CBS = conveyor belt speed
WS = wheel speed of the airplane
AS = airplane speed in relation to the earth (which weŽll assume is the same as the air-speed, since there is no wind)


The following is a stipulation of the original problem; the conveyor belt is designed to match the speed of the wheels in the opposite direction:

CBS = -WS



The conveyor belt is what drives the free-rolling airplane wheels, so for every inch the conveyor belt moves, the wheels move an inch in the opposite direction. The wheels are also fixed to the landing gear of the airplane and the conveyor belt is fixed to the earth, so for every inch the plane moves in relation to the earth, the wheels will move forward one inch on the conveyor belt. So:

WS = -CBS + AS



So if we substitute the conveyor belt speed (CBS) with the first equation, then we have:

WS = -(-WS) + AS

or

WS = WS + AS


So if the earth/air speed of the plane (AS) is not zero, then you can clearly see that the above equation is impossible.

The impossibility comes from the equation to determine the conveyor belt speed (CBS = -WS) which was a stipulation of the original problem -- that the conveyor belt is designed to match the speed of the wheels at all times. So this conveyor belt is impossible to build, unless you can build it so that it accelerates from zero to infinity instantaneously, since zero and infinity are the only valid values for WS and CBS.