A rocket of initial total mass (m_{0}) is fired vertically up from the north pole and accelerates

Question:

A rocket of initial total mass $m_{0}$ is fired vertically up from the north pole and accelerates until the fuel, which burns at a constant rate, is exhausted. The relative nozzle velocity of the exhaust gas has a constant value $u$, and the nozzle exhausts at atmospheric pressure throughout the flight. If the residual mass of the rocket structure and machinery is $m_{b}$ when burnout occurs, determine the expression for the maximum velocity reached by the rocket. Neglect atmospheric resistance and the variation of gravity with altitude.

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