Comments by "Winnetou17" (@Winnetou17) on "A Physics Prof Bet Me $10,000 I'm Wrong" video.
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@DimitrisKanakis "It s still unclear where the power moving the system forward is generated"
I have some doubts too, but it's all from the wind. Let me see if I can make an argumentation:
Let's say we have a 10m/s wind.
The thing is that the wind is so strong, that it will push the cart no matter if it has 10 kg or 100 kg. In that sense, we can say that it's force and power is uncapped, for our purposes (since we don't know at which weight it won't push the cart to 10 m/s speed). But even if the cart has only 0.1 grams, the wind will only transfer enough power to get it to 10 m/s. In all cases it imposes a 10 m/s^2 acceleration, but the mass differs, so the force differs.
Now, because of that, the cart can, normally, only use (is this the correct term?) the power required for it's own weight. Let's assume it has 10 kg. And that is on perfect ice (no friction with the ground). The power needed to push 10 kg at 10 m/s means that it will need to have a 10 m/s^2 acceleration directed forward, so the power needed would be m * a * v = 10 * 10 * 10 = 1000 kg * m^2 / s^3 which is 1000 W.
And that is the main thing. At the wheel level, they have the same power, since they are forced to move with the cart, at the 10 m/s velocity and the torque is ... I guess, the same force of the cart. But, the wheels are also connected to the propeller. So they transfer (for simplicity) all the 1000W to it. But the propeller has a bigger "wheel", so while it has the same power & force, because it's bigger, it will result in lower speed. It's the torque formula which determines the correlation between force, radius and power. Basically at a given, constant, power, the bigger the radius, the smaller the force and viceversa. And likewise, at a given, constant force, the bigger the radius the smaller the power, since bigger radius means lower velocity. And because friction is a thing, at a low enough force, the thing won't rotate.
In a way, I think that the propeller acts as extra weight, extra friction resistance, because it's connected to the wheels, which "saps" more power from the wind, which in turn it uses to propel the cart faster. So the initial 1000W that I said, some of it will be used by the propeller, so there's less power for the wheels. But, the wind will compensate the extra power so the cart still has 10 m/s^2 acceleration forward. So now (well, at the same time) the cart will receive power for both moving the wheels of the cart and the propeller. So you can say that the wind provides the cart with, say, 1500 W, which 1000W are used by the wheels to push the cart forward and 500W used by the propeller. Hope it makes sense.
There's also the thing that, if the car is faster than the wind, then theoretically it doesn't receive any power from the wind. I'm still not sure how to explain that. In a way I can say that both the propellere and the wind are, combined, pushing the cart. But then again, if the cart has been faster than the wind for some time, then it should only go on the propeller, and maybe inertia... dunno.
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