Initially a total mass of a rocket is M, of which kM is the mass of the fuel. Starting from rest, the rocket gives itself a constant vertical acceleration of magnitude g by ejecting fuel with constant speed u relative to itself.

a) If m denotes its remaining mass at time t, show that the rate of decrease of m with respect to t is 2mg/u and deduce that m=Me^(-2gt/u).

b) Show that the kinetic energy of the rocket when the fuel is exhausted is (1/8)Mu^2(1-k)[ln(1-k)]^2

c) Show the value of k for which energy is a maximum is 1-e^-2