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even taking into account friction, it's not a big deal because the relative speed between plane and belt doesn't need to approach infinity. The relative speed ends up being twice the plane's takeoff speed


Tony, you´re wrong on that point. If the plane has a positive earth-ground speed, then the wheels are rotating faster than the conveyor belt.

The conveyor belt can be seen as a giant wheel attached to the earth and the plane wheels are of course attached to the plane, so we have two sets of wheels rolling against eachother. If both sets of wheels turn at the same speed (not RPM´s, but the speed of the outside diameter of the wheels) in opposite directions, then the objects they are attached to will remain stationary in relation to eachother. So that means the earth and the airplane must be stationary in relation to eachother for the plane wheels and the conveyor belt to move at the same speed in opposite directions. But of course, as we all agree, the plane WILL move forward in relation to the earth (and the air), due to the thrust from the engines, so it´s impossible for the conveyor belt to match the speed of the plane wheels.