[QUOTE=bouv]
Can the plane take off?
The fact that the plane has wheels that spin is irrelevant. The fact that people are confusing the wording of the problem is completely relevant as well as the minor logical imperfections and areas open to interpretation of the original poster’s question, so here are few thoughts I’d like to add without digressing into any particularly confusing or cryptic examples.
The most common interpretation is that the tires are spinning as the treadmill is spinning, and there is the assumption that applying forward thrust to the airplane makes the tires spin, which makes the treadmill spin under them at the same speed. We can make different assumptions for a powered treadmill, and intelligent treadmill that adjusts itself to the speed of the wheels automatically (take speed of zero for instance, as another posted suggested, and play with that for a bit).
The real problem here is that some don’t grasp is that the wheels have nothing to do with the power source of the airplane, so let’s try this:
Put a glider with wheels on a treadmill. Put Jim and Larry and Bob and Fran on the treadmill. Jim, Bob and Larry and Fran push on the glider from behind…what happens to the glider as the ground speed of the treadmill increases? The glider wheels move and so do Jim, Bob, Larry and Fran until they get tired and can’t push anymore. The glider will stay there as long as the wheels continue to spin under the glider.
With the Jet airplane, the thrust is external to the treadmill/wheel system, so pushing with a stick at the back of the airplane will move the airplane. pushing with a stick while on the treadmill yourself and while attached to the airplane will have no affect. If you are on the treadmill and on rollerblades and push on the airplane with a stick, the opposing forces equal about zero (given friction, etc.) and you will be propelled backward because you weight much less than the airplane you pushed on.
With the jet airplane, the only thing that matters is that the jet is pushing on the plane. The wheels can do whatever they want and spin as fast as they want, but it won’t make the plane go any slower or stay still unless the jet engine had absolutely no thrust and was instead used to drive a belt or shaft that drove the wheels. This would be a huge problem because as soon as the plane hit take-off speed, the wheels would not have anything to accelerate against (the ground or treadmill) and the plane would crash after briefly taking flight. So of the plane were driven only by wheels, it would not fly, and a plane of this type would never be designed in the first place, except for an hang glider, a kite, or a parasail.
Now, you could fly a kite while sitting in a wagon on a treadmill, but you would have a hard time launching a hang gliding.
If we clamp the plane to the tarmac or the base assembly of the treadmill will the plane take off? No. It has all the additional weight of the treadmill to try to push forward and we assume the thrust of the engines is not that great? This will also pose some serious aerodynamic problems if the plane were eventually able to drage the entire treadmill assembly down the runway until it achieved take-off speed. Will the ropes holding the plane to the base of the treadmill be pulled taught? Yes. What will happen if the ropes are released? The plane will take off.
For those that are so certain that the plane will not take, off try this: stand behind the airplane with a rope tied to it and tied to your waist (or neck or arms for the more disgusting among you) while your feet are tied to the tarmac. Is this something you are willing to do try to prove your point as the engines go to max thrust?
What would happen if we put another treadmill on the top of the plane attached to another set of wheels protruding from the top of the plane and the bottom treadmill and top treadmill were operated in opposite directions thereby cancelling each other out? The wheels will just spin under the friction of the airplane and the airplane will stand still. When the engines are lit and the air is blown out the back of them at high speed, the plane will then move and will take off after the plane overcomes the friction of the wheels against the treadmills, which is very small.