This question came up a few months ago at work. After much discussion, everyone at work agreed that the question was flawed.
tracks the plane speed and tunes the speed of the conveyer to be exactly the same.
What speed are we talking about here? Are we talking about the speed of rotation of the aircraft's wheels? If the aircraft moves forwards at 5kt, the conveyor would then have to start moving backwards at 5kt. But this would of course increase the speed of the aircraft's wheels to 10kt, which would then cause the conveyor to speed up... and before long, the conveyor would be going infinitely fast. This is the answer that is given on the website.
Or are we talking about the airspeed of the aircraft? In which case, if the aircraft starts moving forwards through the air, the conveyor would start moving backwards. Now we need to know some more about the aircraft's wheels. If they are perfect, i.e. no friction, the wheels would be spun backwards, but this would not slow the aircraft's movement through the air, so it would take off at its normal rotate speed.
If the wheels have friction (as all real-world wheels do), then the action of the conveyor on the wheels would cause the aircraft to move backwards, whilst the propellor would continue to move the aircraft forwards. (If you don't believe me, think about an aircraft without its engine running sat stationary on the conveyor, and get the conveyor to move backwards - the aircraft moves with the conveyor. Having the prop running would not affect this.) Since most wheels only have a little friction in them the prop would probably overcome the conveyor, i.e. close to the no-friction case. The aircraft would still rotate at its normal rotate speed, but it might take a little longer to reach this speed than it would if the conveyor wasn't slowing it down. So your conveyor would have to be at least as long as a runway to avoid the aircraft going off the end of it.
FFF
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