A rocket propels itself forward by "burning" fuel (mixing fuel with oxygen) and emitting the resulting hot gases at high velocity out of a nozzle at the rear of the rocket. As a result of the combustion process the mass of the rocket continuously decreases.

1. Show that the motion of the rocket is governed by the following equation:

m(dv/dt) + v* _ex dm/dt = F(t),

where v = v(t) is the velocity of the rocket, m = m(t) is the mass of the rocket, v* _ex , is the velocity of the exhaust gas relative to that of the rocket, and F(t) is the external force acting on the rocket. Hint: Compute the momentum of the rocket at time t and time t + Δt. i.e. p(t + Δt) = p + Δp. The mass of the rocket will be reduced by Δm during this interval. Account for the momentum of the exhaust gas. Obtain dp/dt through a limiting operation.

2. Compute the maximum velocity, v _max , that the rocket can achieve under the following conditions. There is no external force acting on the rocket, F(t) = 0, the relative exhaust velocity v* _ex and rate of change of mass m are constant, the initial velocity is zero, v(0) = 0 the initial mass of the rocket is m _init , the final mass of the rocket (after fuel is expended) is m _{fm .} Given your result, what is the best way for a racket engineer to increase the maximum velocity?