Math Is Fun Forum

crazy 8's

Member

Registered: 2008-01-18

Posts: 1

Will this plane take off?

This (trick) question was posted in the "general" section of a photography forum that I frequent. It has turned into a very long thread.

"Let's say you have a single engine airplane sitting on a runway that is 3000 feet long, and through some great engineering the runway is actually a giant treadmill. It's engineered so that when the plane starts to move forward the runway moves in the opposite direction at exactly the same speed., i.e. as the airplane's wheels pick up speed the runway matches it.

Will the plane get airborne?"

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Will this plane take off?

I would guess no, because normally planes get airborne due to air moving quickly over and under the wings, creating differences in air pressure or something like that.

If the treadmill moves in the opposite direction to the plane, then the plane is effectively staying still. Therefore the wings won't pass through any air and so it won't get lifted.

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Why did the vector cross the road?
It wanted to be normal.

PHP Genius

Member

Registered: 2007-12-04

Posts: 5

Re: Will this plane take off?

I think it would get airborne. Seen as it's not the wheels that drive it, it's the jet engines. So it's still going to be 'pushed' along at the same rate, no matter how quick the treadmill is moving.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Will this plane take off?

There is no question that the treadmill will put a force on the plane.  The question is whether or not this force is signficant compared to the mass of the plane and the force produced by the engines.  Unless someone actually knows the equations behind these things, all debate is going to be pure speculation.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Will this plane take off?

If it was a rocket engine so powerful that it could break the static friction between the rubber
and the runway, then the plane probably wouldn't need any wheels anyway!!
So the wheels would spin until the rockets got up to near full power, and then
the force on the air would exceed the static friction to the ground and wham, it
would skid and move forward and the wheels would still turn but might slow down
a bit, but still turn, and then it would take off eventually, if the single rocket was strong enough!!!!

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igloo myrtilles fourmis

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: Will this plane take off?

mathsy is right, it doesn't matter how fast the plane is moving relative to the ground, it needs to move relative to the air in order to produce lift.  The only way a plane in such a scenario could take off would be if the engines were angled downward, like a Harrier jet.  If the engines are aligned horizontally like normal then the plane can't possibly take off, no matter how fast it moves relative to the runway.

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Wrap it in bacon

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Will this plane take off?

Due to post#3, I think it would take off now.

The wheels are presumably just tyres on an axle, with nothing to drive them (or at least, nothing that would be driving them during takeoff).
They'd have brakes, but those wouldn't be applied during takeoff either.
Therefore, they wouldn't resist circular motion.

I think that the treadmill will largely just cause the wheels to spin around, without significantly affecting the overall motion of the plane. Thus, the jet engines would still drive the plane forward and so it would take off normally.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Will this plane take off?

Therefore, they wouldn't resist circular motion.

Therefore, their resistance to circular motion would be negligible.

Is that what you meant?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Will this plane take off?

Picky Ricky.

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Why did the vector cross the road?
It wanted to be normal.

Laterally Speaking

Real Member

Registered: 2007-05-21

Posts: 356

Re: Will this plane take off?

It would obviously take off. The force generated by the engine is pushing the plane forward through the air, not turning the wheels. Therefore, if you take the option of poorly lubricated wheels out of the picture, there is no difference that the moving runway should make.

The only purpose of the landing gear is to reduce the amount of damage caused by the ground (otherwise, the belly of the plane would be torn to shreds by that force), by keeping the plane off the ground.

Last edited by Laterally Speaking (2008-01-29 08:43:26)

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"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

Laterally Speaking

Real Member

Registered: 2007-05-21

Posts: 356

Re: Will this plane take off?

After more reflection, I think that, as it was said earlier, the only way it could take off is if the thrust provided were enough to break the static friction between the runway and the wheels; that is to say, it would have to be enough to drag the wheels along the runway at speeds sufficient to take off.

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"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"