Paper 4, Section I, C

Dynamics | Part IA, 2007

A rocket, moving vertically upwards, ejects gas vertically downwards at speed uu relative to the rocket. Derive, giving careful explanations, the equation of motion

mdvdt=udmdtgmm \frac{d v}{d t}=-u \frac{d m}{d t}-g m

where vv and mm are the speed and total mass of the rocket (including fuel) at time tt.

If uu is constant and the rocket starts from rest with total mass m0m_{0}, show that

m=m0e(gt+v)/um=m_{0} e^{-(g t+v) / u}

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