A research rocket is launched from rest at an altitude above the surface of a moon
PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Energy and Momentum of Rotating Systems · Energy and Momentum of Rotating Systems
Problem
A research rocket is launched from rest at an altitude h above the surface of a moon. The moon has mass M and radius R. Neglect any atmosphere and assume the rocket’s mass is negligible compared with the moon’s mass.
(a) Derive an expression, in terms of G, M, R, and h, for the minimum launch speed that allows the rocket to escape to ∞ (that is, the speed corresponding to E=0).
(b) Derive an expression for the smallest speed that will place the rocket in a closed (elliptical) orbit that never intersects the moon’s surface.
(c) The rocket is instead launched with a speed exactly half-way between the two speeds you derived in parts (a) and (b). Determine the maximum radial distance (apoapsis) the rocket will reach from the moon’s center, expressed symbolically in terms of G, M, R, and h, and show all reasoning.