivalries.

Conference championship week could be more flexible. Sometimes it's strategic to have a conference championship game and sometimes not. Sometimes the conference championship is already decided before the championship game, in which case it's strategic for the top teams to play teams from other conferences. A pre-bowl game.

The Pac 12 is now the Pac 2. Washington State and Oregon State need to make a move. An option is to go independent and dare more teams to follow. Another option is to invite the best of the Mountain West, and this would consist of San Jose State, San Diego State, UNLV, Utah State, Colorado State, Boise State, and Hawaii. Another option is to join the Mountain West.

Bowls

The bowl system needs reform. There will be teams from the Big Ten and SEC that have losing records but are worthy of bowls. The problem can be solved by letting everyone play bowls.

Traditionally, bowl games are between teams that are equally-matched in strength. College football will adopt a 12-team bracket in 2026, and the first two rounds will typically be mismatches. The college football postseason will look like the college basketball postseason.

In the 12-team bracket, many teams will play multiple postseason games. Teams outside the bracket might demand to do so as well.

To decide bowl matchups, there is rarely enough information from interconfence games to properly compare conferences, suggesting that we need more than 1 round of bowls. Notre Dame is often a vital benchmark for interconference strength. It would be nice to have more independents.

The Australian Football League has a bracket that's better than single-elimination. It has big games early on, and high-seeded teams can take a loss without being eliminated.

Australian Football League

Recruiting

The Big Ten and SEC dominate recruiting. The plot shows recruiting ranks averaged over the past 4 years, along with the AP rank averaged over the past 18 years.

For 2022, the SEC has 9 of the top 16 recruiting classes, and 15 SEC teams are in the top 32.

Among the top 64 recruiting teams, all are in the Power5. The strongest conferences outside the Power5 are the AAC, the Mountain West, and the Sun Belt.

Minor leagues

College football and college basketball dominate the pro minor leagues. For most other sports, pro minor leagues dominate colleges.

Basketball oligarch conferences

College football conferences are more oligarchic than college basketball conferences.

The plots show the history of college football and college basketball, and the past 18 seasons are used to rate teams and conferences.

Poach history

1990  Indep  → Big 10   Penn State
2004  Big E  → ACC      Miami       Virginia T
2005  Big E  → ACC      Boston C
2010  M West → Big 12   TCU
2011  Big 12 → Big 10   Nebraska
2011  Big 12 → Pac 12   Colorado
2011  M West → Pac 12   Utah
2012  Big 12 → SEC      Missouri     Texas AM
2012  Big E  → Big 12   W Virginia
2013  Big E  → ACC      Pittsburgh   Syracuse
2014  Big E  → ACC      Louisville
2023  AAC    → Big 12   Houston      Cincinnati    UCF
2023  Indep  → Big 12   BYU
2023  CUSA   → AAC      UAB          Florida Atl   Carolina Charlotte   N Texas    Rice    UTSA
2023  Indep  → CUSA     Liberty      New Mex St
2023  Ohio V → CUSA     Jacksonville St
2023  WAC    → CUSA     Sam Houston St
2023  AAC    → Sun Belt UCF
2024  Pac 12 → Big 10   UCLA         USC           Washington           Oregon
2024  ASUN   → CUSA     Kennesaw St
2024  Big 12 → SEC      Texas        Oklahoma
2024  Pac 12 → Big 12   Arizona      Arizona St    Colorado             Utah

U. Chicago has 7 Big Ten titles. They withdrew from the Big Ten in 1946 and canceled the football team. Football was a distraction to academics.


Glorify the regular season

In college football, the regular season matters. The postseason consists of only one bowl game.

Home field advantage matters more for the NFL than for the NBA, hence the NFL regular season is more important than the NBA regular season.

For any given sport, the fewer teams in the playoffs, the more important the regular season. MLB has only 12 postseason teams.

The English Premier League has no postseason. The winner is decided purely by the regular season.

     Playoff   Total
      teams    teams

MLB     12     30
NFL     14     32
NHL     16     32
NBA     16     30
NFL playoffs

In the NFL, each regular season win tends to increase the playoff seed by 1.

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 for the quarterfinal and semifinal.  Wildcard bye
  2    12.1   Home field for the quarterfinal.                Wildcard bye
  3    10.9   Home field for the wildcard game.
  4     9.2   Home field 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

Home field advanage can be especially important for cold-weather teams like the Packers.

NBA 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 team. 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

Opium

Opium poppies
Opium poppy

              Strength   Half life   Dose
                           hours      mg

Carfentanil     30000        7.7       .0003
Ohmefentanil     6300
Dihydroetorphine 4000                  .03
Etorphine        2000                  .006
Sufentanil        750        4.4       .015
Ocfentanil        180                  .06
Fentanyl           75         .04      .1
Oxymorphone         7        8       10
Hydromorphone       5        2.5      1.5    Dilaudid
Heroine             4.5      <.6      2.2    Diamorphine
Methadone           3.5     30       35
Oxycodone           1.5      4        6.7
Morphine            1        2.5     10
Hydrocodone         1        5       10      Vicodin
Codeine              .1      2.8    180
Naproxen             .0072  18     1380
Ibuprofen            .0045   2     2220
Aspirin              .0028   6     3600

Strength data


Poppy

The composition of a typical opium poppy is:

                %    First isolated

Morphine        10       1817          Used to produce heroine
Codeine          2       1832
Thebaine         8                     Used to produce hydrocodone and hydromorphone
Papaverine      14       1848          Not psychoactive
Noscapine        5       1820          Not psychoactive
Other alkaloiods  .1

Opioids

Codein
Hydrocodone
Hydrocodone
Morphine
Morphine
Oxycodone
Oxycodone

Methadone
methadone
Heroine
Heroine
Hydromorphone
Oxymorphone
Oxymorphone

Fentanyl
Fentanyl
Ocfentanil
Sufentanil
Sufentanil
Etorphine
Etorphine

Dihydroetorphine
Carfentanil


Opium addiction

The brain calculates from the rear forward. The hindbrain (subconscious) is like the foundation of a building and the forebrain (conscious) constitutes the upper stories. If you overemphasize the upper stories and they become disconnected from the foundation then the brain fractures into bifurcation. Opium encourages this bifurcation by making it easier to maintain conscious while neglecting the foundation. Solutions include:

Meditation can help restore the foundations, but you have to know what your doing. Many techniques exist, such as Alexander Technique, The Feldenkrais Method, Reiki, Kundalini Yoga, but Shaolin meditation is the most powerful. This is a textbook on the art of Shaolin meditation.

Marijuana has the potential to substitute for opium craving. Marijuana cannot be overdosed where as opium is easily overdosable.


Half life

The half life of a drug should neither be too short nor too long. If it's too short then you can't use it for long-term meditation. If it's too long then there is a risk of overdose. The drug should also not be stronger than morphine. Morphine is strong enough. The opoids with moderate half lives are:

            Strength   Half life   Dose
                         hours      mg

Oxycodone     1.5         4         6.7                                              
Morphine      1           2.5      10                                                
Hydrocodone   1           5        10      Vicodin                                   
Codeine        .1         2.8     180
Vicodin and codeine appear to be the ideal meditation opoids.
Caffeine

               Caffeine  Density  Volume
                  mg      mg/oz   mg/oz

Coffee, brewed     163     20.4      8
Mtn. Dew Game Fuel 121      6.0     20
Red Bull            80      9.5      8.5
Espresso            77     51        1.5
Mountain Dew        54      4.5     12
Mello Yello         51      4.2     12
Tea (black)         42      5.2      8
Sunkist             41      3.4     12
Pepsi Cola          38      3.2     12
Arizona iced tea    38      1.9     20
Coca Cola           34      2.8     12
Coffee, decaf        6       .7      8
Sprite               0      0       12

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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