Now that I've closed the comments on the question of the airplane and the conveyor belt, I'm still getting emails calling me an idiot for thinking that the plane will take off. Having believed that after first hearing the question and formulating several reasons reinforcing my belief, I can sympathize with that POV, but that doesn't change the fact that I was initially wrong and that if you believe the plane won't take off, you're wrong too[1].
The only thing is, I'm not sure how to prove it to you if you don't understand the problem and the physics involved. I guess I could urge you to read the question and answer again carefully. I could tell you that not only does the conveyor belt not keep the plane stationary with respect to the ground but it *can't* keep that plane stationary with respect to the ground[2] and once you know that, of course it'll take off. My pal Mouser has a Ph.d in Physics from MIT and he says the plane will take off:
The airplane would take off normally, with the wheels spinning twice as fast as normal and a *slight* reduction in acceleration due to added friction.
Is that enough to convince you?
[1] This situation reminds me of Richard Dawkins' and Jerry Coyne's assertion that "one side can be wrong".
[2] The motion of the conveyor belt does nothing[3] to affect the movement of the plane when the plane is in motion...it doesn't matter if it's moving forward, backward, at 2 MPH, or at 400 MPH. If the plane were on castors that could spin freely from side to side as well as front to back, that treadmill could be spinning 100 MPH to the left and the plane would take off.
[3] Well, almost nothing. The friction of the turning wheels will slow things down a bit, but not enough to not make the plane take off. After all, the main function of the wheels of a plane is to provide a near-frictionless interface with the ground (or whatever the plane happens to be taking off from).