The "Plane on a Treadmill" Question

[Happy Monkey]

Originally Posted by Pie

The plane is not moving ith respect to the AIR,

Yes it is. The plane is moving with respect to the air at the speed that its engines propel it. The treadmill is irrelevant. The speed of the treadmill does not affect the forward motion of the plane relative to the air or the ground.

[SteveDallas]

OK, let's try this one. The engine is not on at all. The plane is just sitting there. Now the treadmill/runway starts to turn, let's say at 10 MPH. What happens to the plane?

[glatt]

Originally Posted by SteveDallas

OK, let's try this one. The engine is not on at all. The plane is just sitting there. Now the treadmill/runway starts to turn, let's say at 10 MPH. What happens to the plane?

Friction or no friction?

[Kitsune]

Originally Posted by SteveDallas

OK, let's try this one. The engine is not on at all. The plane is just sitting there. Now the treadmill/runway starts to turn, let's say at 10 MPH. What happens to the plane?

It moves backwards because of friction. The important aspect here is that it takes very little force to overcome this friction because the wheels roll.

[glatt]

In Steve's scenario, the plane moves along with the runway if there is friction, and it stays stationary if there is no friction.

[SteveDallas]

It's physics class so let's ignore friction.

[glatt]

It's the old pull-the-tablecloth-out-from-under-the-dishes trick.

[Flint]

Originally Posted by Happy Monkey

Because the speed of the plane is not determined by its wheels.

Forget the wheels.

Wheel speed is not a factor in the original question, so you must assume that it is accounted for, or you are adding extra information.

Originally Posted by Undertoad

The conjunction does not change the meaning of the question.

We need to leave the original question alone. But, if we did do what you said, I agree.

Originally Posted by Happy Monkey

The speed of the treadmill does not affect the forward motion of the plane relative to the air or the ground.

But it does, necessarily, affect the speed of the plane relative to the treadmill (see: Interpretation #1).

Originally Posted by SteveDallas

OK, let's try this one.

Or, let's just stick to the original question, as stated.

[Kitsune]

Flint, you make about as much sense as Pie's diagram.

This scientific model is proof it works.

[LabRat]

From the straight dope:
An airplane taxies in one direction on a moving conveyor belt going the opposite direction. Can the plane take off?
03-Feb-2006


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Dear Cecil:

Please, please, please settle this question. The discussion has been going on for ages, and any time someone mentions the words "airplane" or "conveyor belt" everyone starts right back up. Here's the original problem essentially as it was posed to us: "A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?"

There are some difficulties with the wording of the problem, specifically regarding how we define speed, but the spirit of the situation is clear. The solution is also clear to me (and many others), but a staunch group of unbelievers won't accept it. My conclusion is that the plane does take off. Planes, whether jet or propeller, work by pulling themselves through the air. The rotation of their tires results from this forward movement, and has no bearing on the behavior of a plane during takeoff. I claim the only difference between a regular plane and one on a conveyor belt is that the conveyor belt plane's wheels will spin twice as fast during takeoff. Please, Cecil, show us that it's not only theoretically possible (with frictionless wheels) but it's actually possible too. --Berj A. Doudian, via e-mail

Cecil replies:

Excuse me--did I hear somebody say Monty Hall?

On first encounter this question, which has been showing up all over the Net, seems inane because the answer seems so obvious. However, as with the infamous Monty-Hall-three-doors-and-one-prize-problem (see The Straight Dope: "On Let's Make a Deal" you pick Door #1, 02-Nov-1990), the obvious answer is wrong, and you, Berj, are right--the plane takes off normally, with no need to specify frictionless wheels or any other such foolishness. You're also right that the question is often worded badly, leading to confusion, arguments, etc. In short, we've got a topic screaming for the Straight Dope.

First the obvious-but-wrong answer. The unwary tend to reason by analogy to a car on a conveyor belt--if the conveyor moves backward at the same rate that the car's wheels rotate forward, the net result is that the car remains stationary. An aircraft in the same situation, they figure, would stay planted on the ground, since there'd be no air rushing over the wings to give it lift. But of course cars and planes don't work the same way. A car's wheels are its means of propulsion--they push the road backwards (relatively speaking), and the car moves forward. In contrast, a plane's wheels aren't motorized; their purpose is to reduce friction during takeoff (and add it, by braking, when landing). What gets a plane moving are its propellers or jet turbines, which shove the air backward and thereby impel the plane forward. What the wheels, conveyor belt, etc, are up to is largely irrelevant. Let me repeat: Once the pilot fires up the engines, the plane moves forward at pretty much the usual speed relative to the ground--and more importantly the air--regardless of how fast the conveyor belt is moving backward. This generates lift on the wings, and the plane takes off. All the conveyor belt does is, as you correctly conclude, make the plane's wheels spin madly.

A thought experiment commonly cited in discussions of this question is to imagine you're standing on a health-club treadmill in rollerblades while holding a rope attached to the wall in front of you. The treadmill starts; simultaneously you begin to haul in the rope. Although you'll have to overcome some initial friction tugging you backward, in short order you'll be able to pull yourself forward easily.

As you point out, one problem here is the wording of the question. Your version straightforwardly states that the conveyor moves backward at the same rate that the plane moves forward. If the plane's forward speed is 100 miles per hour, the conveyor rolls 100 MPH backward, and the wheels rotate at 200 MPH. Assuming you've got Indy-car-quality tires and wheel bearings, no problem. However, some versions put matters this way: "The conveyer belt is designed to exactly match the speed of the wheels at any given time, moving in the opposite direction of rotation." This language leads to a paradox: If the plane moves forward at 5 MPH, then its wheels will do likewise, and the treadmill will go 5 MPH backward. But if the treadmill is going 5 MPH backward, then the wheels are really turning 10 MPH forward. But if the wheels are going 10 MPH forward . . . Soon the foolish have persuaded themselves that the treadmill must operate at infinite speed. Nonsense. The question thus stated asks the impossible -- simply put, that A = A + 5 -- and so cannot be framed in this way. Everything clear now? Maybe not. But believe this: The plane takes off.

--CECIL ADAMS

[Happy Monkey]

Originally Posted by Flint

Forget the wheels. Wheel speed is not a factor in the original question, so you must assume that it is accounted for, or you are adding extra information.

No, because wheel speed is identical to the speed of the plane relative to the treadmill. They are the same thing.

But it does, necessarily, affect the speed of the plane relative to the treadmill (see: Interpretation #1).

But the speed of the plane relative to the treadmill does not affect the speed of the plane relative to the ground. The plane's engine does. If the engine is going, then the plane moves forward relative to the ground.

[9th Engineer]

My God I can't believe this problem generated 6 pages of posts.
THE PLANE TAKES OFF!!!!
Flint, you are saying we need to forget HOW a plane works and just assume that because of the word 'but' the plane cannot move forward, even though the question then goes on to say that all the treadmill is doing is matching its speed with the forward motion of the aircraft. Flint, you are correct if this was a car with wings, but because the speed of the ground has no effect on the speed of the AIR around the plane, it will take off normally.
Asking to forget everything about how the objects in question work and make an assumption based on a conjunction in one of the sentences is rediculous, there's no point to even making it a plane at that point then, we can't even assume the wings generate lift because the problem doesn't say that they do.

[glatt]

Flint knows the plane takes off from a "normal" runway treadmill. He's saying this is a special treadmill that somehow holds the plane back. He's deliberately looking for a way to misinterpret the meaning of the question just so he can argue. I don't know if it's face-saving on his part or just bullshitting on the internet. I assume it's the latter.

[LabRat]

Pie, compare your diagram to mine in post 67. See the difference? This is critical. The treadmill does not move the axle, it moves the wheel around the axle. Thus, the planes forward thrust has no opposite force, and moves the plane (attached to the axle) forward until lift overcomes gravity and the plane takes off.

This has been fun, but I didn't get a damn thing done this afternoon. We need to add a NSFP* warning on these.

*Not Safe For Productivity.

[Kitsune]

Originally Posted by LabRat

This has been fun, but I didn't get a damn thing done this afternoon.

Alright, mission accomplished!