Strictly speaking, the problem is meaningless if you don't assume it can take off - because if you believe it makes it so the plane can't take off because the air is stationary around it, the speed of the plane is zero and thus the ground isn't moving. Of course, instantaneously, the plane has a speed relative to the air again, and so you get an infinitely fast progression.
The only sensible way of looking at it, if you want to consider what people think it means, is that if the ground has the same speed as the speed of the outer surface of the wheels. Assuming that...
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Originally Posted by Tomkat
Hmmm there isn't actually an answer to it apparently (I looked it up out of curiosity).
The argument given is the same as mine - that in theory the plane is completely stationary (due to the treadmill exerting the exact same force against the wheels). In practice, I expect it would as it'd be impossible to regulate the treadmill's speed to match the plane's speed exactly.
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Under a frictionless environment, then nothing changes. Because friction is zero (either between the wheel and its axel, or between the wheel and the road) there is no way in which the track can interact with the plane. You get a reaction force and nothing else.
I was going to do an analysis of different frictional environments, but then I realised that it was trivially obvious that the amount of friction would not be sufficient because airplanes need brakes and whatnot to stop them when landing on a runway. The necessity of the use of these things to stop a plane
when the engine isn't running shows that the force is significantly less than half that of the maximum output of the engines when going at a speed sufficient to take off, and thus the friction between the wheels and the track alone in the above example would be not be able to stop the plane moving.