Ok Pat,
I’ll take one more stab at it as I promised I would.
Here are the general ideas that I am assuming.
1. The plane is being propelled by thrust, i.e. jet engine, or by propellers.
2. The wheels on the plane are free spinning, no brakes applied, similar to a bicycle.
3. The plane is attempting to take off just like a normal takeoff, from a stop, full thrust applied until it begins to move with enough air speed to create the required lift to fly.
4. The plane needs 2000 feet to gain a speed of 200 mph, the required air speed on a normal runway for this particular model plane X to fly.
5. The MCB is 3000 feet long and the plane is at one end of the MCB.
6. The question “Will it take off, means will it leave the ground and remain in the air.
Now, with these assumption made, the only way for our plane to take off and remain flying is for it to travel, or move, from the start of the MCB toward the opposite end of the MCB, all the while gaining speed until it reaches 200 mph with relation to the ground. If it remains in one place on the MCB, I think we can assume it will not take off and continue to fly.
Now, the original parameters state – “as the plane moves the conveyor moves but in the opposite direction.the conveyor has a system that tracks the speed of the plane and matches it exactly in the opposite direction”
As thrust is applied by the engines, they begin to push the plane forward along the MCB. The only way to stop the plane from moving is to apply an equal amount of force against the plane in the opposite direction. Now, lets factor in the movement of the MCB. Our plane is moving forward at 10 mph using X amount of force from the engines. Our MCB begins moving in the opposite direction as the plane at 10mph. The planes wheels are now spinning at 20 mph, but what force has been applied to the plane to slow it down. The only force is the amount of friction created by the moving wheels. There's nothing else on the MCB creating force against the airplane to slow it down. I think it’s safe to say that the amount of friction from the wheels is less than the X amount of force applied from the engines. In fact, the friction of the wheels would be minimal. Therefore the plane will continue to move forward. Since max thrust is being applied, the plane continues to gain speed and move toward the other end of the MCB. As it approaches 200 mph, the conveyor is moving at 200 mph in the opposite direction, but little or very little additional force is being applied against the airplane. It cannot and will not stay in place due to the movement of the MCB.
Here’s another real world example you could actually test. Take a remote controlled airplane. Let’s assume is has a max flying speed of 20 mph. Place it on a treadmill going 30 mph in the opposite direction the plane is flying. Hold it in place on the treadmill with your hand so that the wheels are touching and moving with the treadmill. It takes virtually no effort to hold it in place. Now apply the engine at full throttle. You will feel the plane start to pull forward. Let go, it will move up the treadmill even though the treadmill is moving in the opposite direction faster than the ability of the plane to fly.
As for your theory, I'm sorry but I'll admit, your way over my head on most of it. Sorry for the long post, but this is my last shot at this.
jk