JK96, et al...
JK you are quite close to being a convert!
I have copied in all your assumptions even the ones about lengths and speeds and all are right on!
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.
The following paragraph is also dead on!
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”
The above is also correct.
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.
OK here is where you start to wander off course big time. This next paragraph of yours is based on unwarranted assumtions and assumes facts not in evidence and violates the conditions imposed by the original problem statement!
Our plane is moving forward at 10 mph using X amount of force from the engines.
How did the plane suddenly get to be going 10MPH???
Wouldn't it have to go 9MPH first and 8MPH before that and 7,6,5,4,3,2,1,,,0.0000001MPH before that? The MCB is a very good device and doesn't wait for the A/C to get to 10MPH before it starts to work. The MCB starts countering the planes attempt to move as soon as the motion is detected which can be at a very low value a tiny fraction of an inch per hour or whatever (it might use LASER interferometry or more advanced motion detectors)
If there were no traction between the plane's wheels and the MCB and if the plane's wheel bearings were frictionless, and if the tires had no rolling friction and the wheels and tires were massless (no weight at all) then the MCB could not counter the planes thrust.
I am not reproducing the rest of your argument because it does not matter since it assumes the plane can move forward and it can't.
NOTE: This is not a practical real world situation. If it were then of course the plane would take off (I'm a pilot, do you suppose I would want the plane to fail if it were avoidable???)
Apparently your statement regarding not understanding "MY THEORY" is a major part of the gap in our understandings. I will try to explain how the MCB can counter the thrust of those powerful Binford Belchfire 5000 Tim The Tool Man Taylor signature model engines!
As soon as the sensitive detectors of the MCB detect the smallest motion they start the conveyor moving in the opposite direction of the plane's desired motioin at just enough speed to match the plane so as to keep it motionless. You will agree won't you that it is possible to move the conveyor such as to stop the planes initial attempt to move, even a microscopic distance? If you have a hard time visualizing that then think that the pilot only gave just a little throttle to move just a very little bit. Then the MCB would move the conveyor just a little bit to exactly match the plane and the net result is the plane stands still with respect to the world.
Here is were the MCB borders on SciFi... As the pilot advances the throttle the MCB accelerates as required to prevent the A/C from advancing. OK, lets take this part slowly and carefully.
In the real world the conveyor would speed up as much as it could and if the tires didn't explode the A/C would roll forward and probably get enouigh speed and lift to take off.)
This however, is not the real world it is the world with an MCB in it.
Ever seen the parlor trick where you yank the table cloth out from under a fancy setting of fine china and crystal? The objects on the table move just a little bit in the directon of the moving cloth but their inertia holds them mostly in place. If you could yank the cloth the cloth fast enough (magic table cloth) you would not detect motion on the part of the table setting.
OK, I'm back... If you "yank" the conveyor belt backwards the plane tends to stand still but the wheels, due to friction between the conveyor and the tires, tend to rotate.
You CAN try this at home kids.
Place something with wheels on something you can yank out from under it. The wheels spin a lot more than the object moves! Why does the object move at all? Why don't the wheels just spin and the object stand still? The wheels have mass (inertia) and the energy required to spin them is supplied by the force on the conveyor you yanked out from under the wheeled gadget. This places a force on the (lets say toy car) and that is why the toy car moves in the direction of the moving conveyor. If the axles were frictionless, there were no roling resistance in the tires, and the whole rotating wheel assembly were massless (weighed nothing) then you could jerk the cloth out at any speed and the wheels would rotate but the toy car would remain motionless.
In your home experiment and in our thought experiment here the wheels are real wheels with mass, roling resistance, and bearing friction. When the MCB moves it puts a real force on the A/C tending to move it in the same direction as the MCB.
In your response you said something aboiut the friction not being too great. Well that depends on the speed of the MCB doesn't it? The faster you run the wheels the more force is transfered due to friction. Since there is no upper limit to the speed of the MCB, it runs as fast as is required to concel the thrust of the engines. All the frictions; roling resistance, bearing friction, whatever contribute to the force holding the plane back. The iinertia of the wheels helps hold the plane back too.
Real wheels have weight and spining them takes energy. Where does the energy come from and how is it transmitted to the wheels? Lets say the pilot firewalls the Binford Belchfire 5000 Tim The Tool Man Taylor signature model engines and produces X amount of lbs of static thrust. This levies a requirement on the MCB. It must continuously accelerate to faster and faster speeds at a rate that will impart a force on the A/C that tries to move it rearward. This force must equal the thrust of those powerful engines. This means for a constant thrust by the engines there must be a constant acceleration of the MCB.
Before the plane runs out of fuel the MCB will have accelerated to extreme speeds and real world tires and bearings and such will have long since failed but that is outside of the realm of the way the problem was stated.
If anyone has a SPECIFIC question they need answered to further their understanding of this explanation I would be happy to provide it here or by PM (your choice)
If someone can point out a serious flaw in this explanation, please do. I am equally pleased to lead, follow, or stay the **** out of the way! IF My description is not valid I sure would like a clear explanation of why not.
I do not represent all my explanation as being unique. Many other posters said essentially the same thing, over and over and over.
I will answer specific direct questions but there will be no additional cradle to grave analysis inflicted on the readers here.
I tried to do this straight forward. There is no tongue in cheek or otherwise humor intended (unless you laugh at the mention of Tim Taylor) Also I forgot to mention that the MCB is a Binford Industries model 7700 and coincidently is also a Tim Taylor signature model.
Pat