The question is confused:
A plane is standing on a runway that can move like a giant conveyor belt. The plane applies full forward power and attempts to take off. This conveyor has a control system that tracks the plane's wheel speed and tunes the speed of the conveyor to be exactly the same but in the opposite direction, similar to a treadmill.
This can mean:
A. Wheels are rotating clockwise; conveyer is rotating opposite direction (counter-clockwise),
or
B. Wheels and Conveyer are rotating clockwise, and surface of conveyer is moving in direction opposite to that of the contact point of the wheels -- the conveyer is moving in the direction of the aircraft.
The question invites us to suppose a condition that does not exist. Hence, we have to put aside objections of impossibility, and make it work. But the conveyer belt is the only thing we are allowed to consider in a contrary-to-fact and -physical law situation.
Yet we can assume that a "control system" would have some sort of frequency between sensor and adjustment, as all mechanical control systems have. It would have to be reasonably fast, but it would be there.
If you deny me a frequency, then yeah, in A it solves to infinity, and the question doesn't make sense. In B, at the first instant of change, the beginning of the motion, the motion is sensed, and corrected for, and you will have a perfectly motionless set of wheels attached to an aircraft that accelerates down the runway and takes off: that motion is increasing in velocity, but always compensated for. In this case, B creates all kinds of interesting philosophical discussions of the instant of change, incipit/desinit, and motion in general, the kinda crap Aristotle threw into his
physics, but not stuff that affects our problem.
So, we have to have some sort of frequency in the control system. The "Exactly" means "Exactly" the same speed as the previous measurement, given the mechanical contraption's ability to accelerate to it. And we'll have to admit that that frequency is pretty high.
Okay, so now, in A, one of several situations can occur, depending on the aircraft:
1. As the wheels are not frictionless, the moment it starts moving forward, the aircraft will be moved backwards, and at an accelerating rate. It will then be dumped off the threshhold, tearing up the PAPI lights: so NO
2. The air moved along the surface of the conveyor belt will provide enough of a headwind that the jet (presumably one of them biplane models) will be able to take off: YES
3. The aircraft will start its forward roll, and the conveyor belt will spin the tires madly. As it starts moving backwards, the engines have spectacular compressor stalls. The lateral vibration, coupled with the severe shakes coming off the overspeeding tires, will induce the gear to collapse. The jet will fall onto the conveyer belt, and the fuselage will be propelled at a couple hundred miles an hour, spattering against an ILS array. Rescue comes on the scene and extricates the pilot from tangled, burning mess: NO
4. The gear stay on through the compressor stall, but as the tires hit the sound barrier, they explode, and the shockwave coming down the conveyor belt jolts the aircraft. It drops onto the conveyor belt, and is instantly shredded into tiny little pieces that are propelled backwards. Subsequent mist bars landing to all but Cat. 3B Aircraft, and wake turbulence is a *****: YES
In B, on the other hand, the wheels would turn, but slower than normal, and the aircraft would move down the runway and take off.
The mistake made in this formulation was to tie the conveyor to the wheel speed of the aircraft, and not to its ground speed.
given the natural order of the universe (aka, physics, not theology), of course.
