I stumbled across this seemingly simple question that really stumped me on further thought:
A rocket is intended to leave the Earth's gravitational field. The fuel in its main engine is a little less than necessary, and an auxiliary engine, only capable of operating for a short amount of time, has to be used as well. When is it best to switch on the auxiliary engine: at take-off, or when the rocket has nearly stopped with respect to the Earth, or does it not matter?
My understanding is that to escape the earth's field, there must be sufficient kinetic energy so that the total energy is positive. My first instinct was that, assuming the auxiliary engine operates at a constant power, it will cause the same change in energy regardless of how far it is from the earth, therefore it doesn't matter. However, it seems that where this change in energy occurs is actually important. Can someone help me understand what the correct answer is?