Connecting the prop and the wheels changes completely how the cart works and will interact with the air and treadmill. I don't see what you're testing other than the bluff forces.
I confess, I didn't grok what you were meaning here.
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I am interested in this because it's something opposite from common encounters. Windmilling props are not beneficial.

And here's a prop hooked up to move something, but... it moves the thing the "wrong" way.

To get a handle on what happens when, I started with the cart on wheels, and the prop freewheeling.
Then noted what occurred.
The prop rotates/windmills as a normal prop does, and creates only drag doing this. Just like real life.

This drag moves the cart downwind...only.. the direction the cart faces, or the orientation of the prop on the shaft machs nichts to that. Downwind it goes.
Fixing the prop so it can't rotate, same thing.
The prop in that configuration is a sail.
On the treadmill running to oppose the motion of the cart pushing it away from the fan, the car will move downwind regardless of the treadmill belt speed.
It needn't run off the far end of the belt, the motion can be restrained to keep it on the belt.
The cart should keep the thread tight.
Connecting the prop and the wheels moves the whole thing out of simple aerodynamics and forces the interaction with the ground.
Then, the usual rotation of the prop with the wind becomes tricky.
That rotation is "wrong". Turning the prop in the expected windmilling rotation direction, the cart backs up.
So there's an initial resistance to moving, when only the air's effect and the rotation of the prop are considered. But... the prop being constrained from rotating in the expected direction by the connection to the wheels and the ground can act as a sail, and without rotating, the wind force on the prop can overcome the rolling friction of the wheels, and the cart begin to move downwind. The wheels then relieve the rotation constraint on the normal direction the prop would naturally rotate when windmilling, and the cart accelerates. The combined effect of the sail force and the wheels driving the cart add to each other until the ground speed can exceed the wind speed, and the propellor begin to operate as a propellor instead of a sail.
This point still makes my head hurt.

I'd kinda expect the cart to get to wind speed and no faster.
That it goes faster is obvious.
Why it should.. that ain't so obvious.

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The Treadmill-airplane test.
http://www.youtube.com/watch?v=4owlyCOzDiE
Note the added marks on the belt..
