Spark123 said:
For an aircraft to take off (i.e. have lift) it must have forward momentum through the air which is theoretically still. This forwards force is caused by the engines thrust pushing the aeropane forwards against the still air (not the moving runway). This force needs to overcome the air resistance and any other forces acting against it for the plane to move forwards. Now comes the tricky bit when you consider what was being said about the runway moving, it depends on how you calculate the speed, wether it is done relative to a still point (the air or terra firma) where the wheels will spin at twice the rate which they would normally or wether it is calculated between the wheel and the moving runway. It is an impossible concept for the runway to be moving at the opposite but equal speed to the wheel and for the aircraft moving forwards, the only time this holds true is when the aircraft is stationary so this is where the treadmill falls down. As soon as the wheel moves forward say 1m, the treadmill will have to move backwards 1m to compensate causing the wheel to move 2m which means at the same time the treadmill has to move 2m to compensate etc etc etc....
The thing is though, the wheel is a free-spinning object at the bottom of the plane, so moving the wheels circumference round by 1m does not necessarily equate to moving the plane back by 1m.
Imagine the plane is on the runway, but tethered to fixed ground points either side of the runway on non-moving bits of earth.
Start up the runway, and the plane will move backwards a bit, but then the tethers will take up the slack and will in effect act as a force moving in the forward direction. The conveyor is still moving, the wheels are spinning, and the plane stays still.
Now, replace the tethers with thrust. Once the thrust is greater than the forward force those tethers were exhibiting, its going to go forward, regardless of what the conveyor belt and wheels are doing. They will both spin faster, but they do not relate to the position of theplane, as the backwards-direction force (caused by friction between the wheels and the conveyor) is going into spinning the wheels round, not into moving the plane backwards.
It is kind of the same difference as you would get between having a car in neutral on a moving conveyor belt, and a car in 1st gear. The Car in neutral would not move backards at the speed of the conveyor belt, even though the wheels were turning at the same speed as the conveyor was moving underneath it.