The "Plane on a Treadmill" Question

[Phage0070]

Originally Posted by Kitsune

A plane is standing on a runway that can move, like a giant treadmill. When the plane's engines throttle up, it begins to move forward, but the treadmill is made to match the forward speed of the plane, only in the opposite direction. So, as the plane moves forward, it moves backwards beneath the aircraft.

As the engines throttle up, does the plane take off?

I will place my answer and explanation in the context of a few assumptions which follow:
1) We will assume that every piece of equipment will function as intended, regardless of the stresses placed on it. This means that the treadmill and the plane's landing gear will not simply fail by being misused.

We need to assume this because if we don't, the problem is subject to wild speculation on the capability of different elements of the model. One person may think the hypothetical treadmill is only capable of 35mph, another than the engines will overheat and explode if run at higher than taxi speeds without adequate airflow.

2) All other factors that are unstated are assumed not to exist, and will never become important to the conclusion. This means there is no headwind or tailwind, and the action of the treadmill will not create one.

Obviously adding elements not stated can alter the outcome enormously.

3) The airplane's engines are capable of finite thrust, equivalent to any real world example you choose.

4) The hypothetical treadmill is capable of infinite exertion within its role; it will move as quickly as required to keep the plane from moving forward.

We cannot simply say that the hypothetical treadmill is not capable of doing something the model specifically states it will do. That would change the original question itself. (BTW, the particular way the treadmill is stated in the problem is incorrect, but this interpretation attempts to follow the spirit of the question)

5) While equipment is assumed to function as intended, they are still subject to physical limitations. The landing gear are not perfectly frictionless, etc.

So, within this context, the plane can never take off. Lets first look at the plane at rest and the forces acting on it.

At rest, the plane is held in place by inertia and the friction of the landing gear. The landing gear is designed to reduce friction (compared to the belly of the plane), but it is not enough to simply glide across the runway with the slightest push. During use the landing gear will heat up due to this friction, with all the vibration and heat coming from the energy sapped by said friction.

Now if we start up the system the plane must be able to roll forward in order to create the lift required to take off. So, the question becomes "Can the airplane roll forward?" The plane pushes against the air and is held back by the friction of the landing gear, which will scale according to a small percentage of the speed of the wheels against the treadmill. Since the landing gear will always be able to provide a little more resistance if the treadmill's speed increases, the plane cannot roll forward and thus will never take off.


Practical issues with this model include the potential friction absorption capacity of the landing gear itself. The average jet engine can probably output enough thrust to exceed the capacity of the landing gear to shed energy, and they would fail spectacularly at some point. The treadmill will also end up traveling at astounding speed, and might end up creating significant airflow even supposing it could be constructed. But, that isn't the question we were asked.