Why can't I jump or fly if I pull my hair upwards, while I can jump using my legs?
The way I see it, when jumping someone lifts its body using the muscles in the legs (while the feet are standing still). If this lift generates enough speed, it continues even after the body is stretched. The fact that one cannot fly by pulling up his own hair I think it has something to do with the energy of an isolated system (jumping can be seen as pushing the earth away from you, while pulling hair doesn't interact directly with the earth).
There are also similar situations when there is some energy that seems to appear from nowhere, like when swinging alone. Using feet, one can accumulate potential energy and release it quickly, but why does this make the swing move?