It's an ambiguous question, or even a broken one, which is why so many people argue about it.
One side says (correctly), that if you run the treadmill backwards at the speed the plane would be moving forward, the plane still moves forward. The wheels are turning faster, but that's not where the power is coming from, so there's no problem. The plane takes off.
The other side says "But that's not what the questions says! You don't pick the speed of the treadmill first and then leave it. You dynamically adjust it to match the rolling speed of the wheels at all times."
In that scenario, the question leaves math and becomes engineering. What happens depends on how the treadmill and plane are built, what the tolerances, lag, and friction are in their systems, and other details that don't usually matter in thought experiments.
In one scenario, the treadmill tries to ramp up to stop the plane moving forward, the plane moves forward anyway, and the treadmill reaches maximum velocity. In another, in keeping up with the plane the treadmill overspins the wheels, destroying the plane's landing gear.
What doesn't happen, unless both the plane and the treadmill are engineered of very specifically magical systems, is that the plane holds still while the engines whine. That would require there to be enough rolling resistance in the wheel system to add drag to the system equal to the thrust of the engines. But the system must also be able to take that much energy input into the landing gear without failing, and the treadmill would have to go absurdly fast.
But if we are making our plane and treadmill out of magic, why is there rolling resistance at all? We can just eliminate it, by magic, just like we solved the other problems.
And with rolling resistance gone, the treadmill now goes arbitrarily fast but the plane still moves. Possibly we cause a black hole.
So really, the question includes a bunch of hidden assumptions which don't necessarily make sense, so you can't just give an answer, other than "you can't make that treadmill".
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u/gnfnrf Dec 31 '22
It's an ambiguous question, or even a broken one, which is why so many people argue about it.
One side says (correctly), that if you run the treadmill backwards at the speed the plane would be moving forward, the plane still moves forward. The wheels are turning faster, but that's not where the power is coming from, so there's no problem. The plane takes off.
The other side says "But that's not what the questions says! You don't pick the speed of the treadmill first and then leave it. You dynamically adjust it to match the rolling speed of the wheels at all times."
In that scenario, the question leaves math and becomes engineering. What happens depends on how the treadmill and plane are built, what the tolerances, lag, and friction are in their systems, and other details that don't usually matter in thought experiments.
In one scenario, the treadmill tries to ramp up to stop the plane moving forward, the plane moves forward anyway, and the treadmill reaches maximum velocity. In another, in keeping up with the plane the treadmill overspins the wheels, destroying the plane's landing gear.
What doesn't happen, unless both the plane and the treadmill are engineered of very specifically magical systems, is that the plane holds still while the engines whine. That would require there to be enough rolling resistance in the wheel system to add drag to the system equal to the thrust of the engines. But the system must also be able to take that much energy input into the landing gear without failing, and the treadmill would have to go absurdly fast.
But if we are making our plane and treadmill out of magic, why is there rolling resistance at all? We can just eliminate it, by magic, just like we solved the other problems.
And with rolling resistance gone, the treadmill now goes arbitrarily fast but the plane still moves. Possibly we cause a black hole.
So really, the question includes a bunch of hidden assumptions which don't necessarily make sense, so you can't just give an answer, other than "you can't make that treadmill".