Equations of motion of a rocket

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Q9. At time t, the mass of a rocket is M(1-kt), where M and k are constants. At time t, the rocket is moving with speed v vertically upwards near the Earth's surface against constant gravity. Burnt fuel is expelled vertically downwards at speed u relative to the rocket.
a) Show that (1-kt)dv/dt = ku-g(1-kt)
b) Given that v=0 when t=0 show that v = -u ln(1-kt)-gt

Solution : As the fuel burns, total mass of the rocket decreases gradually and its velocity increases. Total mass at any instant t is given by m(t) ...