What's the difference between running up a hill and running up an inclined treadmill?

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 $\theta$ 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.

The word 'difference' may be ambiguous, but let's look at the situation from several points of view.

Energy balance: Indeed, your potential energy does increase in case 1 and not in case 2. Muscles clearly perform the same work, so the energy must go somewhere? Yes, to the electric grid. The treadmill device's engine, to maintain constant velocity, will consume less electrical power to do so (or might even push energy back into the grid, in case of an efficient motor) because your legs are actually pulling it now downwards. If you do the math, you see it exactly compensates.

Muscle work: work, in thermodynamical sense, is not just F*dx. One has to take a machine and consider all interfaces. For example, a spring or a muscle have two ends, and dx in the formula is actually the difference between two paths. Muscle expansion/contraction will be the same, and so is the force. Therefore, they are doing the same work. This work is, the amount of chemical internal energy stored within the muscle converted to mechanical work.

Assume that the hill and the treadmill have the same angle of elevation (are inclined identically), and that two identical persons A and B are running on them at the same speed $v$. Here the speed of the person B running on the treadmill is obviously zero with respect to the ground, but we will consider the speed of the treadmill's belt to be $-v$.

Let's assume that B and the treadmill are now in a truck which is running up the hill parallel to A, and with the same speed $v$ as A. The truck should be arranged so that B and the treadmill are not inclined while the truck is running up the hill. By our hypothesis, the speed of the upper part of the belt is zero with respect to the ground. This will not affect the effort done by the runner B, because the truck is moving with constant velocity. That is, there are no extra forces caused by inertia, because the truck doesn't accelerate, decelerate or change direction.

By looking now at both runners A and B we see that they are moving parallel with the same speed. They can even do the same moves in synchrony. The angle elevation is the same for both of them too. But B may even not realize that he is climbing a hill, he may think that he is in a room with no windows, which doesn't move. The conditions are identical. So, there is no difference between them.

If our intuition still saids that the treadmill guy is burning less calories, let's imagine that the road on which the person A is running up the hill is a very long treadmill belt. Imagine that underneath the belt there is a treadmill which is doing two things: is moving with the same speed $v$ towards the top of the hill, and the upper side of the belt is moving backwards with $-v$, so that the belt appear to be fixed with respect to the ground. From the outside, the belt doesn't move (and for runner A too). Now, it should be clear that there is no difference.

In the above I assumed that there is no wind, the treadmill and the runners are moving with uniform speed, there is no difference between the runners. I also assumed that gravity is not weaker towards the top of the hill.