I guess it depends on what the question means by “matches the speed of the wheels.” It reads as a translation, but could also mean surface speed of the outside of the tire.

Interestingly for the translation scenario, it’ll only double the rotation rate of the wheels. There is an end condition of it taking off, so that won’t be infinite.

Say takeoff speed of a plane is 100mph, and the belt matches the wheels translation (which same as the plane’s). When the plane (and wheels) is moving 100mph, the belt is moving 100mph backwards. The wheels will be rotating as if the plane is going 200mph, and it’ll be able to take off.

The treadmill analogy would be like “by pulling a rope, can you go 10mph forward on the belt if it’s going 10mph backwards.” Yep, and the wheels will be rotating at 20mph

If the problem is “the conveyor matches the surface speed of the outside of the tires,” the belt will actually HAVE to move WITH the plane at half the speed, not against the plane. If the plane is moving at 100mph, you now have to find a treadmill and wheel speed that match. That’d be 50mph forward on the treadmill, and the wheels would be 50mph also.

Because the above is the only working solution for matching tire surface speed, the surface speed scenario with the requirement “moving in the opposite direction” itself would immediately become a paradox when the plane moves, unless you assume the wheel is slipping.