Posted by WolstonFlyer on 12/10/2016 21:06:43:
This simple little question has caused a lot of healthy discussion on Facebook today.
Edited By WolstonFlyer on 12/10/2016 21:07:49
This is extremely badly phrased. "Exactly match the speed of the wheels". What is the speed of a rotating wheel? A wheel typically has two speeds: one translational and the other rotational. I think it means there is no slip between the tyre contact patch and the belt.
Short answer: the plane will take off if the engines provide normal take-off thrust.
Long answer.
Start with no motion and no engine thrust. The situation is stable and nothing will move, ever. There is no problem.
If, somehow, the initial conditions are such that the wheels are spinning and the belt moving such that the plane as a whole is stationary, then. in the absence of friction, that state will persist for ever. With friction, what happens depends on the inertia in the wheels /plane and the conveyor belt systems. Unless matched magically, I would expect some movement.
Now begin again and start the engines. If there is no friction between the contact area of the tyres and the belt, the wheels will not rotate and just slide contradicting the conditions implied in the question. The plane will accelerate and take off.
With friction, there will be a force in the backwards direction trying to rotate the wheel. This force must be supplied by the belt. As the plane accelerates, the wheel rotational speed has to increase so a force must persist. The belt system has to supply rotational energy to the wheel. Normally, the Earth does that.
Now consider a very large wheel. If large enough, the contact area becomes almost indistinguishable from a caterpillar track in contact with the ground and with no relative motions. (Alternatively, zoom in on this area). The backwards velocity of the tyre at the contact patch relative to the hub exactly matches the forward velocity of the wheel hub, the plane velocity, So there is no problem. The belt does not need to move at all though it must be constrained. The Earth doesn't when a plane takes off (at least to an observer with poor sensors standing on the Earth).