If the wheels of the airplane were ‘locked’ and the engines are off, and the treadmill accelerated the plane forward to takeoff speed relative to an off-treadmill, off-plane stationery observer, the plane would lift off – however momentarily…(This is the catapult scenario. A significant component, albeit using a steam powered sling rather than a conveyor, of an aircraft carrier takeoff, but they give the pilot a fighting chance by not locking his plane’s wheels and granting him permission to burn his engines).
If the conveyor belt was ‘locked’ and the plane’s wheels were allowed to freely rotate and it fired up its engines and achieved take-off speed it would take off and continue to fly for as long as the engines maintained sufficient thrust. (This is the general case normal runway takeoff scenario).
If the conveyor belt and plane wheels are free running, and the planes engines fired up and the entire system was operating under the normal laws of physics and aerodynamics, the plane would achieve take off at take off speed. (This is a variant of the normal runway takeoff and we can largely discount any frictional forces between the plane wheels and the conveyor).
If (and this is the question as normally posed) the conveyor belt is suitably modified so that any and all forward movement of the plane as evidenced by imminent rotation of the planes wheels is immediately and exactly matched by a reverse motion of the conveyor, the plane can never take off as it will never achieve the necessary airflow over its wings for takeoff to be possible. (This is the theoretical scenario which would be tough to engineer in reality, but from a theoretical perspective, is perfectly valid and quite correct.
I dispute that both answers are correct based on whether one takes a theoretical or realist point of view.
If the conveyor belt can be controlled to exactly counteract any and all rotational motion of the plane’s wheels, the plane isn’t going anywhere.
Stephen W. Hawking
