Resolved: The plane will NOT take off.
(See also here, here, here, and here.)
FOLLOW-UP:
More here, here and here.
ANOTHER FOLLOW-UP:
I'm still thinking about it, and now I can see the argument for it taking off, too. Arrgh.
YET ANOTHER FOLLOW-UP:
I think this is the best explanation I've seen so far. Here's the key bit:
(See also here, here, here, and here.)
FOLLOW-UP:
More here, here and here.
ANOTHER FOLLOW-UP:
I'm still thinking about it, and now I can see the argument for it taking off, too. Arrgh.
YET ANOTHER FOLLOW-UP:
I think this is the best explanation I've seen so far. Here's the key bit:
While the conveyor does exert some modest backward force on the plane, that force is easily overcome by the thrust of the engines pulling the plane ahead. The plane moves forward at roughly its usual speed relative to the ground and air, generates lift, and takes off.I'd say the engines are pushing, not pulling, but otherwise, I agree.
IF the conveyer belt exactly matches the speed of the wheels moving in the opposite direction, that's essentially saying that the conveyer belt accelerates with the engine thrust to provide more friction, so that the plane is locked in place.
Of course, as a practical matter the moving runway will itself carry a stream of air towards the plane that will provide some lift, but unless the runway conveyor belt is very long and wide, the airspeed in the middle of the runway will never reach takeoff speed, so the place will not rise from its still position.
Of course theoretically a plane can take off from a still position relative to the ground - that's what a wind tunnel is. Relative airspeed is all that matters for purposes of getting lift.
So we agree. But do we agree for the same reasons?
However, if the aircraft remains fixed reletive to the frame of the treadmill, it will remain fixed reletive to the air, and no lift will be generated, except the incidental lift caused by the local slipstream at the engine intake and outlet, which is not sufficient.
I think the mistake everybody is making is thinking that the wheels have anything to do with moving the aircraft forward. If the aircraft does not move in relation to the ground, whether the aircraft is on a treadmill or anchored to massive concrete blocks does not matter as far as the engines or wings are concerned.
Now if you put wings on a Chevy Corvette (or a Honda X4...) and ran it down the runway, you could probably drive fast enough to get it to lift off the ground, for a while, until the forward momentum was dissipated by parasitic and induced drag. But if you put that same winged Vette back on the treadmill, we would be right back where we started, with negligible airflow over the wings and no lift.
Try working the problem in reverse. Say that the treadmill is already at full speed, with the plane being held over it by several cranes, without its engines on. If the plane is lowered onto the tarmac, friction will burn off rubber but then the tires will come up to the speed of the runway, with the cranes still holding the plane in place against friction. If the engines are turned on and brought up to full thrust, it will be possible to release the plane from the cranes, but it will not beable to advance - unless you break the premise that the conveyor can accelerate to precisely match the speed of the tires.
The only way the plane gets up is if we assume that the moving conveyor will also accelerate a path of air up to the plane's takeoff speed. Only if the conveyor generates a sufficient wind tunnel effect will the relative airspeed provide enough lift to get the plain off the ground.
If the treadmill matched the speed of the wheels, the treadmill would not move at all. There is no force making the wheels turn, so they would be stationary, along with the treadmill.
Now if the aircraft were held in a fixed position reletive to the frame of the treadmill, much as a runner is when he holds on to the grab bar on a treadmill, the treadmill could be turned to any speed, and the wheels will match the speed of the treamill. But the treadmill will be the driving force that sets the speed, not the wheels.
As to the matter of frictional losses at the wheels... Without the application of brakes, the frictional losses from the wheels will never be enought to balance the thrust of the engines, even at very low thrust setting. Aircraft wheels are designed to spin freely until the brakes are applied. After all, if losses at the wheel were equal to the thrust of the engines at take-of velocities, an aircraft would never be able to accelerate enough to get off the ground.
So the correct frame is to think that the treadmill makes a force ONLY to the wheel and the effect is the spin and the wheel makes a force on his axis, proportional to the friction wheel-axis. Nothing else. Everything is clear then.
The problem is expressed in a tricky way, but it's quite clear that if you suppose ideal (no friction) or reasonable (some friction) parameters the plane will take off.
If you suppose infinite friction between the wheels and the treadmill AND infinite friction between the wheel and his axis the plane will NOT take off and the treadmill is still. No movement at all. But that is an extreme and unrealistic case and obviously not what the author of the problem had in mind.
In each case, however, this force equals the mass of the plane times the acceleration of the plane.
I haven't even tried to crunch through the math on this, and my high-school physics is rusty, but I suspect the treadmill would have to accelerate backwards even more rapidly than the plane itself is accelerating forwards, if it is to keep the plane from taking off. The main reason is that some of the treadmill's force is being used to rotate the wheels. Not all of it is being used to push the plane backward linearly.
The general consensus seems to be that in any remotely realistic situation, the plane takes off, because the backward force of the treadmill could not possibly counter the forward force of the plane's engines. But, as a pure thought experiment, it's possible to imagine a theoretical treadmill accelerating infinitely at such an ungodly rate as to counter the engines' thrust.
As a practical matter, there is no way in hell that a conveyor belt could accelerate with a plane. But definitionally, the plane is stuck. Those who have the plane take off are ignoring the premises.
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