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 $u$ relative to itself. Let $v(t)$ be the speed of the rocket, $M(t)$ its total mass, including fuel and any dust collected, and $m(t)$ the total mass of gas that has been ejected. Show that

$$M \frac{\mathrm{d} v}{\mathrm{~d} t}+v \frac{\mathrm{d} M}{\mathrm{~d} t}+(v-u) \frac{\mathrm{d} m}{\mathrm{dt}}=0$$

assuming that all external forces are negligible.

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

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

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

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

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