(a) A rocket moves in a straight line with speed $v(t)$ and is subject to no external forces. The rocket is composed of a body of mass $M$ and fuel of mass $m(t)$, which is burnt at constant rate $\alpha$ and the exhaust is ejected with constant speed $u$ relative to the rocket. Show that

$$(M+m) \frac{d v}{d t}-\alpha u=0 .$$

Show that the speed of the rocket when all its fuel is burnt is

$$v_{0}+u \log \left(1+\frac{m_{0}}{M}\right)$$

where $v_{0}$ and $m_{0}$ are the speed of the rocket and the mass of the fuel at $t=0$.

(b) A two-stage rocket moves in a straight line and is subject to no external forces. The rocket is initially at rest. The masses of the bodies of the two stages are $k M$ and $(1-k) M$, with $0 \leqslant k \leqslant 1$, and they initially carry masses $k m_{0}$ and $(1-k) m_{0}$ of fuel. Both stages burn fuel at a constant rate $\alpha$ when operating and the exhaust is ejected with constant speed $u$ relative to the rocket. The first stage operates first, until all its fuel is burnt. The body of the first stage is then detached with negligible force and the second stage ignites.

Find the speed of the second stage when all its fuel is burnt. For $0 \leqslant k<1$ compare it with the speed of the rocket in part (a) in the case $v_{0}=0$. Comment on the case $k=1$.