A rocket carries equipment to collect samples from a stationary cloud of cosmic dust. The rocket moves in a straight line, burning fuel and ejecting gas at constant speed relative to itself. Let be the speed of the rocket, its total mass, including fuel and any dust collected, and the total mass of gas that has been ejected. Show that
assuming that all external forces are negligible.
(a) If no dust is collected and the rocket starts from rest with mass , deduce that
(b) If cosmic dust is collected at a constant rate of units of mass per unit time and fuel is consumed at a constant rate , show that, with the same initial conditions as in (a),
Verify that the solution in (a) is recovered in the limit .