Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to [try to ?] exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off? (different answers if you consider the words between brackets or not)
Hover to show the answer.
Yes. (Please don’t hit me).
If by speed of the wheels, we consider the speed of the the center of the wheels relative to the ground, then the answer is yes. The plane doesn’t use its wheel to take off. The motor thrust don’t care that the wheels are spinning or sliding or doing nothing. You have to realize that this is not like a car, the force is no pushing at the wheels, the force making the plane go forward is pushing where the motors are attached. So the wheels are just going to spin faster, and if the conveyor belt is long enough, the plane will take off.
Another way to see it is that the wheels are just there to keep the plane above the ground. They can spin at the speed they want, if the plane can move forward, it will take off.
Now, if by speed of the wheels, we consider the angular speed of the wheels then it gets complicated. This would basically be like saying “the conveyor belt does everything possible to make the plane not move”.
At that point, if the plane moves forward, the problem becomes mathematically impossible. When the plane is not moving, the angular speed is 0. When the plane moves, the angular speed is, let’s say, 10 ms/s. Now if the conveyor belt matches that speed, it will spin at 10m/s. But since the plane is still moving forward, this will make the wheels go at 20m/s (We consider here that the friction of the wheels is 0, they don’t affect the plane). So now the conveyor belt will have to match that speed. If the conveyor belt just tries to match the angular speed of the wheels, it will just go faster and faster up to infinity. If the friction of the wheels is 0, you could still argue in that situation that the plane will take off, with the wheel spinning infinitely fast in the opposite direction.
If the friction of the wheels is bigger than 0, you could argue that the threadmill manages to stop the plane.
If instead of the treadmill “trying to match” you have the treadmill “exactly matching” the speed of the wheels, than this becomes impossible. As soon as the plane starts going forward, this becomes an impossible situation because the treadmill has to match a speed that grows as it would try to match it.
Note that it doesn’t mean that the plane does not take off, but that the problem is mathematically impossible.
You can have a look at the excellent explanation from Randall Monroe on this problem here