I posted this already but it was an edit of a previous post, so maybe people didn't see it. Consider the "mechanical" equivalent, with a turning turntable at the bottom, and a stationary turntable at the top (the mechanical equivalent of a static air mass to which one can couple).
Now, assume a cart between the two disks of the turntable, with two wheels: one which rolls on the top disk, and one on the lower disk.
Now, consider that the two wheels are linked with a gearbox which specifies that the turning rate of the top wheel is p times the turning rate of the the lower wheel (both wheels have same diameter). p can be any number between -inf and +inf (given by the gearbox ratio). - means that the wheels run in the same direction (meaning that the outer velocities are opposite) ; + means that the wheels run in opposite directions (meaning that the outer velocities are in the same direction).
The top disk has a velocity (wrt the lab) of 0, and the lower (turntable) disk has a velocity v_A (wrt the lab). The cart has a velocity v_B. We want to find v_B as a function of v_A and the ratio p.
We move to the cart reference frame (which has velocity v_B wrt to the lab frame). Now the top disk has velocity -v_B and the lower disk has velocity v_A - v_B.
As the wheels are running smoothly on those disks, we have to have that:
(- v_B) / (v_A - v_B) = p.
Indeed, that's the only way to not have any slipping wheels, which we assumed, they wouldn't.
From that, we derive: (p - 1) v_B = p v_A, or:
v_B = p/(p - 1) x v_A
If p = 0 (the top wheel never turns, whatever the lower wheel does), we find v_B = 0
(the cart doesn't move: it is fixed to the top wheel).
If p = -1, then v_B = (-1 / -2) x v_A: the cart goes in the same direction, and at half the speed, of the bottom turntable. That's like in a ball bearing. The top wheel goes as fast, and in the same direction, as the bottom wheel.
If p = +1 there is no finite solution: the thing is blocked. Indeed, at no point, from the cart's PoV, the top and bottom disks can have the same velocity.
If p = + 1/2, then v_B = - v_A. The cart goes as fast, but in the opposite direction, as the turntable.
If p = +2/3, then v_B = - 2 v_A. The cart goes twice as fast backward, as the turntable is going forward.
This is the kind of demonstration that has been given, but instead of having a massive top disk and a wheel, we have an airmass and a propeller.
(for a picture, see the second attachment in post
https://www.physicsforums.com/showpost.php?p=2031426&postcount=96