Question

A single-stage rocket of mass 314 metric tons (not including fuel) carries a payload of 3150 kg to low-Earth orbit. The exhaust speed of the rocket's cryogenic propellant is 3.13 ? 10³ m/s. (Ignore the effects of gravity. Assume the payload is counted in the given mass of the rocket.)

(a) If the speed of the rocket as it enters orbit is 7.70 km/s, what is the mass of propellant used during the rocket's burn?

(b) The rocket is redesigned to boost its exhaust speed by a factor of two. What is the mass of propellant used in the redesigned rocket to carry the same payload to low-Earth orbit?

Answer 1

a)

For a single stage rocket,

velocity change of rocket V = V_e ln (M+m/M)

where V_e is exhaust velocity = 3.13 x 10³ m/s

dry mass of rocket (not including fuel) M = 314 x 10³ kg

m is mass of propellen

V = V_e ln (M+m/M)

7.70 x 10³ = 3.13 x10³ ln (314 x 10³ + m / 314 x 10³)

2.46 = ln (313 x10³ + m / 313 x 10³)

e^2.46 = (313 x 10³ + m / 313 x 10³ )

11.71 =(313 x 10³ +m / 313 x 10³ )

m = 3351 x 10³ kg =  3.351 x 10^6 kg

b)

new exhaust speed v_e= 2 x 3.13 x 10³ m/s

V = V_e ln(M+m/M)

7.17 x 10³ = 2 x 3.13 x 10³ ln(314 x10³ + m / 314 x 10³)

1.145 = ln(314 x 10³ + m / 314 x 10³)

e^1.145 = (314 x 10³ + m / 314 x 10³ )

3.144 = (314 x 10³ + m / 314 x 10³ )

m = 987.1 x 10³ kg = 9.871 x 10^5 kg