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There are lots of sites, like which foster truth. We should honor these sites when they honor truth and consider how WE might better distinguish truth from popular belief.

I love the Straight Dope. It cuts through the bull and gets down to the facts. While much of it may be trivia, the tough questions are also addressed.

Written my the mysterious Cecil Adams, the smartest person in the world, who is never wrong, and never lies.

Well. That is a lie.

It is a fantasy, as Cecil Adams is not the smartest may in the world, I am. :-p

And, He is not always right as I found in the first article I looked at.

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 conveyor). The plane moves in one direction, while the conveyor moves in the opposite direction. This conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor 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:

... But believe this: The plane takes off.

The key thing to understand here is that the plane uses thrust to move and the wheels are not powered. the plane can take off no matter how fast the wheel are spinning as long a friction in the wheel bearings and angular acceleration of the wheels is negligible.

I was impressed that Cecil got to the heart of the matter, that the problem was worded improperly, and he is correct, assuming speed relative to the airport the plane will take off most certainly. If the belt only moves as fast as the plane, then it does not move at all unless the plane is moving and at takeoff the belt is only moving at take off speed opposite the motion of the plane and the wheels are spinning at twice that speed.

However, the nasty trick is in the case where the conveyor moves at fast the wheels are turn, as he says this problem is often framed. In fact, the problem wording does not exclude this case. I was also impressed that he got to the heart of the matter in this case because it is an impossible problem it the plane moves. If the plane is moving at all the wheels must spin faster than the conveyor. He also properly dismissed the infinite speed argument. However, he implies the plane will still take off!

That is not correct. The impossibility assums the plane moves, maybe it doesn't move and is not impossible.

If we consider the resistance to motion due to the angular acceleration of the wheels, the conveyor simply needs to be accelerating fast enough to keep the wheel from moving forward. At all times the wheels and the conveyor will be moving at the same speed such that the plane will not move relative to the airport.

In the theoretical case it makes no difference how fast the wheels and conveyer are spinning. In proportion to mass the energy accererating one dedelerates the other with little loss for zero net motion.

If the plane took off then the wheels certainly tuned farther than the conveyor travelled.

Therefore the wheels moved faster than the conveyor.

The case of speed relative to the conveyor implies that the conveyor travels the same speed as the wheels are turning.

At some point acceleration would become impractical and the system would fail, but for as long as it ran the plane would not move relative to the airport.

In NO case can the plane move if the wheels and the conveyer are moving the same speed. If you allow the wheels to spin faster than the conveyor is moving, and thus allow the plane to move forward, you have violated a condition of the problem.

Thus, the plane will not move much, and will not take off, in the theoretical case.

In the practical case the wheels can only spin so fast before they will explode. When the wheels explode, the fuel tanks will ignight and the whole plane will explode. Thus it will not take off.

If the conveyor moves at the speed of the airplane relative to the airport the plane takes off. If the conveyor moves at the speed of the airplane relative to the conveyor, the plane does not fly.

And, I must be the smartest guy in the world :-p

Or maybe not, I got it wrong a bunch of times before I got it right... My first hunch was correct as was yours most probably. That makes me average I suppose.

I like this problem because there are so many ways that classical logic fails and proves that proofs that are not holistic prove nothing.

The point of the problem is that the plane will take off if speed is taken relative to the earth. It is an interesting problem when worded explicitly. But it can solicite fearse senceless dissagreements when it is not worded explicit as to whether the speed of the airplane is relative to the airport or the conveyor (wheel speed) making the result ambiguous.

JimScarver 13:55, 4 January 2007 (EST)

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