This is a pure mechanics problem, and we must examine it in terms of forces applied on the plane.

While still on the conveyor/runway, the plane is subjected to the following forces:

Gravity, oriented downwards, constant.

Lift, oriented upwards, function of airspeed.

Thrust, oriented forwards, assumed constant.

Drag, oriented backwards, superposition of air resistance and friction in the wheels.

Obviously the plane will not take off until the force due to lift overcomes the force due to gravity. For this to happen, the plane must achieve some minimum forward velocity relative to the air. Forward acceleration will only occur as long as the force of thrust overcomes the drag force.

Because the conveyor belt speed is the inverse of the plane's airspeed, we can say that the conveyor belt does not move until the plane begins to accelerate forward. Thus, rolling resistance has already been overcome by thrust when this problem really begins.

Initially, the engine thrust is considerably higher than the drag—this is what allows aircraft to take off on regular runways. So the question is, once we start moving and the conveyor belt starts up, does it impose some force on the aircraft that can overcome the force of thrust?

Certainly not initially. Consider the case where the aircraft has just begun to move through the air at, say, 1 m/s. The conveyor belt is then moving backwards at 1 m/s and therefore the speed of the aircraft relative to the conveyor is 2 m/s. If a 2 m/s rolling speed created enough friction to overcome the engines, then no aircraft could ever accelerate beyond this speed on a regular runway.

In fact, we know that at the moment of takeoff the wheels are still on the ground and rolling and the aircraft is still accelerating. So up to that rolling speed at least, the engines still win the thrust vs. friction competition. Would rolling at twice that speed cause sufficient friction?

Consider the thrust of an aircraft engine. I won't give you figures here because it depends widely on the type of engine. But... it's a lot of force. Now contrast this with rolling friction. I'm guessing here, but I think that before a wheel could provide enough friction (without using the brakes) to completely counteract an aircraft engine at full throttle, the wheel would be spinning so fast that it would come apart. At any rate, it certainly would not reach this tremendous amount of friction at a rolling rate twice that of a standard takeoff speed.

The airplane would take off normally, with the wheels spinning twice as fast as normal and a *slight* reduction in acceleration due to added friction.

I think where people get thrown off is the idea that if the conveyor is moving backwards and the plane is on the conveyor, then this must translate to some backwards momentum that has to be overcome by the thrust. But of course the conveyor doesn't start up until the plane is already moving forward. It is the conveyor that must overcome the forward momentum imparted by the engines—and the forces just aren't comparable.

http://mouser.org/log/archives/2006/02/001003.html