Can an airplane takeoff from a treadmill?

For another mental image; imagine a light bulb is on the hub/axle of the jets wheel. Turn the lights off in the room. You see the light stationary; the jet engines kick on. What happens? The light moves forward as the jet accelerates.

The contact between the jet and the treadmill is essentially meaningless.
 
Mr. Orange has nicely articulated the correct answer to a different question. If an airplane accelerates past the velocity at which the treadmill is moving in the opposite direction, it will eventually achieve the ground speed necessary to take off. But that isn't what is being asked here, because that is exactly what a plane is doing when it takes off from a runway. It is accelerating past the speed at which the runway is moving, which is zero.
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Oh, and the frictionless environment point is also relevant. If we are in a frictionless environment, the plane will take off no matter how fast the treadmill belt is moving in the opposite direction. But I think the purpose of the question must also assume that we are not in a frictionless environment because if we were, the wheels wouldn't actually spin, they would just slide forward on the treadmill belt (but they would also do that on a runway). With friction, even the most expertly crafted wheels would create some friction on the landing gear which would act against the forward force the plane's jets generate.
 
Wow.

This is so simple.

If the plane isn't moving relative to the ground (actually, the air), then it won't take off.

If it is moving faster than its takeoff speed relative to the air/ground, it will take off.

However you formulate the question, those two possibilities govern every answer.

As a side note, the obvious implication of this question is that the plane is stationary with respect to the ground/frame of the treadmill (the very nature of a treadmill). Any other formulation of the question is the exact same as a traditional runway.
 
Then what is the point of the question? How is it any different than taking off from a runway if the treadmill exerts no force on the system?

Given friction between the axle and the wheel of the plane (or wherever it is supposed to turn), some of the torque would be transmitted as a force backwards instead of purely becoming rotation, which is why a car placed on a treadmill would move backwards. Go grab a matchbox car and head to the gym. Turn the thing on then place the car on it (or start it with the car on). The car won't stay still, because of friction with the axle.

Given that reaction, with which we are all familiar, the plane's thrust must first overcome that backwards force. There is a point at which those forces will equal each other and the plane will not move. The lower the friction, the lower this amount of necessary force from the engines. With no friction you reach the idealized situation where it requires no force to keep the plane stationary relative to the ground. Sure, once it overcomes this force it can move forward, but by that point you have a regular plane with less powerful engines.
 
Okay, so we can all agree that however you interpret the question, it is completely pointless. I'm glad it took 6 pages to reach that determination.
 
how about this. redefine the idea of a treadmill and i submit that every plane already does take off from a treadmill. say a plane is sitting on the ground. instead of saying the ground is not moving, which is incorrect, lets say the runway is oriented so that the rotation of the earth is directly negative to the direction the plane is facing. in this scenario, the plane is moving backwards at a high rate of speed, but is able to take off because it can positively accelerate in relation to the air around the engines. in this scenario, planes take off from all directions relative to the earth's rotation. that doesn't really answer the question asked since the earth rotates the plane, ground, and air at the same rate, but still its something.

back to the original question - the wheels of the plane will spin at a rate equal and opposite to that of the treadmill, minus a coefficient of friction for the wheel bearings. so the engine only has to overcome the coefficient of friction to keep the plane from going backwards, and then apply the normal amount of force required to accelerate the plane to takeoff airspeed. the engine does not have to counter the entire force of the treadmill.

so on a long enough runway, a plane will slowly accelerate backwards at the coefficient of friction until it reaches the same velocity as the treadmill. but once the engines are fired up, all they have to do is counter the friction and the plane will begin losing negative velocity and slowly become stationary in relation to the air and then accelerate positively in relation to the air until the airsppeed is sufficient to generate enough lift for takeoff.

of course the easy solution is to say that the wheel bearings have no rotational coefficient of friction and if we assume that, then the plane does not move no matter how fast the treadmill moves in either direction and the body of the plane does not feel any movement at all. so the plane does not care that the treadmill is going, its engines apply a force and it goes forward.

can a plane take off when the pilots are discussing politics or religion?
 
yet another thing to think about:

there is no conceivable treadmill that could generate friction on the planes wheels to match what a seaplane experiences on every takeoff.
 
holy cow. a whole lot of people seem to have bad info on how the wheels of planes work. as far as i can tell, the wheels of planes don't really have driving force or not much of it at all. they only brake and basically keep the rest of the plane from grinding into the ground. you can't compare this to a car or a bike or a person running in any way because the wheels on an airplane basically freewheel. and the force that makes airplanes move forward in no way requires the movement of the wheels to create thrust.
 
Airplanes are no different than humans. Infact, I was driving by the airport the other day and spotted a "brand new" treadmill out by the dumpster.
 
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