The amount of fuel of the Saturn V was not sufficient

Using exhaust_velocity2.c source code, we write these values at the beginning:


#define AA 49.05 // aceleración soportada por los astronautas 5*g (letal en pocos segundos)
#define EV 3.0E3 // "exhaust velocity" de los gases 3 km/s (2.5 km/s en 1969)
#define MC 2941.12E3 // masa del cohete Saturn V (misión Apollo 11) 2941.12 toneladas
#define DT 0.001 // incremento de tiempo por iteración de cálculo 0.001 segundos


This means that we calculate the case much more favorable (although not possible) in which the Saturn V moves according to a single stage while has fuel with constant maximum acceleration 5*g supported by the astronauts, that is an acceleration surely too high so that the human being can survive more than 30 seconds, and with an "exhaust velocity" of the gases of 3 km/s, that is higher than the value used by the Saturn V engines in 1969 (about 2.5 km/s).

Specific impulse (Isp)
F-1 engines: Isp = 263 [s]
J-2 engines: Isp = 421 [s]
"Exhaust velocity" of the gases (EV)
F-1 engines: EV = 263*9.81 = 2580.03 [m/s]
J-2 engines: EV = 421*9.81 = 4130.01 [m/s]
F-1 engines uses about 78.7 % of the fuel


In this case much more favorable the inert weight that would be released in the stages if the rocket was multistage is considered fuel that we continue using with a single stage.

Remember that the gravitational force is conservative, and the increase of mechanical energy (potential energy + kinetic energy) of the rocket only comes from the fuel consumption of the rocket.


We compile it:

[Windows VC++]
cl /EHsc exhaust_velocity2.c

[Linux]
gcc exhaust_velocity2.c -o exhaust_velocity2


We execute it:

exhaust_velocity2 > results.txt


We obtain a file results.txt of 215 MBytes. At the end of this file we can read:

Tiempo : 250.07 [s]
Velocidad de salida de los gases respecto al cohete : 3000 [m/s]
Masa de gas expulsada por segundo : 806.064 [kg/s]
Masa restante del cohete : 49300.5 [kg]
Masa restante del cohete (porcentaje) : 1.67625 [%]
Empuje del cohete : 2.41819e+006 [N]
Fuerza total sobre el cohete : 2.08034e+006 [N]
Aceleración sufrida por los astronautas : 49.05 [m/(s^2)]
Aceleración del cohete : 42.1972 [m/(s^2)]
Velocidad del cohete : 10087.9 [m/s]
Velocidad del cohete (km/h) : 36316.3 [km/h]
Velocidad de escape : 10087.9 [m/s] MAL
Velocidad de escape (km/h) : 36316.3 [km/h]
Posición del cohete : 7.62431e+006 [m]
Altura del cohete sobre la superficie de la Tierra : 1.24531e+006 [m]
Altura del cohete sobre la superficie de la Tierra (km) : 1245.31 [km]
Distancia del cohete a la superficie de la Luna : 3.47037e+008 [m]
Distancia del cohete a la superficie de la Luna (km) : 347037 [km]


NOTE: The rocket thrust in this case is much greater than the supported by the engines during much of the ascent. We can reduce the thrust reducing the ascent acceleration, but then we increase the fuel consumption necessary to reach the Moon.


If we add the inert weight remaining and the fuel for the return we obtain:

25000+4305+4045+33200+51160+12250+8910 = 138870 [pounds] = 62990.4 [kg]

But

49300.5 [kg] < 62990.4 [kg]

Then we can say that in this case much more favorable there is not sufficient fuel to reach the Moon.

As consequence immediate in any case less favorable than this there is not sufficient fuel to reach the Moon.

Therefore in the Apollo 11 mission there was not sufficient fuel to reach the Moon.

Saturn V inert weight or dry weight or weight without fuel

S-IC Stage: 288750 [pounds] = 130975 [kg]
S-IC/S-II Interstage: 11465 [pounds] = 5200.44 [kg]
S-II Stage: 79920 [pounds] = 36251.1 [kg]
S-II/S-IVB Interstage: 8080 [pounds] = 3665.03 [kg]
S-IVB Stage: 25000 [pounds] = 11339.8 [kg]
Instrument Unit: 4305 [pounds] = 1952.72 [kg]

Spacecraft-LM Adapter: 4045 [pounds] = 1834.78 [kg]
Lunar Module: 9520 [pounds] = 4318.2 [kg]
Service Module: 10555 [pounds] = 4787.67 [kg]
Command Module: 12250 [pounds] = 5556.51 [kg]
Launch Escape System: 8910 [pounds] = 4041.51 [kg]

Saturn V total inert weight: 462800 [pounds] = 209923 [kg]


Saturn V total weight at ignition: 6484070 [pounds] = 2941.12E3 [kg]


462800/6484070*100 = 7.13749 [%]


AT LEAST A 7 % OF THE SATURN V MASS AT IGNITION WAS NOT FUEL.

Saturn V

exhaust_velocity2.c

#define AA 49.05 // aceleración soportada por los astronautas 5*g (letal en pocos segundos)
#define EV 3.0E3 // "exhaust velocity" de los gases 3 km/s (2.5 km/s en 1969)
#define MC 2941.12E3 // masa del cohete Saturn V (misión Apollo 11) 2941.12 toneladas
#define DT 0.1 // incremento de tiempo por iteración de cálculo 0.1 segundos

#define MT 5.97E24 // masa de la Tierra
#define RT 6.379E6 // radio de la Tierra
#define ML 7.35E22 // masa de la Luna
#define RL 1.739E6 // radio de la Luna
#define DTL 356400E3 // distancia Tierra-Luna (entre los centros de masas)
#define GG 6.67259E-11 // constante de gravitación universal



#include <math.h>
#include <stdio.h>
#include <stdlib.h>



int main(int argc, char **argv) {
  double mm; // masa restante del cohete
  double mg; // masa de gas expulsada en iteración
  double ff; // fuerza sobre el cohete
  double aa; // aceleración del cohete
  double vvi; // velocidad inicial del cohete
  double vvf; // velocidad final del cohete
  double rri; // posición inicial del cohete
  double rrf; // posición final del cohete
  double tt; // tiempo
  double rr2; // posición donde se anulan las fuerzas
  double vescape; // velocidad de escape

  rr2 = (DTL*(MT-sqrt(MT*ML)))/(MT-ML);

  tt = 0.0;
  mm = MC;
  vvi = 0.0;
  rri = RT;
  do {
    mg = (DT*mm*AA)/(EV+DT*AA);

    mm = mm - mg;

    ff = mg*EV/DT + GG*(mm*ML)/pow(DTL-rri,2.0) - GG*(mm*MT)/pow(rri,2.0);
    aa = ff/mm;
    vvf = aa*DT + vvi;
    rrf = aa*pow(DT,2.0)/2.0 + vvi*DT + rri;

    vescape = sqrt(2.0*GG*(MT*(1.0/rrf - 1.0/rr2) + ML*(1.0/(DTL-rrf) - 1.0/(DTL-rr2)))); // MAL

    tt = tt + DT;

    printf("Tiempo : %g [s]\n", tt);
    printf("Velocidad de salida de los gases respecto al cohete : %g [m/s]\n", EV);
    printf("Masa de gas expulsada por segundo : %g [kg/s]\n", mg/DT);
    printf("Masa restante del cohete : %g [kg]\n", mm);
    printf("Masa restante del cohete (porcentaje) : %g [%%]\n", mm/MC*100.0);
    printf("Empuje del cohete : %g [N]\n", mg*EV/DT);
    printf("Fuerza total sobre el cohete : %g [N]\n", ff);
    printf("Aceleración sufrida por los astronautas : %g [m/(s^2)]\n", AA);
    printf("Aceleración del cohete : %g [m/(s^2)]\n", aa);
    printf("Velocidad del cohete : %g [m/s]\n", vvf);
    printf("Velocidad del cohete (km/h) : %g [km/h]\n", vvf/1000.0*3600.0);
    printf("Velocidad de escape : %g [m/s]\n", vescape);
    printf("Velocidad de escape (km/h) : %g [km/h]\n", vescape/1000.0*3600.0);
    printf("Posición del cohete : %g [m]\n", rrf);
    printf("Altura del cohete sobre la superficie de la Tierra : %g [m]\n", rrf-RT);
    printf("Altura del cohete sobre la superficie de la Tierra (km) : %g [km]\n", (rrf-RT)/1000.0);
    printf("Distancia del cohete a la superficie de la Luna : %g [m]\n", DTL-rrf-RL);
    printf("Distancia del cohete a la superficie de la Luna (km) : %g [km]\n\n", (DTL-rrf-RL)/1000.0);

    vvi = vvf;
    rri = rrf;
  } while (vvf < vescape);

  return 0;
}

 

MAL