I have a question concerning gravitation potential energy and rockets under an unusual situation. Let a rocket be in a gravity well of a massive object such as a planet. For simplicity, assume that the rocket is in a vacuum. (we can add air effects later). The rocket engine thrust is dynamically adjusted to exactly counter the pull of gravity. Adjustment is needed as the mass of the rocket body will be changing. Three things could happen depending on initial velocity. The rocket could hover at a fixed altitude. The potential energy of the rocket body will be decreasing as the altitude remains constant and the mass is decreasing. There would be no kinetic energy associated with the rocket body itself. The rocket could be climbing at a constant rate as it coasts upward. The kinetic energy of the rocket body would be decreasing as the mass decreased and the velocity was constant. The potential energy could either increase or decrease depending on whether the altitude increased faster than the mass decreased. The rocket could be falling at a constant rate as it coasts downward. The kinetic energy of the rocket body would be decreasing as the mass decreased and the velocity was constant. The potential energy would be decreasing as both the altitude and mass decreased. I am interested in the change in potential energy of the rocket body. Assuming that the magnitude of the velocities are the same in cases 2 and 3 the kinetic energy/time profiles will be identical. The expenditure of energy from the rocket would appear to be identical. What is the source of sink of the potential energy change? I have tried to puzzle this our considering the mass ejected by the rocket motor but so far have come up blank. I would appreciate any thoughts on this question.