The problem statement refers to the speed of the plane and the conveyor belt. Nowhere does it say use a different speed reference for the plane body as for the conveyor belt body. Thus logically one would assume to use a common inertial reference frame, like that of the fixed ground, for measuring the speed of either. Assuming the wheels are free to roll, they will always match the belt speed (assuming they don't skid). The plane moves left with velocity V, the belt moves right with velocity -V, and the velocity of the plane relative to the belt is 2V.

 

following the upside down car theme earlier in the thread:

you have a car, no fix free spinning wheels to the roof of the car and place it upside down on the same conveyor that was used with the plane. then say someone built a "roof" at just the right height where the tires of the car are touching it. now assuming you have a fuel system set up for running upside down and all that stuff. would the car accelerate? yes it would. regardless of wether the conveyor was set up to match the vehicle speed, or the wheel speed. the wheels are free spinning and would still run at 2x the rate of the car.

lol, this analogy is "out there"

 

I'm aware that speed doesn't not necessarily translate into lift. A plane can sit on the ground without an engine, but if enough air (wind) can flow underneath the wing, it will fly. For example, a kite will fly even though its absolute speed is 0 mph relative to the ground. It will fly because of the lift created by the air pressure underneath its "wings."

It doesn't matter if the plane is going 100 mph, or 1,000 mph in relation to the conveyor belt or. If there is no air pressure under the wings, it can't fly. Even if the plane was going 1,000 mph in relation to the ground, it can't fly in a vacuum because there can be no air pressure for the wings to take advantage of.

It seems to me that you are proposing that all the airplane needs to do is double its speed in order to overcome the speed of the conveyor belt. But, I'm assuming the conveyor belt will increase its speed accordingly so that the plane will not move in reference to the ground, which is stated in the problem.

 

You are missing the constraints of the problem. The free-spinning wheels have to be the same speed as the conveyer belt, as per the statement of the problem. Now, write out all the equations you want, the only way for that to work is for there to be no lateral movement of the axis that the free-spinning wheels are attached to, meaning no lateral movement of the car.

This is not a physics problem, its a problem with the constraints of the system as described in the hypothetical problem. Now, in the real world, where it is not frictionless and the conveyer belt will be subject to accelerative as well as velocity forces, then yes, the plane will take off. But then again, you won't be able to build a conveyer belt that can automagically adjust its velocity without exerting an accelerative force, so the question is invalid.

 

The answer to the question simply comes from whether or not there is a large positive accelerating force on the aircraft that will accelerate it relative to the inertial frame. Your argument assumes that somehow a fast moving conveyor belt can impart a force to the plane through the wheels that will overcome the incredibly large force from the engine thrust. That is simply not the case.

If the plane is sitting on a conveyor belt and you tug on the belt, the plane doesn't magically move backward at the speed at which you tug. Tugging on the belt can only impart some force to the plane through the belt-wheel interface. That force is small, and is basically just the rolling resistance of the wheel. The airplane thrust can always overcome this.

 

Actually a better reference frame for the plane's velocity would be the air around it, if the earth sits still, but there's a tailwind blowing at the plane's maximum velocity, it isn't going to fly either.

But in still air as long as the plane can accellerate to take-off speed relative to that air, regardless of what speed or even for that matter what direction the wheels are turning, (in the case of this servo-driven conveyor belt you could reverse its direction and they will be turning backward) the plane will generate enough lift to fly.

 

I'm in partial agreement with you there. With a heavy tailwind, the tailwind is actually applying drag forces to the plane that are helping accelerate it forward. The "maximum speed" of the plain is a function of the force output from the engines balancing the drag of the plane. The plane will have a positive forward force up until the drag cancels it, and this won't happen until the plane actually exceeds the tailwind by it's "maximum speed" in still air. I think it would still take off, given a long enough runway.

But don't mind me, I like to argue.

 

From the way that I am reading this question, the plane will not fly, and here is my reasoning.

If you were to set this plane on the ground with no wheels, just the body laying on the ground, and put the engine at full throttle, it might move a little due to the force of the props. It will not take off because there would not be enough air passing over the wings to create lift. The prop ITSELF does NOT put enough air across the wings for it to create enough lift to gain flight.

So now here is my reasoning why it will never gain flight:

I am assuming that there is no friction on the conveyor belt and the way it's driven. Therefore, it will speed up/slow down at the same EXACT rate as the wheels. Therefore making the conveyor belt an "infinite" length. I am also assuming that this "conveyor" is exactly level, and that the plane has at least three sets of wheels (to stay level).

So, lets say the plane starts at a stop with the engine off, start the engine, once the prop is started it will, obviously, begin to create a pull effect on the airplane. So, on this system, the conveyor belt would begin to move in the opposite direction of the wheels, of course not very fast, and it would accelerate at the same speed until it reached the max output for this level (say 10% of full throttle). The plane would, therefore, still be in the exact same spot because the conveyor belt is canceling out the forward movement of the plane by spinning in the opposite direction at the same speed. Since the conveyor belt has NO delay in matching the speed of the plane's wheels, the plane will ALWAYS stay stationary.

In order for a plane to take off it needs lift. Lift is created by the uneven lengths (top and bottom) of a plane's wing. The air traveling over the top of the wing does not have to travel as far as the air that is traveling under the bottom of the wing. Because of this, the air that goes under the wing has to travel faster, building pressure under the wing. This pressure creates the desired lift. The only way this can all happen is if the wing has enough air flowing over it. The Engine prop (or jet prop) never creates enough air flow over the wings by itself for this lift to be created. illustrated

The props pull the plane at increasing speeds that moves air around the wings, in turn creating enough lift at a certain speed to lift the plane off the ground. In this example this does not happen. There is barely any air being passed over the wings. Instead of air moving, the conveyor belt is moving. For a plane to take flight, the "ground" has to be moving at a slower rate than that of the airplane itsself. That is not happening in this example.

 

He's got pics, and i understand pics. im with his reason.

 

well you got Beurnolli's (sp?) principle wrong. the airfoil of the wing makes the air OVER the wing travel faster than the air UNDER the wing, causing a higher presure under the wing and creating lift.