Dynamics | Part IA, 2003

Because of an accident on launching, a rocket of unladen mass MM lies horizontally on the ground. It initially contains fuel of mass m0m_{0}, which ignites and is emitted horizontally at a constant rate and at uniform speed uu relative to the rocket. The rocket is initially at rest. If the coefficient of friction between the rocket and the ground is μ\mu, and the fuel is completely burnt in a total time TT, show that the final speed of the rocket is

ulog(M+m0M)μgTu \log \left(\frac{M+m_{0}}{M}\right)-\mu g T

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