But infinity speed would cause a failure of the wheel bearings and/or the tires. If the wheels really would be spinning toward infinity speed then I’d think wheel/tire failure would be a certainty.
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That’s why I said “forward motion-wise”. I think for the sake of the scenario we should postulate wheels/tires/bearings that are indestructible.
Like someone said above, imagine the plane is on a frictionless pane of ice.
After thinking about it more, the plane could move forward even if the wheels don’t move quite as fast as the treadmill because the wheels would start skipping or skidding at some point. Imagine instead that the plane is just on skids and has a powerful enough engine to overcome the friction of the skids vs the treadmill.
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That’s the problem I’ve always had with the question. I get where the thrust is coming from, but I’ve never been able to wrap my head around what happens to a plane on wheels on a treadmill that can match the speed of the wheels perfectly and instantly.
It seems that given a real plane and given the impossible treadmill, the correct answer is, no, the plane would not be able to take off.
I guess that’s true. But an impossible plane (or a plane with impossible landing gear) on an impossible treadmill takes off. IMO the question isn’t asking “does landing gear exist that can withstand infinity speed” because obviously that answer is no.
If you can have infinite speed why can’t you have infinitely robust materials?
Feels like this discussion needs a new thread.
And that xkcd was probably right to ban it.
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You see this is what I’m having a problem with. To me it seems to result in a paradox. I get that the spinning wheels are an effect of the plane’s motion rather than a cause. That’s not my issue. My problem is that so long as the plane is connected to the ground it’s a logical necessity that any forward motion results in the wheels spinning faster than the ground beneath them*. It’s absolutely forced that this occur. Yet as soon as they do so they violate the terms of the problem - which is that the ground matches the speed of the wheels.
This is what I mean about a race to infinity.
*edit: I guess we could go Econophile’s route and just assume they start getting dragged along.
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Infinitely spinning wheels would have infinite mass due to relativity. Hence the plane would not take off.
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You can, but the hypothetical doesn’t say that.
Either way the concept of an infinitely and perfectly accelerating treadmill under the wheels seems more complicated than the idea of a plane on landing skis taking off on ice. I understand where the thrust comes from on a plane and I still can’t help thinking the “race to infinity” would be a problem.
Butting in:
Use the brakes. Lock the wheels so they can’t spin. Now the treadmill doesn’t move either.* Apply power to the engines until you overcome static friction then slide along till you take off.
*If it does move, you can still do the same thing.
If the wheels are free-spinning it doesn’t matter either. The wheels spin at a rate proportional to the forward speed of the plane (relative to the stationary ground). There is no running away to infinity.
I think the confusion comes from not considering a consistent reference frame. Relative velocity problems can be tricky. It doesn’t help if the question is ambiguous.
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Does a plane lift off due to the forward motion of its body, or due to the air passing over the wings creating lift? If the latter, than it will eventually generate lift even if stationary, and as soon as it does so it will move forward from thrust with resistance on the tires no longer a factor keeping it stationary. Not sure if this changes with a prop plane vs a jet engine? Prop plane can theoretically push air over the wings. Jet engines under the wing will create a vacuum of sorts which might be enough to get the wheels off the runway.
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this feels like the heart of the issue. if plane requires some degree of forward momentum, then it should not take off given the perfectly matched treadmill.
if the engine thrust alone is enough to produce lift (bc of the flow of air it creates over the wings?) it should be able to leave the ground from a standing position.
edit: or what brad said
Motion is relative. If you consider the plane to be stationary, then you need the air to move in order to generate lift. If the air is stationary, the plane has to move through it to create lift.
You don’t need an engine to fly. Gliders fly. Paper airplanes fly. Squirrels named Rocky fly. You only need relative motion, which can come from an initial push or just the wind blowing.
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Of course the plane needs to be moving forward relative to the air. Why would being on a treadmill prevent that more than being on a runway? The engines aren’t pushing against the ground, they’re pushing against the air.
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The issue here is that “the speed of the wheels” is an ill-defined term. Not all parts of the wheel have the same velocity, and you have to define velocity relative to some frame of reference for it to make sense. Does the term mean the point at which the wheel makes contact with the treadmill? In an ideal wheel that rolls without slipping, that velocity is always zero relative to the treadmill. Does the term mean the center of mass of the wheels relative to some observer on the ground beside the treadmill? If so, the wheels themselves will spin faster than they would if there were no treadmill underneath moving, but there’s no force involved here, except on the wheels from the treadmill, not on the center of mass of the plane.
It sounds like you’re getting stuck on the idea that “the speed of the wheels” must necessarily mean "the speed of the center of mass of the wheel relative to an observer on the ground beside the treadmill if the rate of revolution of the wheels were equal to what it would take to make that center of mass velocity when there was no treadmill. This is basically nonsense, as you’re trying to have the velocity of the center of mass of the wheels in one frame of reference (the treadmill) equal to the center of mass velocity of the wheels in another frame of reference (the observer on the ground). That’s a logical impossibility unless the two frames of reference are equivalent (the stationary treadmill case).
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as you run on a treadmill, your position relative to the air around you doesn’t change, right? your hair doesn’t blow back in the wind at the gym as you run on the treadmill. so a plane being on a treadmill wouldn’t have that either unless the engines alone are enough produce it?
My method of propulsion is my feet on the treadmill. A plane’s propulsion comes from its engines. If I were running on a treadmill with a rocket strapped to my back, would it push me forward even if my legs could run infinity speed and so could the treadmill?
Yeah, to me the question is saying that the plane can’t move relative to the external reference frame of the ground (and air) outside the treadmill. The treadmill matching the speed of the wheel only makes sense if you’re talking about the linear speed of the surface of the wheel. So if the wheel edge is turning at 100 feet per second, the treadmill is going 100 feet per second in the opposite direction.
If that happens instantaneously then it’s a paradox because the plane’s wheels can only move by the plane moving forward. But it’s easy to think about if you switch it to discrete steps. The plane and treadmill start motionless. The engines start and the plane gets to one mph. This causes the treadmill to move in the opposite direction at two mph. Every time the plane gets to one mph relative to the external reference frame the speed of the treadmill increases by two mph. In this scenario the plane never takes off. And the velocity of the edge of the wheels going one way and the treadmill going the opposite direction will be identical to within two mph.
If the plane takes off then the wheel surface will be going twice as fast as the treadmill, which isn’t the formulation of the question.