## The Airplane-Treadmill Conundrum

I ran across this thought experiment on David Pogue’s blog site. Below is my answer to what would happen if you had a plane on a giant treadmill for a runway. Basically it comes down to how you interpret the movement of the treadmill as the problem is stated.

I think that the plane could take off, given the right circumstances.

The forces acting on the plane are, gravity pushing it down, the surface of the treadmill pushing it up, the thrust from the engines and the friction between the tires and the treadmill as a result of the plane and the treadmill pushing against each other. If thrust is applied horizontally, at some point in time it overcomes friction and the plane begins to move forward. At some speed, the lift generated by the wings is more than gravity and the plane begins to fly.

You could lock the wheels so that they never turned and still take off as long as the forward thrust of the engines could overcome the friction between the wheels and the treadmill surface so that it slid along the treadmill without rotating.

If that is breaking the rules of the puzzle and the treadmill moves to match the forward speed of the plane, then the plane only has to exert enough extra thrust to overcome the extra friction that is transferred to the plane through the wheels. The plane is moving forward at 100 miles an hour, so the treadmill is moving at 100 miles an hour in the opposite direction, but only a portion of the treadmill’s movement is transferred to the plane through the wheels as friction. The wheels, of course, are rotating at 200 miles per hour (assuming they stay coupled to the surface of the treadmill). The wheels are rotating faster than the surface of the treadmill is moving below them.

If the problem means that the surface of the treadmill is moving at the same speed as the rotation of the wheels, then this sets up an infinite feedback loop between the forward thrust of the plane and the motion of the treadmill. As soon as the wheels begin to be pushed forward by the thrust of the plane that forward motion is transferred to rotation of the wheel. Because only a portion of the friction of the wheel is transferred back to the plane the plane would continue to move forward.

But if the treadmill is meant to compensate for the motion of the wheel, then it will continue to accelerate to a point where the friction generated by the movement of the wheel is enough to counteract the forward thrust of the engines. As the treadmill accelerates it will actually turn the wheels on the plane faster and faster. If friction increases as the wheels rotate faster, eventually it will reach a point where the friction pushing against the plane is the same as if the wheels were locked in place. The plane will then slide along the treadmill if the engines can push hard enough. If friction does not increase as the wheels spin faster, then the treadmill will attempt to accelerate to infinity. As long as the wheels on the plane can spin at infinite speed, the plane will continue to move forward and take off. If friction moves towards infinity, then it will eventually reach a point where it balances the forward thrust of the engines and the plane will stop moving forward and will never take off.

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