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

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

12 pound cannonballs
24 pound cannonballs

Bullet speed

25 mm
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
Octane   + O2    10.4                    .70   399
Kerosene + O2    10.3   3.52             .80   410
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   3.1             1.45   423  Hydrogen peroxide
Black powder      2.6   3.08     .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    59      29
Basketball  239    624    11.4    7.24   1.57   .087
Cannonball  220  14000   945   1000       .94  7.9
```
"Drag" is the Newton drag length and "Shot" is the typical distance of a shot, unless otherwise specified. "Density" is the density of the ball.

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

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

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

Bullet distance

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

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

Density

```         g/cm3                                    g/cm3

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

Kinetic energy penetrator

Massive Ordnance Penetrator
Bunker buster

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

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

The massive ordnamce penetrator typically penetrates 61 meters of Earth.

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

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

Drag force

The drag force on an object moving through a fluid is

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

```               Drag coefficient

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

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($\theta$)
```
The force of gravity parallel to the ramp surface is
```Framp = Fgrav sin($\theta$)
```
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($\theta$) = C Fgrav cos($\theta$)

C = tan($\theta$)
```
This is a handy way to measure the coefficient of friction. Tilt the ramp until the object slides and measure the angle.
Fixed-wing flight

Wing lift and drag

```Air density            =  D
Velocity               =  V
Wing area              =  A
Wing drag coefficient  =  Cw
Wing drag force        =  F→   =  ½ Cw A D V2
Wing lift force        =  F↑
Wing Lift-to-drag coef.=  Qw =  F↑ / F→

Cw

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

Qw

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

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 (½ Cw D A)-½   ~  M1/6

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

Gliding

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

```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

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.

Drones

X-47B
X-47B

RQ-170 Sentinel
MQ-9 Reaper

Missiles

Air to air missiles

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

```                Mach   Range  Missile  Warhead  Year  Engine
km      kg       kg

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

Ground to air missiles

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

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

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

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

Ground to ground missiles

Tomahawk
Tomahawk

```                Mach   Range  Missile  Warhead  Year  Engine        Launch
km      kg       kg                         platform

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

Hypersonic missiles

HTV-2
X-51
DARPA Falcon HTV-3

```                   Speed   Mass  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 "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
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.
Drone power system

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

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

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

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

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

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

```              kg

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

If a battery and an aluminum capacitor have equal powers,

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

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

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

Avro Lancaster
B-29 Superfortress
Heinkel He 177

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

focke-Wulf Condor
Mitsubishi Ki-67
Mitsubishi G4M

Yokosuka Ginga
Tupolev Tu-2

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

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

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

World War 2 heavy fighters

A-20 Havoc
F7F Tigercat
P-38 Lightning

P-61
P-38
Airspeed chart

Fairey Firefly
Beaufighter
Mosquito
Fairey Fulmar
Defiant

Messerschmitt 410
Heinkel He-219
Junkers Ju-88

Do-217
Me-110

Kawasaki Ki-45
J1N

Gloster Meteor
Me-262 Swallow
Heinkel He-162

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

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

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

World War 2 light fighters

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

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

F2A Buffalo
F4F
F4U

F6F Hellcat
F8F Bearcat

Ki-27
Ki-43
Ki-44

Ki-61
Ki-84
Ki-100

A5M
Mitsubishi A6M Zero
A6M2

J2M
N1K

Hawker Tempest
Hawker Hurricane
Hawker Typhoon

Submarine Seafire
Submarine Spitfire

Fw-190
Bf-109

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

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

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

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

World War 2 aircraft carriers

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

Shokaku Class
Hiyo Class
Chitose Class

Unryu Class
Zuiho Class

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

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

Military

World military budget

```             B\$/yr   % GDP                        B\$/yr   % GDP

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

Military equipment

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

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

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

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

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

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

Submarines

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

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

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

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

Submarines by country

```      Nuclear  Diesel  Aircraft
subs     subs   carriers

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

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

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

Rockets

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

Thrust

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

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

Orbit

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

Saturn V
Saturn V stage separation
Ariane 5

Rocket science is undergoing a renaissance and we can soon expect things such as asteroid mining and a manned Mars mission. Advances in astronautics include:

Stratolaunch pioneered high-altitude launch, using an aircraft consisting of two 747s fused together. Article.

Bigelow Corporation developed a space module that is substantially better than the International Space Station. Article.

Dr. Chang-Diaz perfected the ion drive, which has a much greater exhaust speed than chemical rockets. Article.

The first step toward solar system exploration is to build a base on the moon and launch lunar ice into space. Ice can be used for rocket fuel, life support, and radiation shielding, and this will enable large interplanetary spaceships to be built. Article.

Asteroid mining will soon become possible and will return trillions of dollars in platinum group metals. Article.

Using lunar ice we can build a manned base station at the L2 Lagrange point and from there build colossal space telescopes. This will revolutionize astronomy. Article.

Chemical rockets

The fuel that generates the fastest exhaust is hydrogen+oxygen and this is usually used for the upper stages. It can't be used for the first stage because of liquid hydrogen's low density. The first stage traditionally uses kerosene, and SpaceX's new methane rocket offers an improvement over kerosene.

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

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

Stratolaunch
Pegasus
Pegasus

Launching a rocket from the air brings several advantages over ground launch, such as:

*) The aircraft's speed adds to the rocket speed.

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

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

These advantages mean that the payload for air launch is a larger fraction of the rocket mass than for ground launch, reducing the launch cost. Current launch cost for ground launch is \$2000/kg.

Launch systems are under development by Vulcan Aerospace (the Stratolaunch aircraft) and Virgin Orbit (the LauncherOne aircraft). The Stratolaunch is constructed from two 747 fuselages and 6 747 engines and can carry a 230 ton rocket.

The Stratolaunch moves at Mach .9. Ramjet aircraft can move at Mach 5, are easy to design, and in the near future these will be used. In the distant future scramjet aircraft will be used, which can reach Mach 12.

For launch to low Earth orbit, every bit of speed helps.

```                                       Speed      Speed
(km/s)     (Mach)

Earth rotation speed at equator             .46    1.6
Stratolaunch aircraft speed                 .27     .9
Speed of low Earth orbit                   7.8    26.4
Speed of hydrogen+oxygen rocket exhaust    4.4    14.9
Ramjet speed                               1.5     5
Scramjet speed                             3.5    12
```
Turbofan, ramjet, and scramjet

Atmospheric reentry

Space shuttle
Apollo mission
Mars rover
A reentry vehicle can have a mass as low as 3 tons. The space shuttle was inefficient because it had a mass of 78 tons. There's no need to bring back to the Earth anything more than necessary.

```                   Mass (tons)   Crew

Space shuttle          78.0       7
SpaceX Dragon V2        4.2       7
Soyuz reentry module    2.9       3
ISRO Reentry Vehicle    3.7       3
```

If the rocket fails during launch and the crew are in a lightweight reentry spacecraft then they have a chance at surviving.

SpaceX Dragon

Soyuz

Moon
Earth
Mars
Moon
Ceres
Sizes to scale.

Manned solar system exploration starts by building a base on the moon to mine ice. Ice can be used for rocket fuel, life support, and radiation shielding, and because of the moon's low gravity it is easily lifted into space. Once in space it can be used to make spaceships and propel them throughout the solar system.

To make rocket fuel, a power source such as solar cells is used to split ice into hydrogen + oxygen.

The biggest hazard to humans in interplanetary space is cosmic ray radiation. 3 meters of ice are required to stop the rays, implying a spaceship mass of at least 400 tons. This much ice is difficult to obtain from the Earth and easy to obtain from the moon. Furthermore, such a massive ship requires a lot of ice fuel to move around.

Ice is present on the moon in polar craters that never see the sun. Everywhere else, the sun boils it off. In the asteroid belt, the sun is weaker and ice is everywhere. Ceres has an ocean's worth of ice.

```     Orbit speed   Gravity    Atmosphere       Distance from
(km/s)      (m/s2)    density (kg/m3)     sun (AU)

Earth    7.8         9.8       1.22             1.00
Mars     3.6         3.7        .020            1.52
Moon     1.68        1.6       0                1.00
Ceres     .36         .27      0                2.77
```
Since the moon has low gravity and no atmosphere, it's ideal for electromagnetic sled launch. This will be the method used to launch ice into space in the distant future.
Future rockets

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

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

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

Solar panels on the space station

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

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

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

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

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

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

At 1 AU from the sun, solar cells have a much larger power/mass than nuclear batteries. The distance from the sun for which solar cells and nuclear batteries have equal power/mass is 8 AU, if we assume 5 Watts/kg for the nuclear battery. Missions within Jupiter are best equipped with solar cells and missions beyond Jupiter are best equipped with nuclear batteries.

Generators

Photoelectric cell
Thermoelectric generator
Stirling engine
Stirling engine

The following methods can convert thermal power to electric power.

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

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

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

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

The efficiency of a generator improves with temperature, and nuclear materials don't have great melting points. One has to encase them in a metal with a higher melting point.

```        Melting point (Kelvin)

Tungsten     3693
Uranium      1405
Strontium    1050
Plutonium     913
Caesium       302
```

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

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

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

For an isotope:

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

Ion drives

Franklin Chang-Diaz

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

```Ion speed                        =  V               =      50 km/s
Mass of ion drive                =  M               =    1000 kg
Mass of ions ejected per second  =  m               = .000096 kg/s
Power consumed by the ion drive  =  Po   =  200000 Watts
Drive efficiency                 =  Q               =      .6     For converting electric to ion power
Power delivered to the ion beam  =  P  = Q Po = ½ m V2 =  120000 Watts
Agility  =  Power/Mass           =  p  = P/M        =     120 Watts/kg
Force generated by the ion beam  =  F  = m V        =     4.8 Newtons
Acceleration of the ion drive    =  A  = F/M = 2p/V =   .0048 m/s2
Gravity constant                 =  G               =6.67e-11 Newton meters2 / kg2
Earth-sun distance                                  =1.496e11 meters
Sun mass                                            =1.989e30 kg
Acceleration from sun                               =  .00593 meters/second2
```

The ion speed V can be customized. It should be at least as large as 10 km/s otherwise you might as well use a hydrogen+oxygen rocket. Increasing V decreases the fuel used and decreases the rocket force.

Pebble bed nuclear reactor

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

Energy and power in space

```                   kWatts/kg    MJoules/kg

Space mirror        200             -
Capacitor           100             .010
Flywheel            100             .5
Battery               1.2           .8
Solar cell, Earth      .10          -
Solar cell, Mars       .043         -
Solar cell, Ceres      .013         -
Nuclear battery        .020   589000
Hydrogen+oxygen        -          13
```
Solar cells are more powerful than nuclear batteries if closer to the sun than Ceres, and weaker if beyond Ceres.
Fission fragment rocket

Fission produces 2 fragments
Fission fragment rocket

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

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

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

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

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

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

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

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

Nuclear thermal rocket

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

```     Exhaust speed (km/s)

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

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

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

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

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

Electromagnetic sled launch

An electomagnetic launch sled rides on rails like a roller coaster and can reach a speed of 3 km/s. The sled releases a rocket which accelerates to orbit. The cost of the electricity is negligible compared to the cost of the rocket and hence sled launch reduces launch cost compared to ground launch.

If energy were the only contributor to launch cost then launch cost would be tiny. It costs typically 2000 \$/kg to launch material into space with rockets. If the kinetic energy comes from electricity then the electricity cost is \$1.

```Orbit speed           =  V  =  7.8  km/s
Cost of electricity   =  q  =   36  MJoules/\$
Orbit energy/mass     =  e  =   30  MJoules/kg  =  ½ V2
Electricity cost/mass =  c  =  .85  \$/kg  =  e/q
Typical launch cost         = 2000  \$/kg       Typical cost to use rockets to launch material to orbit
```

Launch sleds are best suited for inanimate cargo that can handle large acceleration. If an acceleration low enough for humans is used then the track is excessively long. If we use a human-friendly acceleration of 5 g's,

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

V2 = 2 A X                X = ½ A T2
```
If a sled is moving at 3 km/s then a centripetal acceleration of 5 g corresponds to a radius of curvature of 180 km. The last half of the track has to be straight.

If we launch inanimate equipment at an acceleration of 500 m/s2 then the track length is 9 km.

Lunar launch sled

The moon is ideal for sled launch because:

*) The lunar orbit speed is 1.68 km/s, well below the practical maximum of 3 km/s for sleds. The sled can be launched directly to orbit without needing a rocket stage.

*) The moon has no atmosphere and so you can launch horizontally and none of the kinetic energy is wasted on vertical motion.

*) Horizontal launch allows the track to be arbitrarily long, enabling human-friendly low-g launch.

*) The moon has abundant iron from metallic asteroid impacts for constructing the track.

The moon has abundant ice in polar craters and it can be launched into space in bulk with sleds. If the sled power comes from solar cells then the mass launch rate is:

```Launch speed             =  V          =  1.68  km/s
Launch energy/mass       =  e  = ½ V2  =  1.41  MJoules/kg
Solar cell average power =  p          =   200  Watts/meter2
Solar cell area          =  A                   meters2
Launch power             =  P  =  p A  =  e m   Watts
Mass launch rate         =  m  =  p A / e       kg/second
```
A spaceship needs 1000 tons of ice to shield cosmic rays. Launching this much ice in one month requires .5 MegaWatts and 2500 meters2 of solar cells.
Mars launch sled

Mars is ideal for a launch sled because the orbit speed is small, the air is thin, and there is a tall mountain. Launch at near horizontal angle is possible from the mountain.

```Mars launch speed         =  3.6  km
Sled practical max speed  =  3    km/s
Mars airmass              =   .16 tons/meter2
Earth airmass             = 10.1  tons/meter2
Mars Mount Olympus height = 21.2  km
Earth Mount Everest height=  8.8  km
```

Mountains

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

```            Peak         Height   Earth    Airmass
(m)    rotation  (tons)
(km/s)
Nepal       Everest        8848    .41       3.1
China       K2             8611    .37       3.2
Nepal       Kangchenjunga  8586    .41       3.3
Argentina   Aconcagua      6962    .39       4.0
Peru        Huascaran      6768    .46       4.1
Peru        Yerupaja       6634    .46       4.2
California  Mt. Whitney    4421    .37       5.6
Colorado    Mt. Elbert     4401    .36       5.6
-           Equator           0    .46      10.1
```
Huascaran is the tallest peak that is close to the equator.

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

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

Rocket speed

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

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

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

Oberth maneuver

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

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

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

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

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

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

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

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

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

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

Rocket power and the Oberth maneuver

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

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

Momentum conservation:    M Vex  =  F T

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

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

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

The following parameters are for a JPL design of a space mirror composed of aluminum-coated mylar.

```Mylar density      =  1.39 g/cm3
Aluminum density   =  2.70 g/cm3
Mylar thickness    =  .025 mm
Aluminum thickness =  .010 mm
Surface density    =  .006 kg/m2         (JPL design)
Mirror area        = 104 km2
Mirror mass        = 6⋅107 kg
Launch cost per kg = 1000 \$/kg
Launch cost        = 6⋅1010 \$
```

```               Payload  Engine  Fuel  Empty  Total  Payload  Payload   Exhaust  Thrust
tons     tons   tons  tons   tons    \$/kg    fraction    m/s    MNewton

Airbus A380      100      25    200    277    602       4      .17       -       1.24
Stratolaunch     230                          540              .43       -       1.78
Falcon stage 1   111       5.7  411     22.2  433       -      .26      3.05     8.2
Falcon stage 2    22.8      .6  107      4.0  111       -      .21      3.41      .93
Falcon total      22.8     6.3  518       -   549    4100      .042      -        -
```

Spaceship design

Bigelow BA-330 habitat
Bigelow Genesis habitat

The chief obstacle to spaceship design is radiation shielding. At least 3 meters of ice are required to stop cosmic rays. If you have this much ice then everything else is easy, because the ice can also be used for rocket fuel and life support.

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

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

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

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

```            Atmosphere thicknes
(tons/m2)

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

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

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

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

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

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

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

Life support

Aeroponic plants
Space station life support

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

```
Mass fraction
in human body

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

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

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

Artificial gravity

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

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

Tether

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

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

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

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

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

Suppose a spherical spaceship is shielded with ice.

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

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

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

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

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

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

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

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

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

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

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

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

t = T exp(-L/S)

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

Suppose you want to stop 99% of the protons.

```t = .01 T

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

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

Manned Mars mission

Mars Institute base in Northern Canada

Hohmann trajectory

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

The spacecraft starts on the cyan circular orbit.

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

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

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

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

Calculation of the Earth-Mars Hohmann orbit

Oberth maneuver

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

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

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

An Oberth maneuver procedes as:

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

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

V2depart  =  V2rocket + 2 Vrocket Vescape           Derivation
```

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

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

Solar system pinball

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

```
Vescape (km/s)

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

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

```Approach speed from deep space  =  Vapproach

V2depart  =  V2approach + V2rocket + 2 Vrocket Vescape           Derivation
```

Mars Mission

A mission to Mars might use the following strategy:

Mine ice on the moon.

Launch the ice from the moon into space.

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

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

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

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

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

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

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

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

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

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.

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

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

Getting around the solar system

Lagrange points
Earth Lagrange points
Interplanetary transport network

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

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

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

Gravity assists

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

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

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

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

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

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

Appendix

Calculation of the Hohmann trajectory

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

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

Solve for V1 and V2

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

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

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

V1 = 1.09891
V2 =  .72107

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

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

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

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

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

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

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

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

The energy is now

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

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

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

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

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

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

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

Bicycle

A typical set of parameters for a racing bike is

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

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

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

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

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

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

(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 * 2$\pi$)
=  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.

Metallurgy

Stone age
Copper age
Bronze age
Iron age
Carbon age

```      Discovery   Yield    Density
(year)    Strength  (g/cm3)
(GPascals)

Gold     Ancient   .20     19.3
Silver   Ancient   .10     10.5
Copper   -5000     .12      9.0
Bronze   -3200     .20      9      Copper + Tin.   Stronger than copper
Brass    -2000     .20      9      Copper + Zinc.  Stronger than copper
Iron     -1200     .25      7.9    Stronger than bronze and brass
Carbon    1987    1.4       1.75
```
The "yield strength" is the maximum stress a material can sustain before breaking.
The carbon age began in 1987 when Jimmy Connors switched from a steel to a carbon racquet.

Discovery of elements

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

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

```         Discovery   Method of
(year)     discovery

Carbon     Ancient   Naturally occuring     Coal, diamond
Gold       Ancient   Naturally occuring
Silver     Ancient   Naturally occuring
Sulfur     Ancient   Naturally occuring
Copper       -5000   Smelt with carbon
Bronze (As)  -4200                          Copper + Arsenic
Bronze (Sn)  -3200                          Copper + Tin
Tin          -3200   Smelt with carbon
Brass        -2000                          Copper + Zinc
Mercury      -2000   Heat the oxide
Iron         -1200   Smelt with carbon      In the form of steel
Zinc          1300   Smelt with carbon      Date first produced in pure form
Antimony      1540   Smelt with iron
Arsenic       1649   Heat the oxide
Phosphorus    1669   Heat the oxide
Cobalt        1735   Smelt with carbon      First metal discovered since iron
Platinum      1735   Naturally occuring
Nickel        1751   Smelt with carbon
Hydrogen      1766   Hot iron + steam
Oxygen        1771   Heat HgO
Nitrogen      1772   From air
Manganese     1774   Smelt with carbon
Molybdenum    1781   Smelt with carbon
Tungsten      1783   Smelt with carbon
Chromium      1797   Smelt with carbon
Osmium        1803
Iridium       1803
Rhodium       1804   Smelt with zinc        Smelt Na3RhCl6 with zinc
Sodium        1807   Electrolysis
Potassium     1807   Electrolysis
Magnesium     1808   Electrolysis
Lithium       1821   Electrolysis
Zirconium     1824   Smelt with potassium
Aluminum      1827   Smelt with potassium
Silicon       1823   Smelt with potassium
Beryllium     1828   Smelt with potassium
Thorium       1929   Smelt with potassium
Uranium       1841   Smelt with potassium
Ruthenium     1844   Smelt with carbon
Tantalum      1864   Smelt with hydrogen
Niobium       1864   Smelt with hydrogen
Fluorine      1886   Electrolysis
Helium        1895   From uranium ore
Titanium      1910   Smelt with sodium
Hafnium       1924
Rhenium       1928   From molybdenite
Scandium      1937

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

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

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

Discovery of the strong metals

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

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

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

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

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

Metals known since antiquity

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

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

Metals

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

Wootz steel

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

Iron meteorites

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

Alloys

Copper
Orichalcum (gold + copper)
Gold

Alloy of gold, silver, and copper

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
Chromium    1907
```

High-temperature superalloys

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

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

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

Yield strength in GPa as a function of Celsius temperature.

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

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

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

Bells and cymbals

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

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
Copper      80   C   -5000      68
Sulfur     200   *   Ancient   420
Nickel     500   C    1751      90
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
Silicon   1575   K    1823  270000
Titanium  1650   Na   1910   66000
Magnesium 1875   e-   1808   29000
Lithium   1900   e-   1821      17
Aluminum  2000   K    1827   82000
Uranium   2000   K    1841       1.8
Beryllium 2350   K    1828       1.9

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

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

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

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

Data

Thermite

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

```Fe2O3 + 2 Al  →  2 Fe + Al2O3
```

Smelting reactions

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

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

Gibbs energy

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

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

For the smelting of cobalt,

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

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

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

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

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

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

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

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

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

MoO3     -668.0     -445.3
WO3      -764.1     -509.4

V2O4    -1318.4     -659.2

Cu          0
C (gas)   672.8
```

Minerals

We list the most abundant mineral for each element.

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

Quartz: SiO2
Rutile: TiO2
Chromite: FeCr2O4
Pyrolusite: MnO2
Hematite: Fe2O3

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

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

More common minerals:

Fluorite: CaF2
Volcanic sulfur
Alumstone: KAl3(SO4)2(OH)6
Malachite: Cu2CO3(OH)l2
Azurite: Cu3(CO3)2(OH)2
Muscovite mica: KAl2(AlSi3O10)(F,OH)2 or KF2(Al2O3)3(SiO2)6(H2O)

```       grams/cm3

Osmium      22.6
Gold        19.3
Tungsten    19.2
Silver      10.5
Sperrylite  10.6   PtAs2
Thorianite  10     ThO2
Copper       8.9
Calaverite   9.2   AuTe2
Petzite      8.7   Ag3AuTe2
Minium       8.2   Pb3O4
Altaite      8.1   PbTe
Cinnabar     8.1   HgS
Iron         7.9
Epsomite     1.67  MgSO4(H2O)7
Carnallite   1.6   KMgCl3(H2O)6
```

Aliens

 Timeline of the universe

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

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

```                Millions of years ago

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

Starship

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

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

Divinity

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

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).

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

Ruby

Ruby in a green laser
Synthetic rubies

Emerald

Sapphire

Synthetic sapphire

Diamond

Raw diamond
Raw diamond
Synthetic diamond
Synthetic diamonds

Topaz

Quartz

Crystals
Crystal, polycrystal, and amorphous

Diamond
Diamond
Diamond
Diamond
Diamond

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

Corundum
Tungsten Carbide
Metal lattice

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

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

```Metal impurity   Color

Chromium         Red
Iron             Blue
Titanium         Yellow
Copper           Orange
Magnesium        Green
```

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

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

Polymers

Zylon
Vectran
Aramid (Kevlar)
Polyethylene

Aramid
Nylon
Hydrogen bonds in Nylon

Spider silk
Lignin

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

Rope

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

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

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

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

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

Wood

```         Density   Tensile   Young
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

Range of instruments

Green dots indicate the frequencies of open strings.

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

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

Stringed instruments

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

Violin and viola
Cello
Bass
Guitar
Electric guitar

Strings on a violin

Strings on a viola or cello

Violin fingering
Strings on a guitar

Wind and brass instruments

Flute
Oboe
Clarinet
Bassoon

Trumpet
French horn
Trombone
Tuba

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

Orchestra
```              String   Baroque    Classical  Modern
quartet  orchestra  orchestra  orchestra

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

Tuning

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

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

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

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

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

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

A bass guitar is tuned like a string bass.

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

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

Bowmore
Bunnahabhain

Kilchoman
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

ce1.html 0;256;0c

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
Naga Morich     1000000
Habanero         250000
Cayenne           40000
Tabasco           40000
Jalapeno           6000
Pimento             400

Molecule        Relative hotness

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

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

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

Signalling molecules

Alcohol
Caffeine
Tetrahydrocannabinol
Nicotine

Dopamine
Seratonin

Aspirin
Ibuprofen
Hydrocodone
Morphone

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

Time zones

Languages

Monarchies and republics

1815
1914
1930

1950
2015

Gross Domestic Product

Miscellaneous

Speed limit

Airports

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

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

Sports

Rugby
Rugby scrum
Football

```    Forwards                     Backs

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

Football

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

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

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

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

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

Leagues and playoffs

The league and playoff systems in common use tend to prevent interesting games from happening. The Football World Cup shows how things can go wrong and old-school college football shows how things can go right. We summarize below and expand on each point in the chapters that follow.

Ideally, the regular season should be as important as the postseason. Successful examples include college football and the NFL and unsuccessful examples include college basketball and the NBA. The World Cup qualification would be more compelling if it affected the World Cup final.

Leagues where teams can schedule their own games include college football and basketball. In most other leagues the matchups are dictated by the league.

Many sports are dominated by oligarch leagues and plebe leagues have a hard time getting attention. Examples of oligarch leagues include the college football 5 power conferences, the English Football Premiere League, and the NFL.

Knockout tournaments deliver few games between top teams, and even fewer if there are upsets. In the history of the World Cup the two most successful teams are Germany and Brazil and they have played only once.

Group stages suffer the same perils as knockout stages, plus they throw away information. A team can win every group game then lose in the first round of the knockout stage.

College League of Legends is a new sport and the structure of the regular season and playoffs changes every year.

American football playoffs

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

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

```Seed   Wins   Privilege

1    13.3   Home field for the semifinal and quarterfinal.  Wildcard game bye.
2    12.1   Home field for the quarterfinal.                Wildcard game 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

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
```

International football

Tournaments

```                                     Knockout  Group  Teams  Round  Previous  Next
stage    stage   per   robin
teams    teams  group

World             World Cup               16    32      4      1     2014    2018
Europe            European Cup            16    24      4      1     2016    2020
Europe            UEFA Nations League      4    12      3      2        -    2019
South America     Copa America             8    12      4      1     2015    2019
North America     CONCACAF Gold Cup        8    12      4      1     2017    2019
Central America   Copa Centroamericana     2     7      3.5    1     2014    2018
Caribbean         Caribbean Cup            2     8      4      1     2014    2018
Asia              Asian Cup                8    16      4      1     2015    2017
Asia East         EAFF E-1 Football Champ  0     4      4      1     2017    2019
Asia South        SAFF Championship        4     7      3.5    1     2015    2018
Asia ASEAN        AFF Championship         4     8      4      1     2016    2018
Asia West         WAFF Championship        4     9      3      1     2014    Postponed
Oceana            OFC Nations Cup          4     8      4      1     2016    2020
Africa            Africa Cup of Nations    8    16      4      1     2017    2019
N + S America     Copa America Centenario  8    16      4      1     2016    2020
Europe            World Cup Qualifying     0    54      6      2     2017    2021
South America     World Cup Qualifying     0    10     10      2     2017    2021
North America     World Cup Qualifying     0     6      6      2     2017    2021
Asia              World Cup Qualifying     0    12      6      2     2017    2021
Oceana            World Cup Qualifying     0     6      3      2     2017    2021
Africa            World Cup Qualifying     0    20      4      2     2017    2021

Round robin:  Number of times a team plays the other teams in the group.
```

Strength of regions

For the previous 6 World Cups (1994-2014), we tally each team's results and then sum over the teams from each region.

```            Qualifying     Round   Round   Quarter   Semi    Final   Win
places       of 32   of 16    final    final

Europe          13           92      51      30       16       8      4
South America    4.5         28      21      13        7       4      2
North America    3.5         19      11       2
Asia             4.5         23       6       1        1
Oceana            .5          2
Africa           5           28       7       2
```

Group death

Qualifying for each region is a double round robin group. The fraction of teams from each group that qualify is:

```             Teams/group   Teams/group   Fraction of teams
that qualify                that qualify

North America     3.5           6           .58
South America     5            10           .50
Asia              2.5           6           .42
Europe            1.5           6           .25
Africa            1             4           .25
Oceana             .5           3           .17
```

The larger the fraction of qualifying teams the better. If the fraction is low then good teams are at risk of not qualifying. In 2018 Italy and The Netherlands failed to qualify.

The regional qualifier with the most games between elite teams is South America.

For Africa, qualification is largely luck and the best teams don't play each other.

Draws

Most soccer group stages use the following scoring:

```       Points

Win      3
Draw     1
Loss     0
```
This is bull. If a game is tied in the final minute it's in the interest of both teams to flip a coin and let the coin toss winner score a free goal.

This scoring penalizes defensive teams. A low scoring game is more likely to draw than a high scoring game.

Games

For a 4-year World Cup cycle,

```            Regional   Subregional   World Cup   World Cup  Total
tournament  tournament    qualifying    games    games
games       games         games

Europe           9          0           10          4        23
South America    8          0           18          4        30
North America   16          0           10          4        30
Asia             8          8           10          4        30
Oceana           4          0            4          4        12
Africa           8          0            6          4        18
```

We assume that a team qualifies for all tournaments, and we use the average number of games played by a team in the tournament. For example, a team in the World Cup plays 3 group games and on average 1 knockout game (16 games and 16 teams), for a total of 4 games.

Regional tournaments:

```Europe            European Cup
Europe            UEFA Nations League
South America     Copa America
North America     CONCACAF Gold Cup
Asia              Asian Cup
Oceana            OFC Nations Cup
Africa            Africa Cup of Nations
N + S America     Copa America Centenario
```
Sub-regional tournaments:
```Central America   Copa Centroamericana
Caribbean         Caribbean Cup
Asia East         EAFF E-1 Football Champ
Asia South        SAFF Championship
Asia ASEAN        AFF Championship
Asia West         WAFF Championship
```

Games played

The USA averages 16 games per year. From 2014-2017:

```World Cup 2014 final        4
World Cup 2014 qualifying  13
CONCACAF Gold Cup 2015      5
Copa America Cent 2017      5
CONCACAF Gold Cup 2017      4
Friendlies, 2014           11
Friendlies, 2015           11
Friendlies, 2016            7
Friendlies, 2017            5

Total                      63
```

Regions

For the past 9 World Cups, the number of times each team qualified is:

```S America     N America      Asia          Africa          Oceana

Brazil    9   USA        7   S Korea   9   Cameroon    6   New Zealand 1
Argentina 9   Costa Rica 5   Japan     6   Nigeria     6
Uruguay   6   Mexico     3   S Arabia  5   Tunisia     4
Paraguay  5   Honduras   2   Iran      4   Morocco     4
Colombia  5   Panama     1   N Korea   1   S Africa    3
Chile     3   Canada     1   UAE       1   Ghana       3
Ecuador   3   Trin Tob   1   Australia 1   Ivory Coast 3
Bolivia   1   Jamaica    1   China     1   Algeria     3
Peru      1                  Iraq      1   Egypt       2
Senegal     2
Angola      1
Togo        1
```

Oligarch leagues

```                     Regular  Teams  Post-   Oligarch  Postseason        Oligarch
season          season  leagues   format            leagues
games           teams

Football    Pro          16    32     12       1      knockout           NFL
Basketball  Pro          82    30     16       1      knockout           NBA
Hockey      Pro          82    30     16       1      knockout           NHL
Baseball    Pro         162    31     10       1      knockout           MLB
Football    College      12  Many      4       2      knockout           Big10, Big12, Pac10, SEC, ACC
Basketball  College      27  Many     64      10      knockout           Big10, Big12, Pac10, SEC, ACC, East, AAC, MW, WC, A10
Hockey      College      35    59     16       2      knockout           Big10, National Collegiate Hockey Conf
Wrestling   College      12  Many      -       2      -                  Big10, Big12
Soccer      England Pro  38    20      -       1      -                  Premier League
Soccer      England Cup   -     -    763       -      knockout           -
Soccer      World Cup     -     -     32       -      groups, knockout   -
Rugby       World Cup     -     -     20       2      groups, knockout   4-Nations, 6-Nations
Rugby       College          Many     16       2      knockout           Division 1A, Varsity Club
LoL         College       -  Many      8       -      knockout           -
Formula-1                20    24      -       1                         Formula-1
NASCAR                   36    59     16       1                         NASCAR
```
The Rugby Championship is a double round robin between New Zealand, Australia, S. Africa, and Argentina.

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

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

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

College football

National Championship

The bowl system is designed to make the national championship game the last game, an inflexibility that prevents interesting games from happening. Furthermore, one can never be sure that the two teams selected for the championship game are the two best teams because few matchups between non-conference teams occur during the season. There's no reason that premier teams can't play each other early on, and this adds interesting games to the season.

Playoffs

There is no good way to decide the playoff teams because there are few non-conference games to go on. In 2017, the non-conference games featuring top teams were:

```Oklahoma     def.   Ohio St      31 16      Big 12  #1  vs.  Big Ten #1
Clemson      def.   Auburn       14  6      ACC     #1  vs.  SEC #3
Georgia      def.   Notre Dame   20 19      SEC     #1
Notre Dame   def.   Mich St      38 18      Big 10  #4
Notre Dame   def.   USC          49 19      Pac 12  #1
Notre Dame   def.   N Carol St   35 14      ACC     #3
Miami        def.   Notre Dame   41  8      ACC     #2
Stanford     def.   Notre Dame   38 20      Pac 12  #3
```
6 of the 8 games involve Notre Dame. Notre Dame is a benchmark for comparing conferences and this year their strength of schedule was overwhelmingly strongest in the nation. They took on two conference #1s, a #2, three #3s, and a #4.

Once the bowls were played it became clear that the toughest conference in terms of depth is the Big Ten, and the weakest conference is the Pac Twelve.

No team from outside the 5 oligarch conferences achieved any success against oligarch conferences except UCF, which defeated Auburn and Maryland. The toughest non-oligarch conferences are the American Athletic Conference and the Sun Belt Conference.

Who won the conference?

The conference championship game are a way of sneaking in an extra round of bowl games, and so we might as well make the most of it. The teams not in the conference championship should also have a chance to schedule games.

Sometimes a conference championship game makes sense and sometimes it doesn't. Consider the top teams from each conference.

```                    Losses   Teams not played

Big 12    Oklahoma      1    -
TCU           2    -

Big Ten   Wisconsin     0    Ohio St
Ohio State    1    Wisconsin

SEC       Georgia       1    Alabama,  LSU
Alabama       1    Georgia
Auburn        1    -
LSU           2    Georgia

Pac 12    USC           1    Washington
Stanford      2    -
Washington    2    USC

ACC       Clemson       1    Miami
Miami         1    Clemson,  N Carol St
N Carol St    2    Miami
```
The Big Twelve is unambiguous. The top two teams played each other during the regular season and Oklahoma has fewer losses, and so there is no need for a conference championship game. Such a game only gives Oklahoma a chance to lose and miss out on the playoffs. It would be better for Big Twelve teams to schedule out-of-conference games that are compelling to watch and will make money. One might schedule games with teams not involved in conference championship games such as Washington, Penn State, and Alabama.

Wisconsin has the fewest losses in the Big Ten but they didn't play Ohio State, and so a game between them is appropriate. If Ohio State wins then both teams have the same number of conference losses and they could be said to share the Big Ten championship.

The SEC has 3 teams with 1 conference loss, Georgia, Auburn, and Alabama. During the regular season, Auburn played Georgia and Alabama, and Georgia didn't play Alabama. It would be appropriate for Georgia to play Alabama in the postseason. If Georgia defeats Alabama then Georgia and Auburn share the conference championship. If Alabama defeats Georgia then Alabama and Auburn share the conference championship.

In the Pac Twelve season, USC didn't play Washington and so it would be appropriate for them to play. The conference championship instead featured USC vs. Stanford but this was unnecessary because they had alread played.

In the ACC, Clemson and Miami had one conference loss and they didn't play during the season, and so it was appropriate for them to play a conference championship game.

To summarize:

```             What happened                What should have happened

Big Ten      Wisconsin vs. Ohio State     Wisconsin vs. Ohio State
Big Twelve   Oklahoma vs. TCU             Out-of-conference games
SEC          Georgia vs. Auburn           Georgia vs. Alabama
Pac Twelve   USC vs. Stanford             USC vs. Washington
ACC          Clemson vs. Miami            Clemson vs. Miami
```

Conference losses

In the Big Ten,

```Ohio State  def.  Wisconsin
Wisconsin   def.  Iowa
Iowa        def.  Ohio State
```
Ohio State and Wisconsin each have 1 conference loss. Ohio State is regarded as the conference champion because they defeated Wisconsin in the championship game, but this diminishes the importance of the conference. If one regards conference games as important than one should consider Wisconsin and Ohio State to share the Big Ten championship because they have the same number of conference losses. This also provides incentive for teams not in contention for the championship to win spoiler games over contenders.
The Plebes

```Oligarch conferences:    Big Ten, Big Twelve, ACC, SEC, Pac Twelve
Plebe conferences:       AAC, Sun Belt, Mountain West, C-USA, MAC
Independents:            Notre Dame, BYU, Army
```

The gap between oligarch and plebe conferences is vast.

Regular season

The college football season could be improved by:

*) Increase the flexibility of the conference championship game to reflect the best interests of the teams, and have all teams in the conference play a game. Games can be either in-conference or out-of-conference, whatever the teams decide.

*) Ditch conference subleagues and instead schedule games that make sense regionally and matchup-wise.

*) Have a round of bowls early in the season in the north when the weather is warm. This adds interesting games to the season and it fixes the injustice of southern stadiums getting all the bowl games.

For example, the schedule for Wisconsin might look like:

``` Boise State     Away     Play a northern team while the weather is warm
Alabama         Home     Host a prominent southern team
Hawaii          Away
Notre Dame      Home
Minnesota       Away     Nearby team.  Play Minnesota while the weather is warm
Michigan State  Home
Illinois        Home     Nearby team
Michigan        Away
Northwestern    Home     Nearby team
Ohio State      Home
Iowa            Away     Nearby team.  Iowa City has great bars
Oklahoma        Away     Play a non-conference game instead of conf champ game
Stanford        Away     Bowl game
```
The schedule emphasizes conference games with nearby teams. Nebraska is always the last game because Nebraska is consistently highly-ranked. The Wisconsin-Nebraska game is analogous to the Michigan vs. Ohio State game.

With this schedule, Wisconsin plays every prominent Big Ten team except Penn State.

The Protestant Schism

The NCAA football system is awkward and change is inevitable. Some things that could happen are:

Conferences could seceed from the NCAA. Any conference can be self-sufficient.

The Big Twelve has ten members and plenty of room for more. They recently added West Virginia and TCU and additional possibilities include Boise State, Hawaii, and Houston.

Allow players to promote themselves with advertisers. B

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
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

Heroine
Heroine
Hydromorphone
Oxymorphone
Oxymorphone

Fentanyl
Fentanyl
Ocfentanil
Sufentanil
Sufentanil
Etorphine
Etorphine

Dihydroetorphine
Carfentanil

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

```                Caffein  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

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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