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.

[Happy Monkey]

Originally Posted by Phage0070

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)

That's your mistake. The trick in the trick question is that it is phrased in a way that encourages you to interpret its spirit rather than its words. It subtly encourages you to choose the physically impossible interpretation.

If there are two interpretations, and one requires a treadmill that can stop a jet engine using friction in the bearings of the landing gear, pick the other interpretation.

[Phage0070]

Originally Posted by Happy Monkey

If there are two interpretations, and one requires a treadmill that can stop a jet engine using friction in the bearings of the landing gear, pick the other interpretation.

One interpretation requires a treadmill to stop a jet engine using friction in the bearings of the landing gear, which is theoretically possible.

The other interpretation is not internally consistent as it requires the treadmill to be simultaneously stationary and moving. (If the plane rolls forward and the treadmill is matching speed as compared to the ground, to keep it stationary would imply that the treadmill is stationary as well, but also moving at twice the speed the plane is attempting to accelerate) I chose the one that made sense.

[lookout123]

Thanks steve. I really appreciate the social responsibility you showed by re-opening this thread.

[Phage0070]

Originally Posted by Happy Monkey

It subtly encourages you to choose the physically impossible interpretation.

Also, what is so physically impossible about stopping a jet with landing gear friction? I would wager that the friction of even a single 747's landing gear assembly can stop the thrust of my model rocket's engine, even without a treadmill. If you don't dispute that, the question becomes if the friction scales with speed, which should be clear.

[Shawnee123]

Five!

[SteveDallas]

Originally Posted by lookout123

Thanks steve. I really appreciate the social responsibility you showed by re-opening this thread.

Just think of it as my own little contribution to our community.

[HungLikeJesus]

Shawnee seems to be the only one who has given this any serious thought.

[Happy Monkey]

Originally Posted by Phage0070

The other interpretation is not internally consistent as it requires the treadmill to be simultaneously stationary and moving. (If the plane rolls forward and the treadmill is matching speed as compared to the ground, to keep it stationary would imply that the treadmill is stationary as well, but also moving at twice the speed the plane is attempting to accelerate) I chose the one that made sense.

Where are you getting the highlighted bit, from the question?

Plane moves forward, relative to the ground, at speed X.
Treadmill moves backward, relative to the ground, at matching speed X.
Wheels spin at 2 X as plane takes off.

The question tries to trick you into thinking that the "spirit" of the question is that the plane is held stationary relative to the ground, but the words of the question make no such claim.

[Phage0070]

Originally Posted by Happy Monkey

Where are you getting the highlighted bit, from the question?

Plane moves forward, relative to the ground, at speed X.
Treadmill moves backward, relative to the ground, at matching speed X.
Wheels spin at 2 X as plane takes off.

The question tries to trick you into thinking that the "spirit" of the question is that the plane is held stationary relative to the ground, but the words of the question make no such claim.

Plane moves forward, relative to the ground, at speed X.(wheels spin at X, unless you are suggesting chunks of them are departing the plane)
Treadmill moves backward, relative to the ground, at matching speed X.
Wheels spin at 2X - so the treadmill must be moving at 2X.
Wheels spin at 4X - so the treadmill must be moving at 4X.
Wheels.....

And thus the problem with the statement. Obviously someone was trying to say something else.

[Happy Monkey]

Originally Posted by Phage0070

Plane moves forward, relative to the ground, at speed X.(wheels spin at X, unless you are suggesting chunks of them are departing the plane)
Treadmill moves backward, relative to the ground, at matching speed X.
Wheels spin at 2X

Yes...

- so the treadmill must be moving at 2X.

No.

Or, more accurately, the treadmill is moving at 2X, relative to the plane. It it still, however, only moving at X relative to the ground, matching the plane.

The treadmill matches the plane speed, both speeds relative to the ground. The plane moves at X, the treadmill moves at X in the other direction, and the wheels spin as if the plane were moving at 2X.

[glatt]

Originally Posted by Phage0070

Plane moves forward, relative to the ground, at speed X.(wheels spin at X, unless you are suggesting chunks of them are departing the plane)

I disagree. The wheels don't spin at X. As soon as the plane begins to move forward, the treadmill instantaneously kicks into gear, and the wheels have to move 2X just to keep up with the plane. The plane moves X, but the wheels instantaneously move at 2X.

Treadmill moves backward, relative to the ground, at matching speed X.

Yes.

Wheels spin at 2X - so the treadmill must be moving at 2X.

I disagree again. The wheels move at 2X, but the treadmill moves at X. This is why the plane is moving forward relative to the ground.

[Griff]

Treadmills are activated by the force applied to their surface. There is no drive power coming from the wheels. No force
= no treadmill movement. The plane will lift off but won't clear the Universal Gym.








Dirty Damn SteveDallas

[HungLikeJesus]

Originally Posted by Happy Monkey

Yes...

No.

Or, more accurately, the treadmill is moving at 2X, relative to the plane. It it still, however, only moving at X relative to the ground, matching the plane.

The treadmill matches the plane speed, both speeds relative to the ground. The plane moves at X, the treadmill moves at X in the other direction, and the wheels spin as if the plane were moving at 2X.

But then it's a circular question. For those saying that the plane doesn't move, relative to the ground, X = 0 and 2X = 0, so the treadmill doesn't turn and the plane can't take off. But if the treadmill doesn't turn, the plane can move as normal and will take off as from a normal runway. Which proves that the plane will take off.

[glatt]

Originally Posted by glatt

I'm so out of this thread once it passes the 200th post. mark my words.

DAMN IT!