Conceptual Puzzle: Runway Belt

[KWhit]

Originally Posted by MattJones4Heisman

This may or may not help some of you:

http://www.straightdope.com/columns/060203.html

From that article:

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.


That's what I have been trying to say, but I couldn't articulate it as well as this. The puzzle, as stated, is impossible, but it makes total sense that the plane takes off.

[st.cronin]

Right, if the treadmill is not required to match the speed of the wheels, of course the plane will take off.

[MJ4H]

Sigh. Just a note: I may have put the question on this board, but it is not my phrasing.

[rkmsuf]

Originally Posted by st.cronin

Right, if the treadmill is not required to match the speed of the wheels, of course the plane will take off.

lol

[JHandley]

Originally Posted by st.cronin

Right, if the treadmill is not required to match the speed of the wheels, of course the plane will take off.

Ditto this and Ditto KWhit.

[MJ4H]

And again, this is not a semantics problem. The plane will take off either way.

[KWhit]

Originally Posted by MattJones4Heisman

Sigh. Just a note: I may have put the question on this board, but it is not my phrasing.

I'm not blasting you. Just glad that my thinking wasn't totally fucked up.

[st.cronin]

And with that, I'm off to Vegas, beyotches. Merry XMAS fools!

[Bee]

Originally Posted by MattJones4Heisman

And again, this is not a semantics problem. The plane will take off either way.

Although only one way creates a blackhole on takeoff.

[Dutch]

I keep thinking of the free-wheel vs. the conveyor belt as an attempt to define a friction-less enviroment.

In the free wheel vs the conveyer belt matchup, you could get a giant Jolly Green Giant out there as the emcee scratching away at the conveyer belt to little or no effect on a jet plane about to put 5,000 lbs (or whatever) of thrust on. The thrusters aren't challenging the belt or the wheels, but the object it is applying force to (the plane). A free wheel isn't going to stop physics, no matter how the conveyer belt matches it's rotation, much like in space, for instance, it simply isn't a part of the equation.

"Any action has an immediate/opposite reaction" (paraphrased). Those wheels are free, they can be moving frontwards or backwards in complete unison with the belt, but they won't stop the reaction of the plane moving forward vs the jet's thrust. Sort of like I would imagine the zero gravity/frictionless enviroment of space has no effect on a rocket booster pushing an object through space. The acceleration from the jets is created by it's own mass, nothing else.

[Toddzilla]

Originally Posted by MattJones4Heisman

Sigh. Just a note: I may have put the question on this board, but it is not my phrasing.

Well for god's sake man, don't delete the thread. I'll lose about 40 posts.

[KWhit]

Originally Posted by st.cronin

And with that, I'm off to Vegas, beyotches. Merry XMAS fools!

Hit the O.G. and say "Hi" to Jenn Lynn for me.

[gstelmack]

Originally Posted by QuikSand

It seems to me that this phrase is at the center of the apparent paradox. I'm not really sure how to properly interpret this, but I suspect that if you grant this condition as it is intended, you get the intended whimsical answer to the puzzle.

But there's enough am,biguity in this phrase, to me, to leave this as much a puzzle of semantics as it is physics.

Post #28 got the answer correct.

[MJ4H]

Post #5 got the correct answer with the correct reasoning.

[Huckleberry]

Not really. The Straight Dope guy interpreted "speed of the wheels" incorrectly. The speed of the wheels equals the speed of the plane. It's not like the guy hasn't screwed up before, as have most people.

[MJ4H]

Ding! Round 2. Fight.

[Huckleberry]

Originally Posted by MattJones4Heisman

Ding! Round 2. Fight.

No need. I only know two people that have never screwed up. Me and Vince Young.

[KWhit]

Originally Posted by Huckleberry

Not really. The Straight Dope guy interpreted "speed of the wheels" incorrectly. The speed of the wheels equals the speed of the plane. It's not like the guy hasn't screwed up before, as have most people.

I disagree.

[Huckleberry]

Originally Posted by KWhit

I disagree.

Then your plane is broken.

[JHandley]

Originally Posted by Huckleberry

Here's the mistake. He and others here have taken the factor of 2 and multiplied instead of dividing. If the treadmill is going 5 mph backward then the wheels are "turning" 5mph forward. And the wheels are moving 2.5 mph relative to the ground.

Can you tell me the term for this? Obviously you're getting quite frustrated with having to explain this, so I'd like to goodle the term to find out more about it. In my mind, the logical conclusion that if the treadmill is going 5mph backwards and the wheels are going 5mph forward, that equals each other out and the wheels are not moving relative to the ground.

[Huckleberry]

Originally Posted by JHandley

Can you tell me the term for this? Obviously you're getting quite frustrated with having to explain this, so I'd like to goodle the term to find out more about it. In my mind, the logical conclusion that if the treadmill is going 5mph backwards and the wheels are going 5mph forward, that equals each other out and the wheels are not moving relative to the ground.

There's no term I'm aware of. The only thing to grasp with this is that the Straight Dope Guy and st. cronin are trying to describe angular velocity using the unit for linear velocity. That is the error that leads to the impossibility. Certainly the question is phrased in a way that leads people to make that error in their thinking, but it's still an error in the analysis.

Using your values, if the treadmill is going 5 mph backwards relative to the ground, then the wheel is going 5 mph forward relative to the ground, as is the plane to which it is attached. The bottom of the wheel is going 5 mph backward, the center of the wheel is going 5 mph forward, and the top of the wheel is going 15 mph forward. The wheel is not going 5 mph forward relative to the belt, is going 10 mph forward.

[st.cronin]

Originally Posted by KWhit

Hit the O.G. and say "Hi" to Jenn Lynn for me.

OG?

[st.cronin]

Originally Posted by Huckleberry

There's no term I'm aware of. The only thing to grasp with this is that the Straight Dope Guy and st. cronin are trying to describe angular velocity using the unit for linear velocity. That is the error that leads to the impossibility. Certainly the question is phrased in a way that leads people to make that error in their thinking, but it's still an error in the analysis.

I have been very clear over and over that I am not describing angular velocity.

[JHandley]

Originally Posted by Huckleberry

Using your values, if the treadmill is going 5 mph backwards relative to the ground, then the wheel is going 5 mph forward relative to the ground, as is the plane to which it is attached. The bottom of the wheel is going 5 mph backward, the center of the wheel is going 5 mph forward, and the top of the wheel is going 15 mph forward. The wheel is not going 5 mph forward relative to the belt, is going 10 mph forward.

Is this true regardless of the forces moving the wheel 5mph?

[st.cronin]

Originally Posted by Huckleberry

There's no term I'm aware of. The only thing to grasp with this is that the Straight Dope Guy and st. cronin are trying to describe angular velocity using the unit for linear velocity. That is the error that leads to the impossibility. Certainly the question is phrased in a way that leads people to make that error in their thinking, but it's still an error in the analysis.

Using your values, if the treadmill is going 5 mph backwards relative to the ground, then the wheel is going 5 mph forward relative to the ground, as is the plane to which it is attached. The bottom of the wheel is going 5 mph backward, the center of the wheel is going 5 mph forward, and the top of the wheel is going 15 mph forward. The wheel is not going 5 mph forward relative to the belt, is going 10 mph forward.

You do realize there is a way to describe the motion of the wheel which is NOT angular velocity and which is not in relation to a fixed point on the ground?

You keep wanting to say it is angular velocity that I am talking about, but it is NOT angular velocity.

wheel with 1 foot circumference rotates once ... treadmill with 360 foot diameter moves in the opposite direction 1 degree ... their angular speeds are very different, but they have moved in equal and opposite ways.

This is not at all what you are talking about, because you are defining the speed differently whether it is at the top of the wheel or the bottom of the wheel. That's a totally different elephant. The speed of a given point on the wheel is constant, whether it is at 12 oclock, 6 oclock, or wherever.

[JHandley]

Originally Posted by st.cronin

The speed of a given point on the wheel is constant, whether it is at 12 oclock, 6 oclock, or wherever.

That's exactly where I'm getting lost in this. How does the speed along the circumference change?

[Huckleberry]

Originally Posted by JHandley

Is this true regardless of the forces moving the wheel 5mph?

I'm not sure I understand your question. The scenario is specific to our conditions, namely that the propulsion does not rely on or need friction and that the belt is designed to match the speed of the wheels but rotate in the opposite direction of the wheels.

[st.cronin]

Originally Posted by JHandley

That's exactly where I'm getting lost in this. How does the speed along the circumference change?

It depends on your reference. It's why Saturn retrogrades - it doesn't actually change speed (the speed I'm talking about), but to us it actually appears to stop and reverse course.

[Huckleberry]

Originally Posted by st.cronin

wheel with 1 foot circumference rotates once ... treadmill with 360 foot circumference moves in the opposite direction 1 degree ... their angular speeds are very different, but they have moved in equal and opposite ways.

Exactly. Their tangential speeds are equal. Tangential speed is a linear velocity.

This is not at all what you are talking about, because you are defining the speed differently whether it is at the top of the wheel or the bottom of the wheel. That's a totally different elephant. The speed of a given point on the wheel is constant, whether it is at 12 oclock, 6 oclock, or wherever.

Yes, the speed is the same. The velocity is different. The velocity at 6 o'clock is equal in magnitude and opposite in direction to the velocity at 12 o'clock, relative to the center of the wheel.

[st.cronin]

Originally Posted by Huckleberry

Exactly. Their tangential speeds are equal. Tangential speed is a linear velocity.



Yes, the speed is the same. The velocity is different. The velocity at 6 o'clock is equal in magnitude and opposite in direction to the velocity at 12 o'clock, relative to the center of the wheel.

Ah, but the puzzle is about speed. So, I don't know why you wanted to bring velocity into it at all.

[Arctus]

Originally Posted by MattJones4Heisman

This may or may not help some of you:

http://www.straightdope.com/columns/060203.html

It helped immensely. Thank you.

[ANut]

Originally Posted by KWhit

From that article:

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.


That's what I have been trying to say, but I couldn't articulate it as well as this. The puzzle, as stated, is impossible, but it makes total sense that the plane takes off.

There is no paradox in or implied by that wording. There is only a slight misuse of terminology in the "direction of rotation" phrase. Let me explain:

First, the conveyor belt matches the speed of the wheels. Note it does not say the rotational speed. I think it's clear that what is meant is that the belt moves linearly at the speed of the wheel. You might think this is impossible because you are thinking that the wheel only rotates, however this is not the case. As soon as the plane starts moving, the wheel also starts moving linearly down the runway. It just happens to also rotate. If the linear speed of the belt matches the linear speed of the wheel (wheel hub, airplane structure, etc all have the same linear speed) but in the opposite direction, then the wheel would have to turn twice as fast as it would on an unmoving runway, but it will still move forward at the same speed. This interpretation is the only logical one, and all other interpretations render the question ill-posed.

The phrase "direction of rotation" is not the best terminology that could have been used in this question, since "direction of wheel motion" would have been much more clear. However I believe that the question was purposefully stated in a slightly misleading way (after all, it is a trick question even if it does lead to such thrilling discussions of physics). If the question had been stated with the belt moving "opposite the direction of aircraft motion," then anyone who (like most people) initially thinks of an aircraft behaving like a car would soon realize that if the belt had the power to keep the aircraft from moving forward, then the aircraft would have no motion, so the belt would have no motion. This IS a paradox, but it is created by the faulty assumption or miscalculation that the belt has the ability to hold the aircraft back. As has been explained many times in many ways, there is no mechanism for the belt to place such a force on the aircraft structure, so the engine will propel the plane forward as normal. The only difference caused by having such a conveyor belt under it would be that the wheels would spin twice as fast as they would during a takeoff from a normal runway.

KEY POINT: "speed of the wheels" means the LINEAR speed of the wheel moving forward, NOT the rotational speed or any tangential speed of the wheel.

[PilotMan]

You guys are killing me with this. I think that I can answer this and in much simpler terms.

The force that the belt/runway can apply is only in one direction, at 180 degrees, or straight backward relative to the ground. That cannot change.

The airplane is not limited to moving in that frame of reference. It has a rudder. So while the belt will sap all of it's movment at 360 degrees, it can still move back and forth down the runway from 315 degrees to 045 degrees, similar to a sailboat tacking against the wind. It will gain forward movement because the engines will be able to apply a force at an angle to the opposite movment of the belt, and therefore gain forward momentum until it will ultimately take off.

That seems to be the simplest and most resonable answer.

[Brillig]

Originally Posted by KWhit

From that article:

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.


That's what I have been trying to say, but I couldn't articulate it as well as this. The puzzle, as stated, is impossible, but it makes total sense that the plane takes off.

Who is this "Straight Dope" moron anyway?

Of course there's a problem with A=A+5. Duh. However, he starts by assuming that the plane is moving forward at 5 MPH. He never addresses how it gets to 5 MPH. The whole idea behind the problem is whether or not the plane can accelerate from 0 MPH.

[ANut]

I think it stands to reason that if it can accelerate from 5 mph to 10 mph, it can accelerate from 0 to 5. If there is a net force forward on the aircraft it will accelerate by F=ma, and if it accelerates the velocity will increase, regardless of what the starting velocity is.

[Brillig]

Are you Straight Dope?

Seriously. Of course, "I think it stands to reason that if it can accelerate from 5 mph to 10 mph, it can accelerate from 0 to 5"

DUH.

The question is whether the plane can accelerate at all.

Yes, any fool who knows high school physics knows that F=ma. So what? The question is whether there's a net force on the aircraft in the first place. There isn't really enough information in the puzzle as stated to determine that. So depending on what your assumptions are, you can answer the problem either way.

[ANut]

I disagree. There is a forward force, the engine thrust. What force is there to counteract this force? There is only the friction in the wheel bearings which is very small. The result of adding these two forces together yields a net force forward, which causes the acceleration.

The only difference between the situation in this question and a normal takeoff is that the wheel has to spin twice as fast because the ground underneath it is moving backwards. This would increase the force of friction in the bearings slightly, but not enough to stop any normal aircraft from taking off.

[Brillig]

Originally Posted by ANut

I disagree. There is a forward force, the engine thrust. What force is there to counteract this force? There is only the friction in the wheel bearings which is very small. The result of adding these two forces together yields a net force forward, which causes the acceleration.

The only difference between the situation in this question and a normal takeoff is that the wheel has to spin twice as fast because the ground underneath it is moving backwards. This would increase the force of friction in the bearings slightly, but not enough to stop any normal aircraft from taking off.

Here's where your imagination fails. You concede that there is a retarding force exerted by friction in the wheel bearings. However, you arbitrarily label it as "very small". The force exerted by the bearings, though, is proportional to the velocity of the wheels, and hence to the velocity of the treadmill. Since there is no limit to the speed of the treadmill, there is nothing keeping it from moving at an arbitrarily large velocity sufficient to counteract the forward thrust of the engine. In fact, according to the original gist of the puzzle, for the plane to move forward, the velocity of the wheel and the treadmill would no longer be in sync. Therefore, the treadmill must accelerate to a speed sufficient to generate adequate counter-force.

QED

[ANut]

Originally Posted by Brillig

Here's where your imagination fails. You concede that there is a retarding force exerted by friction in the wheel bearings. However, you arbitrarily label it as "very small". The force exerted by the bearings, though, is proportional to the velocity of the wheels, and hence to the velocity of the treadmill. Since there is no limit to the speed of the treadmill, there is nothing keeping it from moving at an arbitrarily large velocity sufficient to counteract the forward thrust of the engine. In fact, according to the original gist of the puzzle, for the plane to move forward, the velocity of the wheel and the treadmill would no longer be in sync. Therefore, the treadmill must accelerate to a speed sufficient to generate adequate counter-force.

QED

First, there is a limit to the speed of the conveyor belt: the speed of the wheels. The speed of the wheels is the same as the speed of the rest of the aircraft structure. Therefore the fastest the belt will go is the speed of the aircraft at liftoff. The whole result of the belt being there serves only to cause the wheels to spin twice as fast as they would during a normal takeoff.

Without breaking out my old Aerodynamics books, I can be fairly certain that the drag of moving through the atmosphere with flaps and gear down at takeoff speed is much, much higher than the drag in the wheel bearings. When you consider that the additional friction due to doubling the wheel rpm (assuming that the bearings don't burn up and the tires don't blow) is negligible compared to this, you can understand why any normal aircraft would take off even if the belt underneath it were moving in the opposite direction at the same speed.

Why do I get the feeling that I will have to break out my books and prove that to you?

[Brillig]

In the stupid hope of putting this to bed, the fundamental problem is semantic. The puzzle can be read in (at least) two different ways.

a) if a plane is restrained in such a fashion that it cannot move forward except by taking off, can it then, take off?

b) if a plane is restrained in such a fashion such that it's wheels have no correlation with it's actual motion, can it then, take off?

Anyone proposing a solution to the puzzle must first decide which of these two puzzles he's answering. However, since there isn't enough information to make a determination which of the two puzzles is actually in play, anyone who says that he knows the answer has a hidden assumption that he has no basis for.

[Brillig]

Originally Posted by ANut

First, there is a limit to the speed of the conveyor belt: the speed of the wheels. The speed of the wheels is the same as the speed of the rest of the aircraft structure. Therefore the fastest the belt will go is the speed of the aircraft at liftoff. The whole result of the belt being there serves only to cause the wheels to spin twice as fast as they would during a normal takeoff.

Without breaking out my old Aerodynamics books, I can be fairly certain that the drag of moving through the atmosphere with flaps and gear down at takeoff speed is much, much higher than the drag in the wheel bearings. When you consider that the additional friction due to doubling the wheel rpm (assuming that the bearings don't burn up and the tires don't blow) is negligible compared to this, you can understand why any normal aircraft would take off even if the belt underneath it were moving in the opposite direction at the same speed.

Why do I get the feeling that I will have to break out my books and prove that to you?

You can't bring reality into this. The materials science necessary to move a treadmill the size of an airplane runway at takeoff speeds doesn't exist. This is a thought experiment, and your books don't matter.

[MJ4H]

Originally Posted by Brillig

You can't bring reality into this. The materials science necessary to move a treadmill the size of an airplane runway at takeoff speeds doesn't exist. This is a thought experiment, and your books don't matter.

Well that's certainly a new approach.

[Brillig]

Well if it is a new approach, it would certainly explain why this has gone on for 300 posts. Clearly, if we were to attempt to do this in the real world, the treadmill would break down, probably before the experiment started, and the plane would take off. Otherwise, what's your beef?

[MJ4H]

It's absurd. Sure the conveyor belt is theoretical. The plane and the mechanics of the plane are certainly not, however. I thought this was pretty clear. I guess not. If you are interested in a discussion along these lines, I am most certainly not interested.

[Pumpy Tudors]

Wow, you all take your conceptual puzzle shit seriously.

[MJ4H]

I am theoretically kicking your ass right now.

[Arctus]

Originally Posted by Brillig

In the stupid hope of putting this to bed, the fundamental problem is semantic. The puzzle can be read in (at least) two different ways.

a) if a plane is restrained in such a fashion that it cannot move forward except by taking off, can it then, take off?

b) if a plane is restrained in such a fashion such that it's wheels have no correlation with it's actual motion, can it then, take off?

Anyone proposing a solution to the puzzle must first decide which of these two puzzles he's answering. However, since there isn't enough information to make a determination which of the two puzzles is actually in play, anyone who says that he knows the answer has a hidden assumption that he has no basis for.

I think what you are saying is more or less correct. If you haven't read this already, I found it very helpful in understanding the intended question.

http://www.straightdope.com/columns/060203.html

From that article:

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.

In regards to this, ANut said:

There is no paradox in or implied by that wording. There is only a slight misuse of terminology in the "direction of rotation" phrase. Let me explain

I disagree with this statement in that I do believe it creates a paradox, but when I viewed the original question (in this thread) as it was (IMO) more properly presented in the article:

"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?"

it helped me wrap my brain around the question more properly.

[Brillig]

Originally Posted by MattJones4Heisman

It's absurd. Sure the conveyor belt is theoretical. The plane and the mechanics of the plane are certainly not, however. I thought this was pretty clear. I guess not. If you are interested in a discussion along these lines, I am most certainly not interested.

So your puzzle reads: If an actual plane were sitting on a theoretical treadmill, could it take off?

*snort*

What kind of dumbass discussion is this?

...

Anyway, if by "mechanics of the plane" you mean things like thrust, lift, and newtonian physics, there's no problem there. However, if you want to start talking about the details of drag caused by bearings in the airplane wheels, then the whole problem is stupid, and I'm not interested in that discussion either.

For that matter, I already posted why people arrive at two different answers, so the discussion is over anyway.

[Pumpy Tudors]

Originally Posted by Brillig

For that matter, I already posted why people arrive at two different answers, so the discussion is over anyway.

IT'S OVER, FOLKS. NOTHING MORE TO SEE HERE.

[MJ4H]

Originally Posted by Brillig

For that matter, I already posted why people arrive at two different answers, so the discussion is over anyway.

I do hope your part in it is over.