Find the minimum energy needed to place a satellite of mass m from the surface of a planet (mass M, radius R) into a circular orbit at altitude 2R (orbital radius 3R).

5GmM/6R
2GmM/3R
GmM/2R
GmM/3R

Correct answer: 5GmM/6R

Solution

Total energy in circular orbit of radius r = 3R is E_orbit = -GMm/(2r) = -GMm/(6R). On the surface (at rest) energy is E_surface = -GMm/R. Minimum energy = E_orbit - E_surface = -GMm/6R + GMm/R = GMm(1 - 1/6)/R = 5GMm/6R.

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