My issue with this problem was that it was presented rather poorly to me the first time. Instead of saying that the treadmill matched the negative of the velocity of the plane relative to the ground, the questioner instead stated that the treadmill "matched the speed of the wheel." I took this to mean that the treadmill matched velocity of the wheel as would be given by a conventional speedometer.
Of course, this is prima facie ridiculous. Consider, at any nonzero speed, the speed of both the treadmill and the wheel would quickly approach infinity. Given that the speedometer speed would essentially be the velocity of the wheel + the tangential velocity of the wheel touching the treadmill, the entire thing would just blow up (remember, non-slip == velocity of the wheel == velocity of treadmill). Therefore, I basically answered c) it's a stupid problem and the plane wouldn't move.
Of course, if you define it this way, than it is obvious that the plane would move.
Of course, this is prima facie ridiculous. Consider, at any nonzero speed, the speed of both the treadmill and the wheel would quickly approach infinity. Given that the speedometer speed would essentially be the velocity of the wheel + the tangential velocity of the wheel touching the treadmill, the entire thing would just blow up (remember, non-slip == velocity of the wheel == velocity of treadmill). Therefore, I basically answered c) it's a stupid problem and the plane wouldn't move.
Of course, if you define it this way, than it is obvious that the plane would move.