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World War 2
Dr. Jay Maron


Nuclear fission

Fission chain reaction

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

Chain reaction simulation at phet.colorado.edu

Critical mass

Less than a
critical mass
Critical
mass
More than a
critical mass
Chain reaction in
a supercritical mass
Almost a critical
mass of plutonium

A fission of uranium-235 releases on average 1.86 neutrons, some of which trigger fission in nearby nuclei and some of which escape without triggering fission. If a sphere of uranium-235 is small then most of the neutrons escape before triggering fission and the sphere doesn't blow up. If the sphere is large then most of the neutrons trigger more fission, a chain reaction occurs and the sphere blows up. The threshold for a chain reaction is the "critical mass".

The nuclei that are capable of undergoing a chain reaction are:

           Protons  Neutrons  Critical   Halflife   Neutrons per
                              mass (kg)  (106 yr)     fission

Uranium-233    92     141        16         .160      2.48
Uranium-235    92     143        52      700          1.86
Plutonium-239  94     145        10         .024      2.16

Uranium detonation

Two pieces of uranium-235, 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.

If you bring two pieces of uranium-235 together too slowly, a chain reaction begins in the near side of each piece, generates heat, and blows the two pieces apart before they can come completely together. Only a small amount of uranium undergoes fission and this is referred to as a "fizzle". Using gunpowder and a cannon is fast enough to properly detonate uranium and this is technologically easy to do.


Plutonium detonation

Plutonium is more difficult to detonate than uranium. Simply bringing two pieces together, no matter how fast, results in a fizzle. To detonate plutonium you have to shape it as a sphere and implode it, which is technologically difficult.

In World War 2 the U.S. produced enough uranium for 1 bomb and enough plutonium for 2 bombs. One of the plutonium bombs was tested in the "Trinity" test before being used in the war, and the second bomb was dropped on Nagasaki. The uranium bomb was dropped on Hiroshima without previously being tested.

When Hans Bethe, a physicist on the Manhattan project, was asked why they didn't test the uranium bomb he replied "Because we were perfectly sure it would work".


Separation of Uranium-235 from Uranium-238

Magnetic separation. Dark blue = uranium-235. Light blue = uranium-238. Yellow = magnetic field.
Magnetic separation machines during the Manhattan Project

Natural Uranium is .72% Uranium-235 and 99.3% Uranium-238. Only Uranium-235 undergoes a chain reaction and so it has to be separated from the Uranium-238. Several methods exist for doing this. In World War 2 the isotopes were separated magnetically with calutrons. Gas diffusion and centrifuges can also be used.


Centrifuge separation of uranium-235

UF6
UF6
Light blue: uranium-235. Dark blue: uranium-238
Centrifuges

Uranium is converted to gas form by forming uranium hexafluoride (HF6). HF6 is a gas above 64 Celsius. In a centrifuge, the lighter uranium-235 concentrates at the center and the heavier uranium-238 concentrates at the edge.


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 and don't exist in nature.

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 .7% Uranium-235 and the rest is Uranium-238.

Plutonium has a short half life and doesn't exist in nature. It can be created by subjecting uranium-238 to neutrons in a nuclear reactor. Fissionable uranium-233 can be created from thorium-232.

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

Detail:

Uranium-238 + Neutron  →  Uranium-239
Uranium-239            →  Neptunium-239 + Electron + Antineutrino          Halflife = 23 minutes
Neptunium-239          →  Plutonium-239 + Electron + Antineutrino          Halflife = 2.4 days

Thorium-232 + Neutron  →  Thorium-233
Thorium-233            →  Protactinium-233 + Electron + Antineutrino       Halflife = 22 minutes
Protactinium-233       →  Uranium-233      + Electron + Antineutrino       Halflife = 27.0 days

Fusion bomb

Fusion bombs use the reactions:

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

Fusion bomb design

Fusion of deuterium and lithium requires high temperature and pressure, which is achieved by compressing the fuel. This is done by detonating a fission bomb and using the generated X-rays to compress the fusion fuel. X-rays strike the outer layer and expel atoms, and the recoil compresses the fuel. This is called "ablation" and the design was developed by Teller and Ulam.

             X-ray     Plasma    Ablation
            pressure  pressure   pressure
              TPa       TPa        TPa

Ivy Mike       7.3       35        530
W-80         140        750       6400
Teller
Ulam
Ulam

Energy

The practical limit for the energy/mass of a fusion bomb = 25 TJoules/kg or .0062 Mtons of TNT per kg.

1 ton of TNT                        =   4⋅109  Joules
1 ton of gasoline                   =   4⋅1010 Joules
Massive Ordnance Air Blast bomb     =   .000011 MTons TNT  (Largest U.S. conventional bomb)
Trinity plutonium-239 test          =   .020 MTons TNT
Hiroshima uranium-235 fission bomb  =   .015 MTons TNT   "Little Boy". 60 kg Uranium-235
Nagasaki plutonium-239 fission bomb =   .021 MTons TNT   "Fat Man".     6 kg Plutonium-239
Ivy King fission bomb               =   .5   MTons TNT   Largest pure fission bomb
B83 fusion bomb                     =  1.2   MTons TNT   Largest bomb in active service
Castle Bravo fusion bomb            = 15     MTons TNT   Largest U.S. test
B41 fusion bomb                     = 25     MTons TNT   Largest U.S. bomb created
Tsar Bomba                          = 50     MTons TNT   Largest USSR test

History of nuclear physics

Leo Szilard
Enrico Fermi
Johnny von Neumann, Robert Oppenheimer, and the EDVAC computer
Niels Bohr

1885        Rontgen discovers X-rays
1899        Rutherford discovers alpha and beta rays
1903        Rutherford discovers gamma rays
1905        Einstein discovers that E=mc2. 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
1938 Dec19  Hahn and Strassmann discover uranium fission
1939 Jan 6  Hahn and Strassmann publish uranium fission
1939 Jan25  Fermi begins conducting nuclear fission experiments at Columbia University
1939 Jan26  Bohr and Fermi report on uranium fission at the Washington Conference
            on theoretical physics
1939        Szilard and Zinn discover that bombarding uranium with neutrons produces
            new neutrons.
1939 Jul 4  Szilard, Wigner, and Einstein discuss nuclear fission
1939 Aug 2  Szilard, Teller, and Einstein discuss nuclear fission. Szilard drafts
            the the "Einstein letter" that is later delivered to President Roosevelt
1939 Oct11  Alexander Sachs briefs President Roosevelt on Einstein's letter.
1939 Oct12  Alexander Sachs meets again with President Roosevelt and this time
            Roosevelt gives the order to commence the development of a nuclear bomb.
1942 Dec 2  Fermi and Szilard achieve the first self-sustaining nuclear fission
            reactor at the University of Chicago
1942 Aug    Manhattan project commences
1942-1945   German nuclear bomb project goes nowhere
1945 Jul16  Trinity test of a plutonium bomb yields a 20 kTon TNT equivalent explosion
1945 Aug 6  A uranium bomb is deployed at Hiroshima, yielding 15 kTons TNT equivalent
1945 Aug 9  A plutonium bomb is deployed at Nagasaki, yielding 21 kTons TNT equivalent
Hans Bethe, a physicist on the Manhattan Project, was asked why the uranium type bomb was not tested before deployment and he replied "Because we were perfectly sure it would work".
World War 2

Trinity plutonium test
Trinity plutonium test
Little Boy
Little Boy
Hiroshima

The Enola Gay, the bomber that deployed "Little Boy"
Fat Man
Nagasaki
neutron.

All of the radioactive fission products decay by beta decay.

If the neutron cross section is 8 barnes or higher then the nucleus can potentially be transmuted into a nonradioactive nucleus.

Strontium-90 is ideal for Radioisotope Thermoelectric Generators (RTGs).

The most troublesome fission products are the ones that can't be transmuted. Chief among these are Caesium-137, Zirconium-93, Niobium-94, Strontium-90, Zirconium-91, and Palladium-107.


World War 2 bombers

Avro Lancaster
B-29 Superfortress
Heinkel He 177

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

focke-Wulf Condor
Mitsubishi Ki-67
Mitsubishi G4M

Yokosuka Ginga
Tupolev Tu-2

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

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

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

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


World War 2 heavy fighters

A-20 Havoc
F7F Tigercat
P-38 Lightning

P-61
P-38
Airspeed chart

Fairey Firefly
Beaufighter
Mosquito
Fairey Fulmar
Defiant

Messerschmitt 410
Heinkel He-219
Junkers Ju-88

Do-217
Me-110

Kawasaki Ki-45
J1N

Gloster Meteor
Me-262 Swallow
Heinkel He-162

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

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

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

World War 2 light fighters

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

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

F2A Buffalo
F4F
F4U

F6F Hellcat
F8F Bearcat

Ki-27
Ki-43
Ki-44

Ki-61
Ki-84
Ki-100

A5M
Mitsubishi A6M Zero
A6M2

J2M
N1K

Hawker Tempest
Hawker Hurricane
Hawker Typhoon

Submarine Seafire
Submarine Spitfire

Fw-190
Bf-109

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

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

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

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

World War 2 aircraft carriers

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

Shokaku Class
Hiyo Class
Chitose Class

Unryu Class
Zuiho Class

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

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

World War 2

1943 July 1
1943 December 1
1944 May 1

1944 November 1
1945 March 1
1945 August 1

1943 July 1
1943 November 1
1944 July 1

1944 September 1
1944 December 15
1945 May 1


Flight

Wing lift

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

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

Wing aspect ratio

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

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

Wing lift-to-drag coefficient

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

Width = ½ Area / Length

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

Aspect ratio is Length/Width.

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

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

Wing angle of attack

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

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


Air drag

The air drag force is

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

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

Maximum speed

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

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


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

Wing drag coefficient

             Cw

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

Gliding

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

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

Level flight

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

F   = Fgrav      Vertical force balance

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

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

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

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

Aircraft energy/mass =  e                              ~  M0

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

Range                =  X  =  V T                      ~  M0

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


Wingtip vortex

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

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


Combat aircraft

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

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

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

SR-71 Blackbird  3.3   30.6   78.0     25.9    302    5400          32   1966
MiG-25 Foxbat    2.83  20.0   36.7     20.7    200.2  1730        1186   1970
MiG-31 Foxhound  2.83  21.8   46.2     20.6    304    1450         519   1981
F-22A Raptor     2.51  19.7   38.0     19.8    312    2960   150   195   2005   *
F-15 Eagle       2.5   12.7   30.8     20.0    211.4  4000    28   192   1976
F-14 Tomcat      2.34  19.8   33.7     15.2    268    2960         712   1974
MiG-29 Fulcrum   2.25  11.0   20.0     18.0    162.8  1430    29  1600   1982
Su-35            2.25  18.4   34.5     18.0    284    3600    40    48   1988
F-4 Phantom II   2.23  13.8   28.0     18.3           1500        5195   1958
Chengdu J-10     2.2    9.8   19.3     18.0    130    1850    28   400   2005
F-16 Falcon      2.0    8.6   19.2     15.2    127    1200    15   957   1978
Chengdu J-7      2.0    5.3    9.1     17.5     64.7   850        2400   1966
Dassault Rafale  1.8   10.3   24.5     15.2    151.2  3700    79   152   2001
Euro Typhoon     1.75  11.0   23.5     19.8    180    2900    90   478   2003
F-35A Lightning  1.61  13.2   31.8     15.2    191    2220    85    77   2006   *
B-52              .99  83.2  220       15.0    608   14080    84   744   1952
B-2 Bomber        .95  71.7  170.6     15.2    308   11100   740    21   1997   *
A-10C Warthog     .83  11.3   23.0     13.7     80.6  1200    19   291   1972
Drone RQ-180          ~15              18.3          ~2200               2015   *
Drone X-47B       .95   6.4   20.2     12.2           3890           2   2011   *  Carrier
Drone Avenger     .70          8.3     15.2     17.8  2900    12     3   2009   *
Drone RQ-4        .60   6.8   14.6     18.3     34   22800   131    42   1998
Drone Reaper      .34   2.2    4.8     15.2      5.0  1852    17   163   2007
Drone RQ-170                           15                           20   2007   *

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

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

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

X-47B
RQ-170 Sentinel
MQ-9 Reaper


Missiles

Air to air missiles

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

                Mach   Range  Missile  Warhead  Year  Engine
                        km      kg       kg

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

Ground to air missiles

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

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

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

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

Ground to ground missiles

Tomahawk
Tomahawk

                Mach   Range  Missile  Warhead  Year  Engine        Launch
                        km      kg       kg                         platform

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

Hypersonic missiles

HTV-2
X-51
DARPA Falcon HTV-3

                   Speed   Mass  Range   Year
                   mach    tons   km

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

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

Trident 2
Peacekeeper
Minuteman 3

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

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