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Note that this is more of a Physics question than an Aviation one.
This topic has been extensively covered on the Physics Stack Exchange site, and the answers there are recommended reading for anyone curious about the forces and physics involved.

If you had a giant treadmill that was an infinite plane, had an infinitely adjustable speed, and could tread along any of its dimensions, could an airplane take off from it?

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    $\begingroup$ blog.xkcd.com/2008/09/09/… $\endgroup$ – Essah Sep 30 '15 at 10:58
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    $\begingroup$ Oh no, not the treadmill! $\endgroup$ – Jörg W Mittag Sep 30 '15 at 11:11
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    $\begingroup$ Ahhhhh! Well we had to have it on here. ;) $\endgroup$ – ryan1618 Sep 30 '15 at 12:06
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    $\begingroup$ Someone did build a giant treadmill to test this. See the Mythbusters video below $\endgroup$ – Pondlife Sep 30 '15 at 13:07
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    $\begingroup$ The treadmill is irrelevant. All it can do is spin the wheels. It cannot move the aircraft. Myth Busters should be considered entertainment and not serious scientific enquiry. They did a lousy job with this one and demonstrated, well, that an aircraft can fly. $\endgroup$ – Simon Sep 30 '15 at 17:19
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Everyone collectively went "Oh god, not this one" because this same question has sparked some intense debates in the past. Aircraft rely on airflow over the airfoil (wings/tail etc) to produce lift - which is independent of the movement of the tires. This means that with enough air going over the wing, the aircraft will fly even if it isn't moving forward at all relative to the ground.

This is why aircraft on ramps at airports need to be tethered to the ground. This isn't only to prevent them from rolling around, but from taking off should the air flow get fast enough over the wing.

If you are interested in an entertaining segment, the MythBusters did a fairly scientific experiment of this.

Although it isn't perfect, I think it does a good job of explaining the concepts.

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    $\begingroup$ It isn't that it won't be moving in reference to the ground, it's that the treadmill can't stop it from moving in reference to the ground. $\endgroup$ – Lnafziger Sep 30 '15 at 15:31
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    $\begingroup$ All that Mythbusters video did was prove that the wheels on the plane could handle rotating at twice the RPM necessary for the plane to take off. $\endgroup$ – FreeMan Sep 30 '15 at 20:36
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    $\begingroup$ @FreeMan: The wheels are free-wheeling and are essentially not connected to the plane. Think of it like this: if you lie on your back wearing rollerblades, can I lift your leg by spinning the wheels? Your're saying I can. $\endgroup$ – slebetman Oct 1 '15 at 1:59
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    $\begingroup$ @slebetman not sure I follow you. The plane was moving forward at X knots, while Jamie pulled the "conveyor belt" backwards at X knots. The wings generated enough lift for take off, but the wheels spun at 2X. Therefore, all the video proved was that the wheels could handle spinning at 2X, where they would normally rotate at X. I fully understand that take-off has nothing to do with the rotational speed of the wheels and that's why I indicated the experiment didn't prove anything else. $\endgroup$ – FreeMan Oct 1 '15 at 14:29
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    $\begingroup$ @FreeMan: The wheels being able to handle spinning at X speed has nothing to do with keeping the airplane from moving forward. The wheels are basically acting as roller bearings. The only way to stop the plane is to make the wheels explode (which may be possible to do with a threadmill in theory). There was a also an answer in the physics SE that pointed out that the rotational inertia of the wheels can impart a tiny force on the plane. But it would take the wheels moving at or beyond the speed of light to have any noticeable effect. $\endgroup$ – slebetman Oct 1 '15 at 22:37
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Yes.

Airplanes get their thrust by using the air. The wheels are not powered. The drag from the wheels will limit how fast the treadmill can go before the plane won't be able to take off anymore.

It's simpler to understand if you pick a different frame of reference. Assume the treadmill is standing still but the air is moving around it in any direction with any speed.

Notice I just described a windy day.

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  • $\begingroup$ Has there ever been an airplane that has flown away, or been lifted away on a windy day? $\endgroup$ – Sargun Dhillon Oct 1 '15 at 0:32
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    $\begingroup$ @SargunDhillon Yes. –slebetman's YouTube link. $\endgroup$ – Mazura Oct 1 '15 at 2:10
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    $\begingroup$ The windy day argument is somewhat flawed. If the wind is blowing along the airfield the planes will take of into the wind. $\endgroup$ – Taemyr Oct 1 '15 at 9:26
  • $\begingroup$ @Taemyr there is nothing stopping the plane from turning on the infinite length conveyor and turning into the wind. $\endgroup$ – ratchet freak Oct 1 '15 at 9:28
  • $\begingroup$ @ratchetfreak I would say that the question implies that in that case the threadmill would run the other way. Essentially turning the wind. $\endgroup$ – Taemyr Oct 1 '15 at 9:32
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This question is at the best, ambiguous. There can be both yes and no answers based on what is being done with the aircraft and treadmill. The point is that for an airplane to lift off, there should be sufficient airspeed. If there is no wind, there airspeed is equal to the ground speed

Assuming that there is no wind (into or against the aircraft), there are two possible solutions.

  • If the airplane is stationary relative to the ground, it won't take off (as wind speed is zero).

  • If the airplane moves relative to the ground (with sufficient speed), it will take off.

Assume that we have a jet airplane (just for sake of argument) and some one pushes the throttle and it begins to move forward. Now, as the treadmill has an infinitely adjustable speed, we can have three conditions:

  • If the treadmill speed is zero, the airplane will eventually generate sufficient lift and take off.

  • If the treadmill speed is adjusted such that the airplane is kept stationary relative to the treadmill, the airplane will take off (as it is moving with respect to ground, and so has some airspeed).

  • If the treadmill speed is adjusted so that the airplane is kept stationary relative to the ground, the airplane cannot takeoff, as the ground and air speeds are both zero. Note that in this case, the aircraft speed relative to the treadmill is twice that of the speed at which the treadmill is being operated.

If there is a wind, the required ground speed can be adjusted accordingly, but the principle remains the same. For example, if the wind speed is equal to the airspeed required for take off, the airplane will lift off even though it is stationary with respect to the ground.

Again, the important concept here is airspeed. It does not matter if the aircraft is on a treadmill, train track or runway.

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    $\begingroup$ Your last bullet point is flawed, and is the whole problem with this stupid "riddle". The treadmill has absolutely no way to overcome the independent thrust generated by the engines - it can not prevent the aircraft from moving relative to the ground, and therefore relative to the wind. $\endgroup$ – Dan Sep 30 '15 at 13:43
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    $\begingroup$ @aeroalias How? The physics don't match - I guess at a certain speed you might have a wheel bearing failure, but while those bearings are functional there's almost no relationship between forward thrust and the rotation of the wheels. $\endgroup$ – Dan Sep 30 '15 at 13:57
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    $\begingroup$ @Dan Please do see the same question in Physics Stackexchange for a much clearer explanation of what I'm trying to say. $\endgroup$ – aeroalias Sep 30 '15 at 14:04
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    $\begingroup$ This issue is that #3 isn't possible. No matter how fast the treadmill is running, the tires on the airplane are free-wheeling so simply spin faster while the aircraft continues doing what it was doing before (accelerating if it is under power). Put another way, if the engines were off and you start the treadmill, (assuming that the brakes are not set), the tires would spin but the aircraft would remain stationary. Note that for clarity this ignores the friction between the axle and the wheel hub and that eventually it would slowly start to move, but this is overcome by thrust in the question. $\endgroup$ – Lnafziger Sep 30 '15 at 15:30
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    $\begingroup$ What everyone misses is that the treadmill, and what it is doing, is totally irrelevant. The only thing it can do is spin the wheels (ignoring friction, which is a rounding error in the forces involved). The only thing that matters is air over the wings and the only way that can be generated (assuming zero wind) is for the aircraft to produce thrust in the normal manner. The first half of your answer is correct. The second half is wrong. The treadmill cannot move the aircraft. $\endgroup$ – Simon Sep 30 '15 at 15:35
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Yes. It does not actually matter much which direction and how fast the treadmill would rotate; the aircraft will take off.

The only requirement for generating lift is to move through the air sufficiently fast. The speed is created by thrust. And the thrust of the aircraft engine does not depend on the ground speed ("ground" would be the surface of the treadmill for this case).

The treadmill can only affect the ground speed so would have no effect on the engine thrust. Hence it would not have significant effect on the air speed either, unless through the friction forces in the bearings of the wheels. I assume that these forces are small in comparison the power of the engine.

The only chance, as the aircraft chassis is the designed for the limited ground speed only, the treadmill may prevent take off by rotating in the opposite direction fast enough to cause the chassis to collapse.

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  • $\begingroup$ If we ignore friction, the treadmill would affect neither ground speed nor airspeed, unless you're defining 'ground speed' to be speed relative to the moving treadmill surface. If you're defining 'ground speed' as being relative to the Earth that the treadmill is sitting on (i.e. normal definition of ground speed,) the ground speed would differ from the airspeed only by whatever the wind happened to be at the time. $\endgroup$ – reirab Sep 30 '15 at 20:46
  • $\begingroup$ What forces would cause the chassis to collapse? $\endgroup$ – Octopus Sep 30 '15 at 22:28
  • $\begingroup$ @Octopus Friction in the wheel bearings leading to a torque on the landing gear struts. $\endgroup$ – David Richerby Oct 1 '15 at 7:46
  • $\begingroup$ "Ground speed" I mean relative to the surface of the treadmill (the "ground" the aircraft is taking off from). The treadmill may force the wheels to spin faster than they bearings could hold by moving fast enough in the opposite direction. $\endgroup$ – h22 Oct 1 '15 at 8:41
  • $\begingroup$ @DavidRicherby Not to mention friction in the wheel bearings and friction on the tire surface (both with air and with the treadmill surface) likely setting the gear on fire. $\endgroup$ – reirab Oct 1 '15 at 16:09
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Theoretically yes. In reality it depends.

Theoretically

We don't account for friction in the landing gear wheel bearings or between the treadmill and the wheels. This would mean that with the plane just sitting idle, if the treadmill moves the plane will remain still. You can try this by putting a toy car on a piece of paper. If you jerk the paper back and forth the car doesn't really move. The only reason the car moves is because of friction. If you eliminated friction in the wheels the car wouldn't move at all. We've now established that the moving runway has no influence over the plane. The pilot is free to start the engine and take off.

In reality

The real answer depends on the design and limits of the plane/treadmill:

  • In real life there is friction in the landing gear. There are limits to how fast the wheels can spin before they fail. But, there would also be a limit to how fast the treadmill could go.
  • There are limits to how fast the treadmill and plane can accelerate and change directions. A pilot may be able to get the treadmill going in one direction then turn around and take off in the other.
  • A very large treadmill moving at high speed would create wind. A strong enough wind can allow a plane to take off even though it's standing still.
    The amount of wind generated would depend on exactly how smooth the surface is and how large it is.
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    $\begingroup$ "or between the treadmill and the wheels." If you don't account for friction between the tires and the treadmill, the wheels won't even turn. The aircraft would just skid across the surface of the treadmill. $\endgroup$ – reirab Oct 1 '15 at 16:05
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I had a thought here: If we are considering a perfect treadmill and perfect wheels/bearings on the plane it doesn't take off.

The plane starts to roll. The treadmill matches the wheel's speed, but this simply spins the wheels faster--so long as the plane is rolling the treadmill is in a never-ending race against the wheel.

Since we are looking at a perfect system this proceeds without limit and infinitely fast--the treadmill (and wheel's outer edge) is going to be approaching lightspeed. The mass grows without limit, the plane is too heavy to take off.

In the real world with imperfect systems something has to give.

1) The wheels have a maximum speed. Exceed that by too much and your landing gear explodes. The plane pancakes onto the treadmill, the friction is too great for it to overcome, it's tossed backwards and then stops.

2) The treadmill has a maximum speed. If the wheels can survive takeoff velocity plus this speed the plane takes off, else #1.

3) The treadmill has a finite acceleration rate. The plane very well might take off before the treadmill has built up serious velocity.

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