OK. In that post you wrote: “I don't believe 9 m/s is the right speed to use here. That's the cart's speed relative to the ground. But the propeller exerts a force on the air, not on the ground. Since the propeller's thrust is moving the propeller through the air at only 2 m/s, only (160 N)(2 m/s) = 320 Watts of power is required.”
If the objective here was to move 72 kg per second of air @ 2.2 m/s, as a fan would, then 320 Watts would definitely be sufficient for that task. But the main objective here is to move the 295 kg cart @ 9 m/s over the ground. To do that requires Work = Force x Distance and in this case that is 160 N x 9 m = 1,440 Joules, or in terms of Power 1,440 J/s (Watts).
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.
Michael wrote: "we have a large truck with the brakes off. We give it a push to set it rolling at, say 5 m/s. We throw a tennis ball in the direction of travel, at, say 10 m/s. The tennis ball hits the rear of the truck. We have now increased the kinetic energy of the truck by hitting it with something that has less kinetic energy."
True, but this does not contradict me at all. I said that an object with less kinetic energy can transfer that energy to another object with greater kinetic energy if work is done. Work was done in throwing the tennis ball at the back of the truck.
We're supposing that the air is already moving, so why can't we suppose that the tennis ball is already moving?
This is also a bad example because it does not reflect the situation we have with the wind moving slower than the cart.
Yes, I agree. This important point is the difference in speed, not the difference in kinetic energy
This is known as over unity. In Newtonian mechanics, there is no way to use energy such that it creates more energy than what you started with.
I certainly agree that we can't end up with more energy than we start with. But the wind contains energy, and that's part of what we start with. There's no reason why we can't get a lot of energy from the wind in exchange for expending a little energy setting things up right. For example, it takes energy to erect a windmill, but once it's up, we can gain that energy back, and much more, from the wind.
I assure you I have examined the treadmill in great detail. [...]
I have to go now. I will try to get to this part of your post later.