phattslagg wrote:But isn't that the nub of it? If the treadmill was to match the wheels' rotation right from the start (and therein lies possibly the biggest problem of all) then the plane would never move. Nothing like the speed of light would be reached anyway, since the engines would max out much earlier than this.
This is, inded, the nub of it. It's the nub of where you are wrong.
Consider a vehicle that is driven along the ground using its wheels. If we match the wheel's speed by moving the surface under the wheels at the same speed, yes, the vehicle will stay in the same place.
Now consider an aeroplane. It is not driven by its wheels, but, instead, by the reaction caused by applying forces to the air around it, one way or another (either by a jet engine, or a propellor). The wheels only rotate whilst it is on the ground because there is a speed differential between the two surfaces. The only way some "magic treadmill" could cause a plane not to take off would be to apply enough rolling resistance to stop the plane moving (very, very, unlikely, given that it would have to somehow apply that resistance through a set of relatively free-moving bearings), or by somehow applying enough force to the air surrounding the airplane in order to keep it under the plane's stall speed relative to the airplane's ground/treadmill speed.
The first of these is highly unlikely, and the second is not stated as being a function of the "magic treadmill", although there would likely be a certain amount of "entrainment" of the air due to the moving treadmill surface itself, the question then becomes "if we run the treadmill in the same direction as the aircraft, will it entrain enough air and fast enough to make it difficult for the plane to take off, and will that near-negligible entrainment be counteracted by the near-negligible accelerative effect of the frictional forces generated by running the wheels faster than they should go?"
So yes, the aircraft would take off.
simon