Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2011

A rocket carries equipment to collect samples from a stationary cloud of cosmic dust. The rocket moves in a straight line, burning fuel and ejecting gas at constant speed uu relative to itself. Let v(t)v(t) be the speed of the rocket, M(t)M(t) its total mass, including fuel and any dust collected, and m(t)m(t) the total mass of gas that has been ejected. Show that

Mdv dt+vdM dt+(vu)dmdt=0M \frac{\mathrm{d} v}{\mathrm{~d} t}+v \frac{\mathrm{d} M}{\mathrm{~d} t}+(v-u) \frac{\mathrm{d} m}{\mathrm{dt}}=0

assuming that all external forces are negligible.

(a) If no dust is collected and the rocket starts from rest with mass M0M_{0}, deduce that

v=ulog(M0/M)v=u \log \left(M_{0} / M\right)

(b) If cosmic dust is collected at a constant rate of α\alpha units of mass per unit time and fuel is consumed at a constant rate dm/dt=β\mathrm{d} m / \mathrm{d} t=\beta, show that, with the same initial conditions as in (a),

v=uβα(1(M/M0)α/(βα))v=\frac{u \beta}{\alpha}\left(1-\left(M / M_{0}\right)^{\alpha /(\beta-\alpha)}\right)

Verify that the solution in (a) is recovered in the limit α0\alpha \rightarrow 0.

Typos? Please submit corrections to this page on GitHub.