Question

Clearly there will be differences like air resistance; I'm not interested in that. It seems like you're working against gravity when you're actually running in a way that you're not if you're on a treadmill, but on the other hand it seems like one should be able to take a piece of the treadmill's belt as an inertial reference point. What's going on here?

Answer 1

For me it is axiomatic that machine miles are easier than real miles, but let's analyze the situation.

Assume the runner maintains a constant velocity up the hill, or remains stationary in the frame of the gym on the treadmill. In both cases the runner's acceleration is zero, so we know that her legs must provide a constant force with upward magnitude mg, and the they have to do this against a surface passing by at an angle ? below the horizontal and moving with a velocity v.

The kinematics in the runners frame of reference look the same. This is not the cause of the difference in perceived difficulty.

I have always assumed that the difference in difficulty was two fold:

Wind resistance is not really negligible.
The treadmill presents a very uniform reliable surface and the runner need not lift her legs as high to insure non-tripping progress.

Also modern treadmill are designed to be relatively easy on the knees, and the accomplish this by having a slightly springy feeling which presumably returns some energy to the runner.