Fter its second booster has been fired, a space vehicle finds itself outside the earth’s atmosphere,

Fter its second booster has been fired, a space vehicle finds itself outside the earth’s atmosphere, moving
vertically upward at a speed v0 against gravity g. its total mass at that point is m0. at t = 0, the vehicle’s
third stage is turned on and the rocket burns propellant at a mass rate mrkg/s, ejecting gas from the exit
plane (area ae) at speed ve relative to the rocket.
show that if the gravitational acceleration remains essentially constant at the vehicle during the rocket firing,
the velocity v (t) of the vehicle after time t will be given by
m0 g[m0 − m(t)] v (t) − v0 = ve ln − m(t) m˙ r
where m(t) is the mass of the system at time t. assume that although the pressure of the gas at the rocket
exit plane is pe (the rocket exhaust is supersonic, and hence the pressure at the exit is not balanced with
the zero pressure of space), the effect of the finite exit plane pressure on the thrust is negligible.

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