A probe of mass m is placed in an elliptical orbit about a planet of mass M. The semi-major axis of the orbit is a, and at an arbitrary point in its motion the radial distance from the planet’s center is r.
In the problems below, treat the probe’s mass as negligible compared with M.
(a) Starting from the expressions for kinetic energy K=21mv2 and gravitational potential energy U=−rGMm, derive an expression for the probe’s kinetic energy K in terms of G, M, m, r, and a only. Show all steps in your derivation.
(b) Using your result from part (a), determine the radial location in the orbit where the probe attains its maximum kinetic energy and the location where it attains its minimum kinetic energy. Explain your reasoning.
(c) For each location identified in part (b), describe the corresponding values (maximum, minimum, or neither) of the gravitational potential energy and of the total mechanical energy. Provide justification based on your expression from part (a) and on energy conservation.