All right for the “plane will take off” people, what is the velocity of the surface of the wheels at takeoff and what is the velocity of the treadmill going backwards?
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Close, The same amount of air goes above and below the wing.
The air moving over the wings moves faster, making an area of lower pressure than the air below the wing.
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Which part of the wheel, and relative to what?
It doesn’t matter. The only thing that affects a plane taking off is air speed over the wings.
But if the treadmill is stationary, and the plane is moving forward, the tire speed will be faster than the treadmill speed.
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An then Laplace’s demon. Laplace's demon - Wikipedia
You know arguments are getting real when they involve maximum computational power of the universe. Can’t really go beyond that in an argument.
There has recently been proposed a limit on the computational power of the universe, i.e. the ability of Laplace’s demon to process an infinite amount of information. The limit is based on the maximum entropy of the universe, the speed of light, and the minimum amount of time taken to move information across the Planck length, and the figure was shown to be about 10120 bits.
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This is what I was going to ask Devil, but I was afraid I was missing something obvious. Well, I still am.
I guess the pro-flight is saying the treadmill is only going to turn backwards at the same speed the plane is moving, not the wheels, so the 747 will accelerate to 150 knots or whatever takeoff speed is, and the wheels will be spinning at the equivalent of 300 knots because the treadmill is going 150 knots backward.
But the question says the wheels and the treadmill will always be at the same speed. Right? In their framing of it, the wheels and the treadmill are not at matching speeds.
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But the increasingly faster wind moving against the plane’s wings should cause lift…no?
Edit: I must be riding the Pony Local.
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The picture is standard. The description is not quite right. The same amount of air flows past the top and bottom of the wing, but at higher velocity over the top of the wing, which creates a lower pressure there compared to below the wing (Bernoulli’s principle). Anyway, that’s more or less the usual explanation. (This came up here a long time ago and it turns out things aren’t quite so simple or even completely understood so 
).
Whatever though. I agree, the plane takes off.
Assume plane wheels have 1m circumference
Treadmill is a belt around two cylinders with 1m circumference
If there is no slippage of the belt on the cylinders, no slippage of the wheels on the belt, and the wheels and cylinders always have the same RPM, how does plane move forward relative to ground?
If the plane were 1 inch above the treadmill with engines at full blast, do you believe it would move forward?
Depends on the plane.
The plane moves forward no matter what due to the thrust of the engines.
There’s no way the wheels and the treadmill match speeds as the question is written.
I believe a plane would take off from any treadmill that is feasible to construct (and long enough), but the question posits an impossible treadmill
It does matter, because as worded the answer is both would be going arbitrarily fast. Clearly the treadmill is magical as it can instantaneously accelerate to an arbitrarily high speed to match the speed of the surface of the wheel. The regular old 747 trying to take off is not designed for its wheels to spin arbitrarily fast and would fucken explode.
It’s a 747, as stated in the question.
The entire point is that it doesn’t need to move forward relative to the ground.
The only reason standard airplanes move forward in takeoff is that it is currently the only way we have to get the airspeed over the wings to generate enough lift to get the plane off the ground.
Once airborne, the instruments track airspeed, relative speed, and ground speed. It might blow your mind to know that a plane’s ground speed can be ZERO or even negative and it can still fly.
Yes, pedants, I know VSTOLs exist. That’s not what we’re talking about.
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Thanks. I know.
I’m trying to keep this at a layman’s level but I can pull out Navier-Stokes if you wish.
The question us poorly written. I think the spirit of the question is whether or not the pane takes off a treadmill moving in the opposite direction of takeoff.
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Sure, this sounds like a paradox because you’re describing a nonsensical situation:
There is no physical law that states that the center of mass of a spinning wheel must move forward relative to a stationary observer at a rate as if that wheel were rolling without slipping on a surface stationary to that observer. Surely you agree that if a wheel were suspended from a helicopter, it could spin at arbitrarily high speeds without moving forward relative to an observer on the ground if the helicopter is stationary relative to the ground too, yes? Similarly, there’s no obligation for a wheel spinning at X rpms to move forwards relative to an observer on the ground at Y velocity for its center of mass when that wheel spins at that same rate both on and off a treadmill.