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

$$m(t) \frac{\mathrm{d} v}{\mathrm{~d} t}+U(t) \frac{\mathrm{d} m}{\mathrm{~d} t}=-m(t) g$$

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

State the dimensions of all the quantities involved in your expression for $v(t)$, 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.]