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The treadmill does not remove the ability of the plane to move forward. If it did, you are absolutely correct, the plane could not fly.

The wheels on an aircraft are not attached to any drive mechanism, they are free spinning. In the scenario described they would act as a bearing and mitigate nearly all of force applied by the treadmill. There simply isn't enough friction transmitted to the airframe to counteract the force created by the propeller.

An airplane with no airspeed sitting on a treadmill would not fly, as you say. An airplane generating enough airspeed will. The treadmill does not prevent the airplane from moving forward with sufficient airspeed.

The wheels not being powered is irrelevant. The engine produces forward power. The plane moving forward causes the wheels to move in relation to the forward power of the engine.

By your logic, putting breaks on aircraft wheels would be irrelevant. But guess what, they can and do stop the aircraft from rolling forward.

If you counter that movement - of the wheels - you counter the forward movement of the aircraft, because the movement at this point is against the ground, not the air.

Ergo, no flight.

The plane will fly, and the wheels will spin at the double of the airspeed. The engines 'push' against the air, not the ground.

This isn't true becasue moving the wheels in the oposite direction does little to slow the plane becasue the bearings in the wheels don't transfer any appreciable force back to the airframe.

Try it with a hot wheels car on a treadmill sometime - it take very little force to keep the car stationary because the wheels act as bearing and keep the surface of the treadmill from applying much force to the car body.

Edit: or sandwich a hot wheels car between your palms, move the bottom palm backwards to simulate the treadmill effect -you wont feel much force acting on the body of the car that is resting on your other palm, no matter how fast you move your lower palm in the other direction.

Either way, this is a really interesting debate :)

When you take off with a tail wind you have basicly the same setup, the wheels need to roll faster relative to the ground to take off. All that happens is you need a longer runway, it does not take significantly longer because the largest limit on acceleration is drag from the airspeed not wheel drag.

Isn't this the difference between the way an automobile moves and an airplane? The automobile applies force through friction of a rotating tire against the ground, a non-moving surface. The fact that the car moves through air is secondary, hence aerodynamic designs...

The aeroplane applies force through movement of air via propeller or jet/turbine engine. The fact that the aeroplane interacts with the ground is a secondary (hence bearings to reduce friction) while the primary thrust is derived from air movement.

Therefore, wouldn't the brakes remain relevant for providing the friction against another form of energy (inertia) of contact with the ground to help "fight" the plane's movement through air as it lands? Now a plane is harnessing both mediums - Air flaps/reversed engines to help create braking push against air flow, and wheel brakes to push against the ground...

The real math on this goes someting like this.

Wheel drag is a function of weight, at 1/2 take off speed you have 1/2 the weight friction so you can get closer to take off speed. (The limit works out so you can still take off.)

The less obvious and more important factor is stall speed of an aircraft = takeoff speed, but because of ground effects you can lift the wheels off the ground below take off speed. You don't do this because it creats the posiblity of bouncing the aircraft as you stall once the height is above the ground effects.

"Any assumption that the aircraft can move forward is predicated on the fact that the treadmill cannot match the engine's output. This is accurate, but like I said, outside of the scope of the problem."

If this is assumed then you are absolutely correct. This, however describes a situation that is different than the reality. If the problem is phrased "Assuming that the treadmill can provide opposite forces sufficient to prevent the airframe from achieving enough airspeed to fly, will the airplane take off?" Then no, the airplane won't take off.

Your example of the floats is a good one. The friction created by the water flowing over the floats would be much greater than that of wheels on a treadmill, so in that case the plane may or may not be able to achieve enough airspeed.

What if a big fan blew air straight into the airplanes wings while it was stationary and those fans were attached to the plane so that they would continue to blow air if the plane gained lift?

I think they call these propellers ;)