Classical Mechanics (Kibble & Berkshire) · Problem 10.10
The angular momentum principle — T.W.B. Kibble, F.H. Berkshire
Physics — Mechanics · rocket-motion
Problem
Two-stage rocket A two-stage rocket has a first stage of initial mass M1, of which (1−η)M1 is fuel, a second stage of initial mass M2, of which (1−η)M2 is fuel, and an inert payload of mass m0. In each stage, the exhaust gases are ejected with the same speed u. The rocket is initially at rest in free space. The first stage is fired and, on completion, the first stage carcass (of mass ηM1) is discarded. The second stage is then fired. Find an expression for the final speed V of the rocket and deduce that V will be maximised when the mass ratio α=M2/(M1+M2) satisfies the equation
α2+2βα−β=0,
where β=m0/(M1+M2). [Messy algebra.]
Show that, when β is small, the optimum value of α is approximately β1/2 and the maximum velocity reached is approximately 2ulnγ, where γ=1/η.