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Fuel
Black: Carbon    White: Hydrogen    Red: Oxygen

Methane (Natural gas)
Ethane
Propane
Butane (Lighter fluid)
Octane (gasoline)
Dodecane (Kerosene)

Hexadecane (Diesel)
Palmitic acid (fat)
Ethanol (alcohol)

Glucose (sugar)
Fructose (sugar)
Galactose (sugar)
Lactose = Glucose + Galactose
Starch (sugar chain)
Leucine (amino acid)

ATP (Adenosine triphosphate)
Phosphocreatine
Nitrocellulose (smokeless powder)
TNT
HMX (plastic explosive)

Lignin (wood)
Coal

Medival-style black powder
Modern smokeless powder
Capacitor
Lithium-ion battery
Nuclear battery (radioactive plutonium-238)
Nuclear fission
Nuclear fusion
Antimatter


Energy and power

Energy          =  E          Joules
Time            =  T          seconds
Power           =  P  =  E/T  Watts
Mass            =  M          kilograms
Energy/Mass     =  e  =  E/M  Joules/kilogram
Power/Mass      =  p  =  P/M  Watts/kilogram

Energy per mass

           Energy per mass   Carbons
             (MJoule/kg)

Antimatter   90000000000
Fusion bomb    250000000                Maximum for a deuterium+tritium fusion bomb
Fission bomb    83000000                Maximum for a uranium bomb
Nuclear battery   589000                Strontium-90, beta decay, 29 year half life
Hydrogen             141.8     0
Methane               55.5     1
Ethane                51.9     2
Propane               50.4     3
Butane                49.5     4
Octane                47.8     8
Gasoline              47       8±2
Kerosene              46      12±2
Diesel                46      16±3
Lubricating oil       46      36±16
Fat                   37      20±10     9 Calories/gram
Ethanol               29       2        7 Calories/gram
Sugar                 17       6        4 Calories/gram
Protein               17                4 Calories/gram
Coal                  32
Wood                  22
Pure carbon           32.8     1
Plastic explos.        8.0
Smokeless powder       5.2              Modern gunpowder
TNT                    4.7
Aluminum-air battery   4.6
Black powder           2.6              Medieval gunpowder
Lithium-ion battery    1.0
Phosphocreatine         .137
ATP                     .057            Adenosine triphosphate
Supercapacitor          .026
Aluminum capacitor      .012
Spring                  .0003


Vehicle power

Tesla Roadster

The energy sources that can be used by vehicles are:

              Energy/Mass   Power/mass   Energy/$   Rechargeable   Charge   Maximum charging
               MJoule/kg     Watt/kg     MJoule/$                  time          cycles

Gasoline            45                   60
Battery, aluminum    4.6       130                      No
Battery, lithium-ion  .8      1200         .010         Yes        1 hour      1000
Supercapacitor        .026   14000         .0005        Yes        Instant     Infinite
Aluminum capacitor    .010   50000         .0001        Yes        Instant     Infinite

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.


Lithium-ion batteries

The properties of the best commercial lithium ion batteries are:

Energy/Mass     =    .8  Joule/kg
Power/Mass      =  1200  Watt/kg
Energy/$        =    .01 MJoule/kg
Density         =   3.5  gram/cm3
Recharges       =1000
Shelf life      =   1.0  year
Voltage         =   3.7  Volt
Energy/Mass and Power/Mass are an engineering tradeoff. One can be increased at the expense of the other.
Battery energy and power

Battery energy is often given in "Watt hours" or "Ampere hours".

Voltage          =  V         Volts
Charge           =  C         Coulombs    (1 Amphour = 3600 Coulombs)
Electric current =  I         Amperes
Electric power   =  P  =  VI  Watts
Time             =  T         seconds
Energy           =  E  =  PT  Joules
                       =  CV  Joules
1 Watt hour = 3600 Joules = 1 Watt * 3600 seconds

1 Amp hour = 3600 Coulombs = 1 Coulombs/second * 3600 seconds

A battery with a voltage of 3.7 Volts that delivers 1 Ampere for 1 hour has an energy of
Energy = 1 Ampere * 3.7 Volts * 3600 seconds = 13320 Joules


Battery packs

A single battery is a "cell" and a set of cells is a "pack". Packs are used to multiply the energy and power of cells.

Battery packs are notorous for catching fire, but cell technology has reached the point where it's now possible to make safe battery packs, and the design is simple enough so that anyone can construct their own packs.

Cells can be combined in series and/or parallel. Connecting in series multiples voltage, and voltage is helpful for achieving high power in a motor.

Connecting in series is easier than in parallel. If it's possible to achieve the required power without parallelization then one should do so, and this is usually possible with modern cells.

Series packs have the advantage that the cells can easily be extracted and charged individually, and cells can be interchanged between packs. One can also construct a set of series packs and swap them in like gun clips.

High power electric bikes use a voltage of 72 Volts. If we use one series array of C cells then a pack provides 4440 Watts and 1.2 MJoules. Any electric device requiring less than this much power can be powered by a series pack.

The properties of a modern high-power cell are:

Type         =  "C"
Voltage      =   3.7 Volts
Energy       =  60   kJoules
Power        = 155   Watts
Mass         =  92   grams
Energy/mass  = 650   kJoules/kg
Power/mass   =1680   Watts/kg
Current      =  42   Amperes
Manufacturer = "Basen"
When the cells are connected in series the values for voltage and power are:
Cells   Voltage    Power
         Volts     kWatts

   1      3.7        .15
   2      7.4        .30
   3     11          .45     Electric kick scooter
   4     15          .60
   6     24          .90     Electric bike
  10     36         1.5
  20     72         3.0      Compact electric car
  96    356        15.0      Large electric car

Battery pack strategy

Electric bike motors use either 36, 48, or 72 Volts. The following table shows how to build a battery pack for each motor power.

Power  Volts  Cells  Series  Parallel  Current   Cell      Cell    Cell  Cell  Cell
kWatt                                  Amperes  Amperes  Amphours   $    type  ID#

   .5   36     10
   .75  36     10      10       1         21      25       2.1      4     A    LG HD4
  1.5   48     13      13       1         31      30       2.0      4.5   A    Sony VTC4
  3     72     20      20       1         42      60       4.5      4.5   C    Basen
  6     72     40      20       2         83     120       4.5      4.5   C    Basen
 12     72     80      20       3        167     180       4.5      4.5   C    Basen

Cells     Total number of cells, equal to the number of cells connected in series
          times the number of cells connected in parallel.
Series    Number of cells connected in series. For example, 20 batteries
          with 3.6 volts each connected in series produces a voltage of 72 Volts.
Parallel  Number of cells connected in parallel.
Current   Current required to provide given power
Cell      Maximum current of a cell

Gigafactory

The Gigafactory in Nevada has a production target for 2020 of:

Battery production   =  200 TeraJoules/year
Energy of one car    =  310 MJoules           (Tesla Model S)
Cars supplied/year   =  .64 million
The Solar City Factory produces solar panels.
Panel production  =  1.0 GWatts/year
Panel duty factor =  .25        (Average solar intensity over peak solar intensity)
Effective power production  =  .25 GWatts/year

Capacitors
Voltage          =  V             Volts
Capacitance      =  C             Farads
Total energy     =  E  =  ½ C V2  Joules
Effective        =  Ee =  ¼ C V2  Joules
Not all of the energy in a capacitor is harnessable because the voltage diminishes as the charge diminishes, hence the effective energy is less than the total energy.
Capacitance
A   =  Plate area
Z   =  Plate spacing
Ke  =  Electric force constant  =  8.9876e9 N m2 / C2
Q   =  Max charge on the plate     (Coulombs)
Emax=  Max electric field       =  4 Pi Ke Q / A
V   =  Voltage between plates   =  E Z     =  4 Pi Ke Q Z / A
En  =  Energy                   =  .5 Q V  =  .5 A Z E2 / (4 π Ke)
e   =  Energy/Volume            =  E / A Z =  .5 E2 / (4 π Ke)
q   =  Charge/Volume            =  Q / A / Z
C   =  Capacitance              =  Q/V     =  (4 Pi Ke)-1 A/Z   (Farads)
c   =  Capacitance/Volume       =  C / A / Z =  (4 Pi Ke)-1 Emax2 / V2
Eair=  Max electric field in air=  3 MVolt/meter
k   =  Dielectric factor        =  Emax / Eair


Continuum                                                 Macroscopic

Energy/Volume  =  .5 E2  / (4 Pi Ke)           <->        Energy = .5 C V2
               =  .5 q V                                         =  .5 Q V
c              =  (4 Pi Ke)-1 Emax2  / V2      <->        C      = (4 Pi Ke)-1 A / Z

A capacitor can be specified by two parameters:
*)   Maximum energy density or maximum electric field
*)   Voltage between the plates

The maximum electric field is equal to the max field for air times a dimensionless number characterizing the dielectric

Eair =  Maximum electric field for air before electical breakdown
Emax =  Maximum electric field in the capacitor
Rbohr=  Bohr radius
     =  Characteristic size of atoms
     =  5.2918e-11 m
     =  hbar2 / (ElectronMass*ElectronCharge2*Ke)
Ebohr=  Bohr electric field
     =  Field generated by a proton at a distance of 1 Bohr radius
     =  5.142e11 Volt/m
Maximum energy density  =  .5 * 8.854e-12 Emax2


                         Emax (MVolt/m)   Energy density
                                            (Joule/kg)
Al electrolyte capacitor     15.0            1000
Supercapacitor               90.2           36000
Bohr limit               510000            1.2e12            Capacitor with a Bohr electric field

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.
Commercial supercapacitors
                Mass     Energy      E/M    Power   P/M   Price  Energy/$    C     Voltage
                 kg      kJoule     kJ/kg   kWatt  kW/kg    $    kJoule/$  Farads   Volts

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

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