I've been meaning to post this for a while, as it's an interesting problem_
A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?
Think about it long and hard before you look at the solution (below the jump).
The answer is yes, the plane will take off. This is fairly unintuitive, because we're so used to thinking in terms of cars and bikes, where the motion of the object depends on pushing against the ground below us.
A plane's momentum, though, is provided by it's engines, which push on the air. During takeoff, the wheels are essentially free-rolling.
The best analogy I've heard is the following; Imagine that you take your skateboard onto a treadmill, and hold on to a rope tied to the wall in front of you. You won't go anywhere, right? Now imagine the same setup, only you're wearing a jetpack instead of holding a rope. The only thing pulling you backwards will be the small amount of friction generated by the wheels turning, and the force of the jetpack will easily overcome that. You'll easily move forward and off the treadmill.
It works the same way with the plane.