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Imagine two identical aircraft with identical drag, wing area and thrust but different payloads, such that one has a max straight and level speed of 205mph and the other 210mph. Initially, at t=0 the slower aircraft is 200ft vertically above the faster aircraft, and then starts to dive at a CONSTANT RATE in the most efficient way (I'll return to what I mean by this in a minute) such that at some point ahead of the starting point, it ends up at the same height as the faster aircraft.

Clearly, immediately after t=0, the faster aircraft will be ahead of the slower one when observed from above or below, but as the slower aircraft descends, it will accelerate such that it MAY overtake the faster one. Its speed willl increase until it tops out at some new higher speed. Once it levels out, it will start to slow again back towards its initial speed of 205 mph.

By "most efficient way", I mean that the slower aircraft uses its excess potential energy such that it stays ahead of the faster one for the longest time possible after t.

The questions I'm trying to answer are: what is the max speed the slower aircraft will reach? How far will it have travelled when it reaches this speed? Will it be ahead of or behind the faster aircraft at this point and by how much ? For a given distance x from the starting point that is beyond the point at which the slower aircraft levels out, how do I plot the relative positions of the two aircraft?

This is actually a real world problem that came about in an air race. If you're interested, I'd be happy to tell you about it, its quite fun!

Thanks,

Craig