# 2.25 Problem Sets - Section 5

## Problem 5.17: Rocket accellerating against gravity

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After its second booster has been fired, a space vehicle finds itself
outside the earth's atmosphere moving vertically upward against gravity
at speed *V _{0}*.
Its total mass at that point is

*M*

_{0}. At t = 0, the vehicle's third stage is turned on and the rocket burns propellant at a mass rate of kg/s. Show that if the gravitational acceleration remains essentially constant at the rocket during the firing, the rocket’s velocity

*V(t)*after time

*t*will be given by

where *M(t)* is the mass of the system at time *t.* Note
that the pressure of the gas at the rocket exit plane, *p*_{e},
will not be zero as it is in the ambient space, since the rocket exhaust
is supersonic and hence the pressure at the exit is not in balance with
the ambient pressure. The answer given above neglects the effect of
the exit plane pressure on thrust.