Paper 4, Section I, A

Dynamics and Relativity | Part IA, 2009

A rocket moves vertically upwards in a uniform gravitational field and emits exhaust gas downwards with time-dependent speed U(t)U(t) relative to the rocket. Derive the rocket equation

m(t)dv dt+U(t)dm dt=m(t)gm(t) \frac{\mathrm{d} v}{\mathrm{~d} t}+U(t) \frac{\mathrm{d} m}{\mathrm{~d} t}=-m(t) g

where m(t)m(t) and v(t)v(t) are respectively the rocket's mass and upward vertical speed at time tt. Suppose now that m(t)=m0αt,U(t)=U0m0/m(t)m(t)=m_{0}-\alpha t, U(t)=U_{0} m_{0} / m(t) and v(0)=0v(0)=0. What is the condition for the rocket to lift off at t=0t=0 ? Assuming that this condition is satisfied, find v(t)v(t).

State the dimensions of all the quantities involved in your expression for v(t)v(t), and verify that the expression is dimensionally consistent.

[ You may assume that all speeds are small compared with the speed of light and neglect any relativistic effects.]

Typos? Please submit corrections to this page on GitHub.