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

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

where $v$ and $m$ are the speed and total mass of the rocket (including fuel) at time $t$.

If $u$ is constant and the rocket starts from rest with total mass $m_{0}$, show that

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