Because of an accident on launching, a rocket of unladen mass $M$ lies horizontally on the ground. It initially contains fuel of mass $m_{0}$, which ignites and is emitted horizontally at a constant rate and at uniform speed $u$ relative to the rocket. The rocket is initially at rest. If the coefficient of friction between the rocket and the ground is $\mu$, and the fuel is completely burnt in a total time $T$, show that the final speed of the rocket is

$$u \log \left(\frac{M+m_{0}}{M}\right)-\mu g T$$