The plane and the conveyor belt

This popular physics question was posted on the tao-gamers.com forum I frequently visit:

A plane is standing on a runway that can move (some sort of huge conveyor belt). The plane moves in one direction, while the conveyor belt moves in the opposite direction. This conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor belt to be exactly the same (but in the opposite direction). Can the plane take off?

Spend some time thinking about the question and go over different scenarios in your head. Try to understand the problem and visualize what's going on. I've heard that even pilots frequently get the answer wrong. I have posted the answer, and my explanation, in a comment to this post. Try to figure it out before you look at the answer!

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  1. I’m guessing your conclusion was that the plane will remain stationary and therefore will be unable to take off. Keep in mind, the question stated the conveyor belt moves in the opposite direction at the same speed of the plane, not the speed of the plane’s wheels!!! If the conveyor belt always moved at the same (opposite) speed as the plane’s wheels, then yes the plane would never move!

    The “speed of the plane”, as far as the conveyor belt is concerned, is totally unrelated to the speed of the wheels. Because the plane’s wheels freely rotate (like the wheels on a toy car), the plane itself would be allowed to move forward as long as the wheels are spinning at exactly double the current speed of the plane.

    — Plane moves at 1mph.
    — Conveyor moves at 1mph.
    — Wheels spin at 2mph.
    — Plane’s engine pushes against the AIR (not the ground) and moves the plane forward.
    (this process continues until the plane has enough speed to take off, at which time the wheels would be spinning at exactly double the current speed of the plane)

    So now:

    — Plane moves at 100mph.
    — Conveyor moves at 100mph.
    — Wheels spin at 200mph (while the plane is actually moving at only 100mph and takes off)

    Think about this: A water plane does not use wheels to take off! It pulls itself through the AIR while gliding on the water.

    On the other hand, if you stated that the AIR was moving in the opposite direction as the plane, with the same force and displacement as the plane’s engines, then the plane would never take off — regardless of how fast it was moving on the ground.

    Plane = air
    Car = ground

    Car on conveyor belt = car doesn’t move. (ground)
    Plane on conveyor belt = plane moves, wheels just spin. (air)
    Car with air in opposite direction = car moves. (ground)
    Plane with air in opposite direction = plane doesn’t move. (air)