[psykotic]

Your conclusion is right but your argument appears to be lacking.

> You integrate this over the path that your center of mass takes, which is going to change linearly with your scale k (d).

But if that's how you define d, then it is the height h! Here you are assuming that d scales linearly with k. Later you are saying that h is invariant with respect to k. Which is it?

Jumping is impulsive, not sustained, so your force times distance formulation doesn't seem appropriate.

Thompson has a nice analysis in his classic treatise On Growth and Form, which is all about dimensional analysis applied to biology. Here is the relevant excerpt:

http://books.google.com/books?id=8FrORfyp7bsC&pg=PA36#v=...

[robobenjie]

I think I wasn't quite clear in what I meant. d is the distance that your center of mass changes while your feet are in contact with the ground (While you are doing work) and h is the height that your center of mass changes while you are in the air.

Saying a force is impulsive means that you making certain assumptions to make your calculation easier. It doesn't change the fact that W = integral of force dotted with displacement. It is true that I am making a big approximation where I say that the force is constant over the jump, making the integral evaluate to f * x.

Thanks for the link. I think this kind of stuff is very interesting.