Showerthought: what's the simplest way to tell that the human body is less than 50% efficient at converting chemical energy to mechanical work via running? I think it's that running uphill makes you warmer than running downhill at the same speed.

When running up a hill at mechanical power p and efficiency f, you have to exert p/f total power and so p(1/f - 1) is dissipated as heat. When running down the hill you convert p to heat. p(1/f - 1) > p implies that f > 0.5.

Maybe this story is wrong somehow. I'm pretty sure your body has no way of recovering your potential energy on the way down; I'd expect most of the waste heat to go in your joints and muscles but maybe some of it goes into your shoes.