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I've heard conflicting answers, and would like to see the record set straight:

An jet/propeller airplane is traveling on a giant treadmill at takeoff speed. Will the plane takeoff, or will it remain on the runway, and why?

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    $\begingroup$ The definitive answer is in the XKCD blog. $\endgroup$ – rodrigo Jul 31 '14 at 10:15
  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/32269/2451 and links therein. $\endgroup$ – Qmechanic♦ Jul 31 '14 at 10:32
  • $\begingroup$ The simplest way to answer this is to imagine replacing the wheels with frictionless skates (after all, that's what wheels are trying to be). Then you see that the premise of the question is not possible. Except for the implied drag due to spinning up the wheels, There's no speed the ground can be moving backward that prevents the plane from moving forward. $\endgroup$ – Mike Dunlavey Aug 23 '14 at 16:03
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An airplane's propulsion does not depend on friction between its wheels and the runway so the relative motion of the runway to the body of the airplane has no effect on the plane's motion$^1$. For example an airplane can take off from ice, where the friction between the wheels and the runway surface is effectively zero.

So the plane would take off normally if you placed it on a giant treadmill running at any speed.

$^1$ there would be some effect because there will be some friction in the wheel axles. However the forces due to friction in the wheel axles will be tiny compared to the thrust generated by the plane's engine.

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    $\begingroup$ When you run on a treadmill, there is no wind resistance. Would a plane's airfoils indeed work in such a situation? $\endgroup$ – Mark Biwojno Jul 31 '14 at 9:27
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    $\begingroup$ When you're running it's the friction between your feet and the ground that provides the propulsion. Suppose you are running on a treadmill wearing a jetpack and you turn the jetpack on. What's going to happen? $\endgroup$ – John Rennie Jul 31 '14 at 10:57
  • $\begingroup$ @MarkBiwojno: John is right. What matters to an airplane is air, not ground. A Cessna 172 requires air to be traveling past the wing at about 55 nautical miles per hour. The wing deflects the air downward, creating more lift than the plane weighs, so it takes off. The ground can be going forward, backward, sideways, whatever. What matters is the motion of air over the wing. The propeller creates thrust against air, to accelerate the aircraft. The wheels, skids, floats, are only there to hold it up when it's not supported by the air. $\endgroup$ – Mike Dunlavey Jul 31 '14 at 13:56
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An jet/propeller airplane is traveling on a giant treadmill at takeoff speed.

With respect to what does the plane have takeoff speed?

I believe the following two statements are uncontroversial:

(1) if the plane has takeoff airspeed (or greater), the plane can takeoff.

(2) if the plane does not have takeoff airspeed (or greater), the plane cannot takeoff.

Now, the plane's airspeed and the speed of the plane with respect to the treadmill belt are, within practical bounds, unrelated.

Thus, if you only specify the speed of the plane with respect to the treadmill belt, we don't know what the plane's airspeed is so we can only refer you to (1) and (2) above.

To be clear, it is possible for the plane to have both takeoff airspeed and, say, twice takeoff speed with respect to the treadmill belt.

For concreteness, assume the takeoff airspeed is 100 mph. Then, assuming low enough friction and strong enough wheels/tires, there is nothing to prevent the plane on the treadmill from having an airspeed of 100 mph and a speed relative to the treadmill belt of 200 mph.

The plane's propeller pulls the plane through the air as normal and the wheels rotate at twice the speed compared to an identical plane alongside on the runway next to the treadmill.

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