Wow. Up to 37 pages, and there's still a handful of morons that think the wheels on an airplane control the airplane's motion...

Guys, it's pretty simple. The original statement says the belt goes the same speed as the plane, but in an opposite direction.

If the plane has no forward motion (speed), then the belt does not move.

If the plane goes forward at 10 mph, then the belt goes in reverse at 10 mph.

If the plane goes forward at 150 mph, then the belt goes in reverse at 150 mph.

But what does that mean to an outside observer, for Madman's benefit? It means that the person, standing still, sees a plane moving forward at 10mph, then accelerating up to 150 mph, and then taking off.

Nowhere in the original problem statement does it say the belt matches the speed of the tires. The belt matches the speed of the AIRPLANE. The speed of the tires are relative to the forward motion of the plane, and the reverse motion of the belt.

In this scenario, the plane's movement forward makes the tires rotate forward. The conveyor belt moving backwards makes the tires rotate forward as well (think of it like two gears in contact with each other; turn one, and the other moves in the opposite direction). The actual rotational speed of the tires is the sum of both inputs, airplane speed plus conveyor speed. So if the plane is moving at 10 mph, and the conveyor belt is moving at 10mph the opposite direction, the tires are rotating at 20 mph.

The thing is, there are two completely independent forces acting on the tires. There is the force from the engines of the plane, pushing air and not directly in contact w/ the tires. Then there's the force of the conveyor belt acting directly on the tires. The tires move based on the two inputs.

The problem that everyone that thinks the plane won't take off is the external power source of the engines. The force of the engines acts on the body of the plane. The tires only spin by the plane if the plane is moving forward and the tires are in contact with the ground. The force of the belt acts directly on the tires, causing them to do nothing more than spin in the forward direction. But again, the tires spinning in the forward direction do not cause the body of the aircraft to move, as they are only free spinning along the axle shaft, with no mechanical linkage to force the aircraft forward.

Guys, the plane has not only taken off on the original page, but it already landed at it's destination, and all the passengers have been sitting at home reading this thread, knowing PERSONALLY that the "won't take off" group are out of their minds...