Imagine the wind as a stream of tennis balls, all moving at close to the same speed. (This is in some ways a poor model for actual wind, where the velocity distribution of the component particles is much wider, and only net pressure matters on the macro scale. But it does simplify things. I'll point out, later, where the simplification might cause some trouble.)
Average molecular speed is 500m/s. So the balls are in 3D random motion at 500m/s.
What you're saying is, you want to see, from the ground reference frame, forward-moving tennis balls (wind) catching up to and hitting some part of the cart, being slowed down and so providing a push to make it go faster or at least maintain speed.
From the ground, there is an average drift velocity called "wind speed".
So, we start by building a cart with a big tennis racket mounted in the back, face-on to the wind. Put this cart into the tennis-ball current, and it will start moving, as tennis balls hit the racket.
The balls will decelerate against the tennis racket. The net force will drive the vehicles in opposition to that force.
But, when it gets close to wind speed, fewer tennis balls per unit time catch up to the racket and hit it, and when they do the relative speed is very small so they no longer give it much of a push. When the total net force is no longer greater than frictional losses, the cart won't go any faster. If you were to push the cart to wind speed, no tennis balls would hit the racket at all, and if you pushed it faster, tennis balls would start hitting the front of the racket and slowing the cart down.
The force will fall with increasing velocity wrt the wind. The cart will reach terminal velocity below WS, wheels or not, because the balls will loose momentum as a result of transfer to the racket, and so slow. The accumulation produces a high pressure zone in front of the racket.
The way to get around this involves two factors: one, the face of the racket has to be placed at an angle; and two, the racket has to be moving laterally relative to the cart; that is, in a plane perpendicular to the cart's trajectory.
We could rig some kind of mechanical forehand-arm to sweep the tennis racket side to side across the back of the cart while holding it at one angle when going left to right and a symmetrical angle when going right to left. But it's easier to just spin the racket around by the handle (on an axis perpendicular to the handle).
As the air has all but come to a stop at WS, to move the ball in any way will require energy from the device that accelerates them away.
If a static deflector were used, then less force will be derived from the former tailwind, as more of its KE is remained with the deflected balls.
If that sounds a bit unbalanced, we can start with a two-headed racket, with a head at each end of the handle, and spin it by the middle of the handle. The axis is perpendicular to the handle, and oriented parallel to the cart's trajectory.
That takes care of the motion, now we must consider the angle. One possible angle (call it zero degrees) is for the racket heads to be flat in the plane that they're spinning in. But that gives us no advantage (or any real difference at all) from the single stationary racket.
All need power for the same reason as above.
Another possibility is for the racket heads to be turned 90 degrees from the previous case. That turns the device into essentially a sideways paddle wheel. That would also be ineffective: it would bat any tennis balls that caught up with the cart (or that the cart caught up with) off in all different directions but it would only ever accelerate them in the plane perpendicular to the cart's trajectory, giving no benefit regardless of the apparent wind speed.
Nature is quite old and won't fall for tricks.
We can use Newton's impact theorem. Not accurate, but good enough.
Each air molecule strikes the prop over impact time, t
f =ma, so
f*t = ma*t (1)
V= at. ( the wind velocity derived from the acceleration)
f = (mV)/t (2)
During the impact time, N molecules will hit the prop, so the force is
Fd = N*f. (3)
To find the force, we need to know N, and the area hit by those molecules.
The prop has the disk area A. Instant deceleration is impossible, so as the molecules decelerate over time 't', the wind moves distance x. That is, deceleration takes place over a small but finite distance x.
Consider an area, A, of the blade.
Distance = velocity*time
x = V*t
Volume = V*t*x
Combining
N = A*V*t (Number/Volume)
Multiply N by the single molecule force, f.
Fd = m*(Number/Volume)*A*V^2
The density of the molecules (number/volume) can be found from p, the air density. The final product is:
So Fd = p*A*V^2.
That is the force that will accelerate the prop (cart) when in a tailwind, but slow it when a headwind, and does not depend upon "how far the prop moves relative to the air", but only on the amount of air over impact time t.
That equation makes many simplifying assumptions, but when refined as Rayleigh and other did, we get the familiar drag equation.
Fd = 1/2*Cd*p*A*V^2