| Here is what I have come up with. Using conservation of energy: When you jump your legs do work that gets converted in to kinetic energy and then potential energy. Assume that when you are shrunk down you maintain density. The potential energy that you have at the top of a jump is mgh. (mass * gravitational-constant * jump-height). When you scale your body down your mass goes down with the cube of the scaling, which I'll call k. So after scaling your energy would be mgh/(k^3) (m is your original mass). So how does the initial work change as you scale. The force (F) you can apply is roughly proportional to the cross section of your muscles. This changes with k^2. You integrate this over the path that your center of mass takes, which is going to change linearly with your scale k (d). That means that the work going in should be proportional to 1/k^3 as well! So we can make two equations: one before scaling: F * d = mgh (Leg force * leg movement = mass * gravity * jump height) and F * d / k^3 = m * g * h' / k^3 Which means that, to first order approximations jump height is independent of scale (h - h') and you should easily be able to jump out of the blender. |
> 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=...