Work is force times distance, but the force and the distance have to be at the same place, so to speak. I don't think it's correct to multiply the force of the propeller on the air by the distance that the cart travels over the ground. You could use the 9 m/s speed of the cart over the ground, but then you'd also have to use the force between the cart and the ground, whatever that is.
Actually, now that I think about it, I don't believe any work is done at the wheel/ground interface, except that needed to overcome rolling resistance. Each wheel rolls along the ground without slipping; the point of contact with the ground does not move relative to the ground, so although there's force, there's no distance. But I am not especially sure about this. I welcome comments.
You should listen to what everyone is trying to tell you, because in this one instance at least, everyone is right! The thrust force generated by the propeller pushes the air one way, at 2.2 m/s, relative to the cart, and pushes the cart the opposite way, at 9 m/s relative to the ground. The 350 Watts in my calculation is adequate to push the air at 2.2 m/s ( 160 N x 2.2 m/s ~ 350 Watts) . But it is not adequate to push the 295 kg car over the ground at 9 m/s (160 N x 9 m/s = 1,440 Watts)
Maybe now you can see why I base my analysis around energy and power as well as forces, as just looking at forces alone does not tell you the whole story. In this case, if there was no other source of energy, it is clear the cart could not work. For example, if the cart was trying to run in still air, it should be very obvious that it isn’t going anywhere for long because it does not have the energy needed to do so.
In the case we are considering, with a wind blowing at 7 m/s and the cart moving along due to the Good Lord towing it up there and releasing it, there is some energy available from slowing down the wind.
The calculation for propeller thrust showed you the mass of air that gets slowed down is about 72 kg and is slows by 2.2 m/s with respect to the ground. From that you can see that the air mass loses about 935 J/s. Earlier, I rounded that up to a nice 1000 Watts of additional power available to the cart, if that power could be 100% transferred to the cart. The 100% transference is absurd, but even if it were allowed to happen, that makes the total power available to the cart 1,350 Watts and it needs 1,440 Watts to move nine meters in one second. If it can’t move that nine meters in one second, it cannot produce the 350 Watts at the propeller, and that is the end of that.
Notice, in this analysis I didn’t even bother to mention the fact that in order to get that 350 Watts on the propeller, the cart needed to come up with somewhere between 750 and 850 Watts to overcome efficiency losses. Remember, we are in the ground frame, so no ground power allowed. The only source of energy is whatever kinetic energy the cart has been given to it by the Holy Tow Vehicle, which has towed and departed. So the cart has to dig into its own kinetic energy to get the propeller working on the air. Some 500 Watts (500 J/s) is being lost in that process, slowing the cart down. Now to get back up to 9 m/s it needs that 1,440 Watts, plus the 500 Watts it burned off in this rather asinine assault on conservation of energy.
Well, as all assaults on conservation of energy are doomed to fail, so it is with this one. There simply is not enough energy available from the wind to power the cart to move the required 9 m/s over the ground to spin the propeller and do the work to slow down the air and get the energy from the wind to power the cart to move the required 9 m/s over the ground to spin the propeller to do the . . . . . .By now maybe you see what the problem is? No matter how many times our carteers deny it, they are in fact proposing just another in a long long line of perpetual motion machines.
The math says they are wrong. The NALSA data says they are wrong, and the treadmill has not one blessed thing to do with wind power.