I created this blog specifically to make this post. It may be the only post I ever write, but since human ignorance is seemingly unbounded, perhaps it won't be.
I thought that today would be a monumental day for this topic. Today, the Mythbusters debuted their long-awaited "Airplane on a treadmill" episode. For years, physics teachers around the world have cringed in horror at heated internet debates concerning a ludicrous thought experiment. Sadly, half of them recoiled in disgust at the correct arguments. Forum posters signed their names with such epithets as "Ph.D. Aerospace Engineer" and "20-year pilot." Somewhat tellingly, these ego-boosters were most often employed by those delivering the wrong answers. Mythbusters finally attempted to end the insanity by performing the experiment themselves.
AND YET...
The debate rages on. Even after being shown seemingly conclusive evidence of the other side's argument, forum-goers from near and far continued to staunchly defend their own theories.
Here and now, the debate will end. I intend this long-winded article to be the definitive answer to the great AOAT conundrum. No further debate is necessary - simply direct the ignorant people to this page, tell them to read it, and let's all get on with answering more intriguing questions, like does P = NP?
For those of you just joining us, "Airplane On A Treadmill," also known as "Airplane on a Conveyor Belt," is a thought experiment in physics. Some consider it a litmus test for assessing one's knowledge of airplane physics. In its most basic form, the experiment is worded thusly:
The question suffers from many rewordings that muddle much of the debate about the thought experiment. The basic idea is that there's a plane, on a treadmill, and we're going to run the treadmill backwards in an attempt to stop the plane from taking off. And here, at the very beginning of this explanation, is the definitive answer. There are in fact two correct answers to this question:
A plane is standing on a large treadmill or conveyor belt. 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?
-No, the plane can't take off.
-Yes, the plane can take off.
Fooled you! But that's just the point. The experiment is meaningless, and the passionate internet debates more so, if we cannot agree on what is truly meant by the question. But don't worry, I won't pull a Lost on you - I do intend to give a truly airtight answer later on. For now though, we need to debate semantics.
Really, we do.
You see, the AOAT confusion all arises from misses - misconceptions, misinterpretations, and misunderstandings. Consider three rewordings of the question:
1) An airplane is sitting at rest on a very powerful treadmill. You are at the controls of the treadmill, while I am at the controls of the airplane. On some signal, I begin to attempt to take flight in the plane, and you attempt to match my speed to try to keep me stationary. Will the plane take off?
2) An airplane is sitting at rest on a very powerful treadmill. You are at the controls of the treadmill, while I am at the controls of the airplane. On some signal, I throttle up the airplane and you turn on the treadmill, and we conspire by our joint effort to try to keep the plane stationary relative to the ground. Will the plane take off?
3) An airplane is sitting at rest on a very powerful treadmill. You are at the controls of the treadmill, while I am at the controls of the airplane. On some signal, I attempt to take flight in the plane, but you match my speed with the treadmill and keep me stationary relative to the ground. Will the plane take off?
Here are the absolute, 100%, bet-your-life-on-it answers to these rewordings:
Yes.
No.
Whoever asked this question is an idiot.
And that's about all this debate comes down to, folks. If we could all agree on one set of rules for the thought experiment, then we ought to be able to make the explanation of the answer clear. As it stands, normally one side has interpretation (1) in mind, and argues vehemently with someone else who has interpretation (2) in mind, and the whole thing blows up into a senseless squabble.
Here are the three core facts that are rock-solid:
A) If the plane remains stationary relative to the ground, it will not take off.
B) If the plane moves relative to the ground, it will take off.
C) The person operating the conveyor belt cannot by himself make the plane remain stationary relative to the ground.
(EDIT: Really, you should substitute the word "air" for ground in the above facts. I use "ground" throughout this post because of a consistent mistake made by "no-flys" in their assumption that the plane remains stationary. It doesn't remain stationary, relative to the ground or the air. The important point is that it moves relative to the air, not the ground, but I'm assuming throughout this post that there is no significant tailwind or headwind. I discuss the implications of this briefly in the section about windtunnels.)
That's about all you need to know to argue whichever interpretation is appropriate. I'll discuss why these facts are true in a moment. In the meantime, look back at the three re-wordings of the question above.
In (1), the key phrase is "you attempt to match my speed to try to keep me stationary." Since we know from fact (C) that you cannot keep me stationary, it follows from (B) that I will take off successfully.
In (2), we conspire together to keep the plane stationary. This is possible, albeit stupid. We know from fact (A) that I will not take off.
In (3) - and this is the important part - the actions being described are impossible. We know from (C) that the conveyor operator cannot keep the plane stationary. The most powerful conveyor belt in the world couldn't do it. David Copperfield couldn't do it. It can't be done. Only if the pilot "plays along" can the plane be made to remain stationary.
Unfortunately, most of the "no-flys" - the label given to those who argue that the plane won't take off - are assuming that interpretation (3) is what is being asked. They accept that the plane remains stationary, and say it won't take off. The "will-flys" know that the plane can't remain stationary, and say it will take off. Add to the mix a few people who see that in one way, the plane could be forced to be stationary by some pilot-conveyor cooperation, and you've got a deadly internet forum explosion cocktail.
Let's examine the physics behind the three key facts, so that there will be no doubt as to their validity. The first two are pretty easy to follow. Airplanes create lift by causing air to flow over their wings. This airflow is caused by the motion of the wings relative to the air - that can happen in two ways. The first way is to move the plane forward through the air. The second is to blow air against the plane and over the wings. As far as the plane is concerned, these two scenarios are equivalent. So you could put a plane in a very powerful wind tunnel, blow air over its wings, and have it fly stationary relative to the ground. But that's another question.
In our treadmill scenario, the air is stationary relative to the ground, so the plane has to move relative to the ground in order to gain flight. If it doesn't move, it simply won't fly. There will be no airflow over the wings, and there will be no lift. A lot of people get confused here, and think that the original thought experiment is some sort of trick question, and that the propeller of the airplane, or possibly the jet engines, will be blowing air backwards over the wings, which will create lift. While there will be a certain amount of airflow created by the propeller or engines, it is not enough to create flight. I promise you, that's not what the question is asking.
Really, I promise. Please, please stop talking about airflow created by the prop. It isn't part of the question.
So we have facts (A) and (B) well taken care of. If the plane moves, it flies. If it doesn't move, it doesn't fly. The real question is, will it move? Again, the answer is unambiguous - if the pilot doesn't try to make the plane stay still, it won't. If he does, it will. This is always, always the part that confuses people, so stick with me for a few more paragraphs.
When a plane is sitting on a runway, it moves by using its engines. It does not move by any sort of motorized wheel. The propellers or jets create thrust that pushes against the surrounding air and causes the plane to move forward. A plane wouldn't move at all in a vacuum chamber. Compare this to a car, which moves by applying torque to the wheels. A car would drive just fine in a vacuum chamber - at least, as long as the driver could survive (and technically, it would need some sort of air reservoir to provide something to mix with the fuel. An electric car wouldn't have this problem.) However, a car could not drive on a frictionless surface - imagine, for example, that you had your car on a super slippery frozen lake. As you hit the gas, the wheels would simply spin and spin in place, and the car wouldn't move forward. You may even have firsthand experience with this situation if you've ever gotten stuck in a snow bank. In contrast, a plane would have no trouble moving on a frictionless surface. The jet engines or propeller would still push against the air, and the plane would still move forward. If it were on a truly frictionless surface, then you would see the wheels sliding along the ground, not rotating.
I hope those two scenarios clearly illustrate the difference in motive force between cars and planes. Cars create their forward movement from torque applied to the wheels, which push against the ground and create forward motion from friction. Planes create their forward movement from thrust applied to the air, which pushes the plane forward regardless of the surface it is on.
Imagine a plane without wheels. The fuselage would sit on the runway, and as you fired up the engines, it would skid spectacularly along the runway, possibly spewing sparks in its wake and doing quite a number on the body of the aircraft. No matter how fast it was going, the frictional force against the airplane would be constant; friction does not depend on speed! If the engines were strong enough to get the plane up to the critical take-off speed, then it would still take off. The only reason planes have wheels is to reduce this sliding friction. The wheels roll along the runway instead of sliding, and the only friction that the plane feels is in the bearings of the wheels. This is substantially less than the friction that a sliding fuselage would create, and it's a much smoother ride for the passengers as well.
(Edit: Technically, there are some factors that would make the friction change with speed. The classic idealized model called "coulomb friction" doesn't really apply to bearings. As the bearings spun faster and faster, they would generate heat, which would increase the friction slightly on the wheels. However, it would never be enough force to prevent take-off. The only time this would prevent take-off is if the wheels locked up or broke off, and then we'd have a much bigger problem and catastrophic failure.)
So what does this all have to do with treadmills? Well, now let's place our plane on that treadmill and see what happens. If the wheels were perfect - that is, there is no friction in the bearings (and no deformation of the wheels as they spin) - then something interesting happens. When we turn on the treadmill, the plane stays stationary on its own. The wheels simply spin along the track, and impart no force to the plane. If you had a car with frictionless axles, and you disconnected the whole drive train, the same thing would happen to your car.
The only reason that a plane or a car moves backwards on a treadmill is that the wheels are somehow partially locked to the axles. In a plane, this is because of minor friction in the bearings. In a car, it's because of the drive train. If you want the car to stay still, you have to turn the drive train at the proper speed. If you want the plane to stay still, you have to overcome the minor bearing friction. And again, since friction does not change with speed, you don't have to exert any more force at higher speeds. If you run the treadmill at 5mph and turn on the plane's engines just slightly, they will provide enough thrust, pushing against the air, to keep the plane still. If you then increase the treadmill speed to 500 mph, you won't need to adjust the throttle on the airplane - it will remain stationary. That's because it's seeing the same frictional force that it was at 5mph. Thus, it doesn't matter how fast the treadmill is moving - if the pilot does not want to remain stationary, then he won't. It only uses the very first bit of power from the engines to keep the plane stationary. As the throttle is increased from that point, it moves forward just as it would on any other runway. It's pushing against the stationary air!
If you don't believe me, imagine this (or even try it at home): you're standing on a skateboard on a treadmill. You hold onto the handrails of the treadmill and turn it on. Of course, you'll remain stationary (relative to the ground). In fact, you only need to use a very light touch to stay stationary - perhaps a few fingers pressed against the handrails. Crank up the treadmill speed as high as you like. You'll still only need the same light touch to remain stationary. At any time you like, you can move forward - closer to the treadmill console - by simple pulling on the handrails. If you had a jet engine, or super-strong hairdryer, you could use this to propel yourself forward instead of holding onto the handrails. In fact, if you're really careful, you might be able to do this at home with a skateboard and a leafbower, but I doubt you'll have a sensitive enough control of your leafblower thrust to get yourself to remain stationary.
So you see (oh please tell me you see), the conveyor operator cannot force the plane to remain stationary. And if the plane isn't stationary, it can take off.
And yes, if we interpret the question in a different way, and assume that for some reason the pilot is colluding with the conveyor operator and keeping the plane stationary, then the plane can't take off.
But what is the question really getting at, anyway? There are really two "spirits" of the question. In the first, we're asking "can a plane take off with no runway, if I replace the runway with a treadmill?" The answer, as we know now, is no. The plane must move relative to the ground in order to take off. In another deep-meaning of the question, we're asking "is it possible to prevent a plane from taking off, by moving the runway backwards under it?" The answer again is no, you can't prevent it from taking off.
The interesting thing about all this is that in both scenarios, you'd wind up with a plane moving relative to the ground. In the first scenario, you might think you're being clever by allowing a plane to take off from a very small field, by using a treadmill runway. If you actually tried it, you'd be attempting to take off, so the plane would move, and would likely crash into something, or fall off a cliff, or suffer some other catastrophe that you were trying to avoid with questionable physics. In the second scenario, you'd give the plane plenty of room and safety to take off, but attempt to play a practical joke on the pilot by moving the runway backwards, and you'd wind up with a plane in flight, much to your chagrin.
When the "no-flys" saw the Mythbusters episode, they all complained that it wasn't done properly, because the plane didn't remain stationary. But think about it for a moment. No, really think about it, don't just spout about Bernoulli's principle and airflow and all that. In what possible scenario would the plane actually stay still? The only way this can happen is if the pilot is trying to stay still, and this only happens if he just barely applies the throttle, making no attempt to take off. This makes no sense. Either you're trying to prevent him from taking off with your clever and misinformed use of a conveyor belt, or he's trying to defy physics by taking off in a too-small area. There is no scenario in which the plane would realistically stay still. We know what would happen if it did - it would sit on the runway, not taking off, and we'd all stare at each other in an all-too-short silence punctuated by loud exclamations of "I told you so!". But that's not really what the thought experiment is getting at, no matter how you reasonably interpret it. Luckily for all of us, if we agree on the interpretation, reasonable or not, we should all agree on the answer.
So let's get back to the next great internet debate, shall we?
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151 comments:
"Whoever asked this question is an idiot."
LOL. I just about dropped out of my chair in a fit of joy after reading this. Thanks for this article... perhaps there is hope for humanity afterall.
"On some signal, I throttle up the airplane and you turn on the treadmill, and we conspire by our joint effort to try to keep the plane stationary relative to the ground. Will the plane take off? No."
No amount of collaboration will prevent the plane from taking off. Given the existence of a free wheel between the treadmill and the plane, the treadmill is incapable of applying any horizontal force to the plane. It cannot slow the plane down.
"No amount of collaboration will prevent the plane from taking off. Given the existence of a free wheel between the treadmill and the plane, the treadmill is incapable of applying any horizontal force to the plane. It cannot slow the plane down."
The plane can always not take off, if the pilot doesn't want it to. If its forward thrust matches the force of friction it will remain stationary. If by "free wheel" you mean frictionless then the pilot doesn't apply any thrust at all and the plain remains stationary.
Ok, I can agree to that. As long as we're clear that in #2 you're arguing a deep corner case. That is, the pilot purposely fails to overcome the (relatively) small friction in the axles of his wheels with the engines, in some cases by keeping them shut off. Of course, you can say the same thing about a runway without a treadmill.
If the pilot wanted to take off, he could.
"If the pilot wanted to take off, he could."
Agreed.
I once saw a UFO take off from a treadmill. I've never been the same.
Simplest way for me to think about it is Newton's laws. You apply a force, F, to the plane with the prop. Its mass, m, relative to the ground, must accelerate, since F is the only force acting along the horizontal axis. F=ma, so eventually it must reach takeoff speed because it is accelerating relative to ground.
I fully agree with the people who said that in situation #2 the pilot would have to keep the engines on stand-by or even shut off, in which case the presence or absence of a conveyor belt doesn't even begin to enter into the equation, at least not for practical purposes.
In the discussion over on kottke.org, I explained it like this:
> "and we conspire by our joint effort to try to keep the plane stationary relative to the ground. Will the plane take off? No."
The conclusion would be correct assuming that premise, but the premise is surreal. In order to keep the plane stationary relative to the ground, the pilot wouldn't be allowed to turn on the engines in the first place. Once they're running and generating thrust, the c-belt operator can't do anything to hold the plane in place. (Except in those situations, already mentioned by many posters, in which the belt would accelerate beyond light speed in a microsecond and the universe violently implodes, or something like that. Which obviously neither works in a thought experiment with ideal, friction-less wheels, nor in the real world.) Note that you can 'prove' anything when you start out from a wrong premise. Consider the following statement: "If the circumference of a circle equals its radius, then the pope is a Protestant." Think about it. It's true, literally!
"In the first, we're asking "can a plane take off with no runway, if I replace the runway with a treadmill?" The answer, as we know now, is no. The plane must move relative to the ground in order to take off."
Not in a good enough headwind!!
Being an MD, PhD, and a Scientologist, I consider myself an expert on everything.
1) The very idea that this Chris character decided to spend so much time on the subject makes me violently angry. It is obvious that the plane will take off, lest the pilot match the applied force of the engines to friction force from the ground. If you needed to read this entire post to get that, you might want to forget how to breath.
2) Find me a perfect frictionless bearing that will allow the plane to remain stationary on a treadmill, and I will pull a leprechaun out of my rear end. Frictionless is a concept used to make physics problems simpler. It is near impossible to create frictionless conditions in the really real world.
The biggest problem with this thought experiment, is that many of the wordings make the assumption that the plane does, in fact, remain stationary. Since this is worded as a part of the statement, and not the question, people do not question it, and assume the statement is correct. Therefore, the plane would not take off, because it's stationary. This assumption is further assisted by images such as this one, when posing the question "will it take off?" http://farm1.static.flickr.com/134/336644021_adf0c4a276_m.jpg
Since the plane is shown on a treadmill only as long as the fuselage, and there is a barrier to keep it from rolling -off- the treadmill, it seems a valid enough assumption that the plane must not be moving, and this assumption is translated to the question, "Will it take off?"
In this case, it couldn't because the wings would hang on the treadmill - but that's just an interpretation of the image.
"The biggest problem with this thought experiment, is that many of the wordings make the assumption that the plane does, in fact, remain stationary."
That would be interpretation (3) in the original post, answered by the authoritative "Whoever asked this question is an idiot."
Simplify. There are 2 interpretations to this question:
1. If you keep the plane from moving, will it be able to take off?
Clearly, the answer is "NO". Movement is necessary to generate airflow and thus lift.
2. Will the treadmill be able to keep the plane from moving, and thus prevent it from taking off?
The answer to number 2 is also "NO". This seems to be the question that most people fail to understand, but the proofs have been posted so I will not repeat them.
The treadmill cannot prevent the plane from moving. It will gain airspeed and thus generate lift, and will fly.
A good quote I came across somewhere:
"The plane is not magically levitating while standing still. It is moving forwards as usual, and ignoring what the ground underneath it is doing. Which is the entire point of having a plane, really."
Is it really true that an airplane moves forward by "pushing against the surrounding air"?
Isn't it really that the forward thrust is generated by the "equal and opposite reaction" of the action of throwing stuff backwards (like air or spent fuel)?
An airplane in a vacuum chamber couldn't move forward, because there is no air to throw backwards.
Unless it was rocket propelled. No air needed in that case.
Is it really true that an airplane moves forward by "pushing against the surrounding air"?
That is a simplification, but yes it's true.
Isn't it really that the forward thrust is generated by the "equal and opposite reaction" of the action of throwing stuff backwards (like air or spent fuel)?
Yes, that's correct. But what is an airplane engine pushing backwards? You said it yourself - air! It's pulling the surrounding air into the engine, and then pushing that air backwards to generate thrust. The fact that there is "spent fuel", or exhaust, in that air now is of little to no consequence.
An airplane in a vacuum chamber couldn't move forward, because there is no air to throw backwards.
Unless it was rocket propelled. No air needed in that case.
True, but no air also means no combustion, so the engine couldn't run anyway. And rockets, believe it or not, actually carry their own "air" with them, in the form of an oxidizer, so they are still pushing air backwards in order to generate thrust forwards.
thank you thank you thank you. now lets see jamie and adam fly a plane in a wind tunnel.
...we conspire by our joint effort to try to keep the plane stationary...
#2 is a specious argument. The only way the plane can remain grounded is to have the pilot purposely powering down his engine, or even never turning the engines on. If this is the case, why even get the pilot out of bed for the experiment? Why use a plane with working engines?
The simplest way to bust this "myth" is to look at the implications of the original question.
It says that the conveyor belt must match the speed of the plane.
What speed? If the plane stays stationary, there is no speed.
To have speed, the plane must already be moving and thus it cannot stay stationary.
Yeah, if you could add a slight edit in your post it would be perfect. Where you mention a vacuum preventing a plane from taking off, a rocket engine could take off even in a vacuum. (I just don't want anyone to be able to poke holes in your explanation ;)
Some figures would be nice but I'm having trouble trying to think of any. I'll send some if I can.
It turns out it _is_ possible for the conveyor operator to keep the plane stationary, but not by matching "speed"; and to do it indefinitely you need a conveyor that can accelerate indefinitely.
See http://www.straightdope.com/columns/060303.html for the details.
So, the thrust of your post is correct w.r.t. the original (silly) question, but your supporting assertion that it is fundamentally impossible for _any_ action on the conveyor's part to prevent the plane from moving is not correct.
you need a conveyor that can accelerate indefinitely.
The original question required the speed of the belt to match the speed of the plane. Accelerating the belt to infinity breaks this condition.
"to do it indefinitely you need a conveyor that can accelerate indefinitely."
In other words, no. You can't keep the plane from taking off.
You can't accelerate the conveyor belt past the speed of light, which means you can't stop the plane from taking off.
What a silly argument.
DUFF has it right... there's two different understandings of the problem:
1) can a plane achieve flight from a stationary position relative to earth, but in motion relative to the treadmill?
2) can the force of the plane overcome the force of the treadmill?
the treadmill is intended as a device that keeps the plane from moving forward, assuming that the plane behaves on the treadmill the same as a car would. but it doesn't. and that sets up problem number 2.
the 1st understanding of the problem is a question about what generates lift. the 2nd understanding of the problem is a question about the origin of force and the influence of friction.
My analysis:
Plane on a Conveyor Belt Takes Off!
As predicted, of course. The Mythbusters proved it in last night's show.
There has been lots of discussion on the internet about whether an airplane on a conveyor belt could take off, if the conveyor belt moved in the opposite direction at the same speed as the airplane. The problem is usually stated something like this:
"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?"
The people who say no, typically reason that for an airplane to take off, it needs airflow over the wings (more than just the amount the propeller blows), but since the airplane can't move forward, there is no airflow over the wings, therefore the airplane can't take off.
But this begs the question. Of course the airplane needs to move forward at a speed great enough to supply the lift in order to take off. That's kind of integral to the idea of "taking off." Whether the airplane can do so is exactly the question we are being asked, I think.
The faulty assumption is that the contraption is designed to prevent the airplane from gaining forward speed. That, however, is not stated in the problem. People make this assumption, I guess, by analogy to walking on a tread mill where one does not go forward, but stays in one spot as the treadmill goes backward at the same speed one is walking forward.
But airplanes don't propel themselves forward by pushing against the ground (or the treadmill in this case) like walking people or cars do. Airplanes' forward thrust comes from the "equal and opposite reaction" of the action of throwing stuff backwards (air, in the case of a propeller plane, or the hot expanding gases of spent fuel in the case of a rocket).
It amazes me that so many people just don't get it, even after having it explained and demonstrated. Many of them claim that the Mythbusters experiment was flawed, because "the plane actually did move forward!" For some reason, they thought that the "wheel speed" of the plane was to match the conveyor belt speed. But the original question only says that the speed of the conveyor belt matches the speed of the plane.
The assumption that the "speed of the plane" means it's wheel speed relative to the belt is an unwarranted assumption, in my opinion. Unwarranted, because it makes the original question illogical. The only way for the wheel speed to match the belt speed is for neither the plane nor the belt to be moving at all; or for the plane's engines to be providing just enough thrust to match the very minimal force of the wheel friction in order to keep it from being dragged backwards. But the question is, "can the plane take off?" This implies that the plane is trying to take off, which means full thrust, not just minimal or no thrust. So it must be that the "speed of the plane" must be relative to the ground.
Wow
I'd never even heard of this debate until now, and i have to say at first, i was on the stupid end of the stick but your explanation was very to the point, and it cleared it right up for me.
and while i felt kind of stupid for getting it wrong at first: (no it can't take off! durrr!) I have to say this is a really elegant thought puzzle. unless you realise where the actuall force is comming from and being applied, your stuck in your own stupidity. good post
"Here are the three core facts that are rock-solid:
...
C) The person operating the conveyor belt cannot by himself make the plane remain stationary relative to the ground."
This is true for planes with wheels and achievable speeds on conveyor belts. The friction with the ground is very small.
However, if the plane has skis or pontoons (on the ground) or you allow the conveyor speed to be unbounded then friction can increase to a degree where it is possible for the conveyor operator to keep the plane grounded.
"or you allow the conveyor speed to be unbounded then friction can increase to a degree where it is possible for the conveyor operator to keep the plane grounded."
In theory, the speed at which the wheels are turning should not increase the force of friction. That is, an ideal wheel would not be affected by the speed. Of course, in the real world, the mechanics of the wheel would break down at high enough speeds and the friction would increase to the point where either the plane would slow down and stop, or the wheels would snap off. Of course, these speeds would be pretty astronomical, and I doubt any treadmill in the real world would be able to withstand those speeds itself.
However, even if we allow an 'ideal wheel' there is still a purely hypothetical scenario where the treadmill could keep the plane stationary. If the treadmill is allowed to have unbounded speed and unbounded acceleration, then I believe the rotational inertia would also be unbounded, because it can be expressed as a function of the acceleraion of the treadmill. And if the rotational inertia is unbounded then the force it exerts on the plane is also unbounded, and the plane could be forced to remain stationary.
....Aaactually, now that I think about it, the friction between the wheel and the treadmill would probably not be great enough, and when the rotational inertia started to build against the turning of the wheel, the wheel would just start to slide along the treadmill.. and the plane would still move forward.
I have FAITH that it won't take off...you're persecuting my religion!!1!11
Seriously though, well done. You've gathered a coherant collection of arguments (none of which should really have been required if you ask me...). There's still gonna be people complaining though. Some people simply can't admit that they are wrong (I think the largest source of confusion is people mistaking the wheels on a car [driven] with the wheels on a plane [freewheeling]).
"So what does this all have to do with treadmills? Well, now let's place our plane on that treadmill and see what happens. If the wheels were perfect - that is, there is no friction in the bearings (and no deformation of the wheels as they spin) - then something interesting happens. When we turn on the treadmill, the plane stays stationary on its own. The wheels simply spin along the track, and impart no force to the plane. If you had a car with frictionless axles, and you disconnected the whole drive train, the same thing would happen to your car."
This part is not true unless you are talking about massless as well as frictionless wheels. The wheels' rotational inertia will resist the treadmill and will apply a backward force on the airplane as the treadmill is accelerating.
Oh, if I could only sage, I would sage you to deletion and beyond.
RAAAAAAAAAAAAAAAAAAGE
Alright, so we've got a 747 sitting at rest on a hypothetical uber-treadmill. the engines fire up, and the plane starts to roll forward.
then the treadmill starts rolling backwards at an equal amount to keep the plane geographically stationary.
guess what guys - the whole reason a plane has engines is so it can move the damn wings through the air, as its the movement of the air over the wings that provides lift.
if there's not sufficient wind flowing over the wings..
....guess what...
IT WONT FLY.
then the treadmill starts rolling backwards at an equal amount to keep the plane geographically stationary.
siztem, did you even READ the webpage? The whole point is that the plane doesn't remain geographically stationary. It doesn't matter how uber your uber treadmill is.
...guess what...
IT WILL FLY.
A 747 taxies to the threshold and lines up, ready for take-off. Each set of wheels sits on a conveyor belt which is free-moving (i.e. not driven and, for the sake of this hypothetical, completely friction-less). Light the fires. Accelerate to take-off thrust. Forward thrust is created but the heavy body doesn't move - just spins its wheels. Just like an engine test bed situation, except the engine isn't tethered down. The thrust has instead been converted into spinning a conveyor belt.
What's happening inside the cockpit? OttoPilot has pushed his thrust levers fully forward. He hears the scream of the engines, but the world outside isn't moving. What does the ASI read? Zip. Because there's no movement through the air.
No airspeed? No lift.
That 747 isn't going anywhere. Least of all up.
Same 747 at the threshold, mains are sitting on the conveyor belts. Engines are off. This time OttoPilot is staring at a whacking great big fan sitting on the runway in front of the nose. The fan starts up with a whirr. Faster and faster it spins, pushing air at OttoPilot and his 747. The heavy bird would move backwards because of the force of the air pressure, but instead the wheels spin backwards on the free-moving belt. Otto looks out of his window, but no movement is discernable. He looks at his ASI. Wow! The needle moves! As the fan speeds up even more, the ASI creeps upwards until - lo and behold! - takeoff speed is reached. With a slight judder, the wings produce enough lift to raise the wheels off the belt. Flying! No engines! WTF?
Grahame said: "... the heavy body doesn't move - just spins its wheels...."
It doesn't spin its wheels. A CAR would spin its wheels, because a CARs engine drives the wheels. A planes engines, however, DO NOT DRIVE THE FUCKING WHEELS. A plane, effectively, pushs itself off from the air behind it. It takes off.
Eoin, yes...engine thrust does not directly power the wheels. Agreed.
1. We know that the engine produces thrust. Undisputed.
2. Engine is attached to plane. Engine thrust directed rearwards. Because of equal and opposite reaction, plane wants to move forward.
3. Thrust develops until it overcomes inertia of the body. As it overcomes that inertia, thrust is then greater than all the forces holding the body to the ground. Something must happen. If the plane's wheels were on concrete, the wheels would enable it to move forward, acting as bearings between two solid bodies, the plane and the ground, converting engine thrust into movement. But now instead of the ground, we have a free-wheeling conveyor belt.
Thrust is therefore converted into movement of the belt (which can be connected via an axle to a belt-driven generator to supplement Eskom's deficiencies - a jumbo electricity supply. Probably still cheaper than accepting quotes from all the power station smouse now flocking to the scene of the disaster).
Plane can't fly. So where is my layman's logic wrong. Help please.
Sorry, please ignore comments about Eskom - this is a reference to a South African power supply issue.
A 747 taxies to the threshold and lines up, ready for take-off. Each set of wheels sits on a conveyor belt which is free-moving (i.e. not driven and, for the sake of this hypothetical, completely friction-less). Light the fires. Accelerate to take-off thrust. Forward thrust is created but the heavy body doesn't move - just spins its wheels.
INCORRECT!!
If the belt is unpowered, it won't move at all. The plane will roll along the belt just as well as along the ground. Now, if the belt is truly frictionless, as you say, and the wheels have any amount of friction in the bearings, the belt will actually move forwards (ie, in the same direction as the plane), but very slowly (since it's only being accelerated by a very minor frictional force in the wheel bearings), and the wheels will roll faster along the belt, causing forward motion of the plane.
Same analogy with the home treadmill and skateboard again - if you stand on a skateboard (or rollerskates) on a treadmill, and pull yourself forward by using the handrails, it doesn't make the belt spin under you, does it? Even if the belt was completely frictionless, surely, surely, you would move. Do you believe that you would be pulling on the handrails as hard as you could, and magically the belt would be spinning underneath you? Of course not! As soon as you pull on the handrails, you will move, regardless of what the belt under you is doing. The plane is doing exactly the same thing, but its handrails are made of stationary air.
The plane will move. I promise.
Chris, I see the light! Hallelujah! Why, though, isn't yet clear. But the analogy of the roller skates on the treadmill made intuitive sense.
Grahame: "...Thrust is therefore converted into movement of the belt..." is where your laymans logic went wrong. The statement is false.
Glad you've seen the light...although the skateboard analogy in the original post really should have made it clear for you...
Graham, the easiest way to picture it for me is if you consider the plane to have super-slick near-frictionless skiis on the bottom, instead of wheels. The wheels thing messes people up because they are used to cars. When you compare the wheels on a plane to skiis, its easy to see that they both serve the same purpose... simply that of reducing friction.
If a plane with super-slick skiis is on a fast-moving treadmill, as long as the thrust of the engine is enough to overcome the friction between the skiis and the treadmill, the plane will accelerate forward to take-off speed. The same is true of a plane with wheels. Wheels are just better at reducing friction than skiis.
All 3 of his interpretations are incorrect because he is measuring the plane speed relative to the conveyor belt, then measuring the conveyor belt speed relative to the ground, then comparing the two speeds measured in different reference frames.
The question does not specify how speeds are measured. But there is no common speed dial on a plane that measures speed by the rotation of the wheel, and the general rule for every situation is to measure speed relative to the Earth (the ground).
Chris,
This site is great! You finally disassemble each side of the argument in a well-thought out article. And yes, I know that the plane takes off (given the original wording of the question).
A long time ago, I was thinking of explaining this whole question in different terms (on the ebaumsworld forum, of course). I'd explain what was meant by "speed of the airplane" and "speed of the treadmill" (as an anonymous poster mentioned above). That is, I'd explain what effects each of the 9 different cases of reference points (or 4 if you consider air and the stationary ground to be equivalent frames of reference) had on the interpretation and the potential answers to the original question.
So let's break it down to just 4 cases for the sake of brevity. The 3 reference points I'm using are airplane, treadmill, and ground (== air). Because an object has 0 speed relative to itself, I won't consider those cases.
1. The airplane's speed is relative to the ground, and the treadmill's speed is relative to the ground.
2. The airplane's speed is relative to the ground, and the treadmill's speed is relative to the airplane.
3. The airplane's speed is relative to the treadmill, and the treadmill's speed is relative to the ground.
4. The airplane's speed is relative to the treadmill, and the treadmill's speed is relative to the airplane.
Case 1 is the most likely interpretation of "speed" for the airplane and treadmill. The reference point is the same for both, so the answer must be "yes", the plane takes off (with little difference from a normal runway takeoff, even).
Case 2 doesn't seem like a good interpretation of "speed" for the treadmill; who would measure the treadmill's speed relative to the airplane? If you did, you would see that it's impossible to keep the speeds the same but opposite except at a speed of 0. But as the original question stated, the plane does in fact move. This case is not possible, and fortunately it's not a popular interpretation of "speed".
Case 3 is similar to case 2, in that the plane must not move for the condition that the speeds must match for the treadmill to do its job. Besides, nobody measures the speed of an airplane by its wheels, and normally an airplane doesn't use the runway below it to measure its speed; an airplane's speed is measured by the air flowing past it, which (I think) can be assumed to be the same as the speed over the stationary ground.
Case 4 is always true by law, as two objects' speeds relative to each other are always equal but opposite. That's a given. Of course, as in case 3, nobody measures the plane's speed with its wheels (which is its speed relative to the treadmill); as in case 2, nobody would measure the treadmill's speed relative to the airplane. So case 4 is right out. (The plane can take off as normal in this case—the runway could move at any speed w.r.t. the ground and the airplane moves as normal, alse w.r.t. the ground/air—but it should be obvious that this interpretation is a silly one).
I think most people use case 3 to argue that the plane does not take off, and in fact you (Chris) don't make it clear that this is not a good set of reference points to use. Case 1 seems the better interpretation of the two, and it's easy to explain that it takes off if a common reference point (the stationary ground) is used to measure the two speeds.
"If the plane remains stationary relative to the ground, it will not take off."
If you want to bury every shred of the no-fly argument, you can counter that statement. It is possible to design a plane that will take off with its brakes on, just from propwash alone. Just give it freakishly huge flaps. It won't fly very well, but on takeoff, it can pop itself straight upward. Nobody's built a physical plane that does it, but there have been a few designs that work in X-plane, a very accurate flight simulator.
Of course, there are many real-life examples of VTOL, but I'm sure the no-fly folks won't accept VTOL as an answer to the question.
I think your model is missing a force: you mention the friction of the wheel bearings, but omit the rolling resistance force created by friction between the wheel and the ground.
Doesn't change the fact that the plane will take off, though.
However, I'm interested in the hypothetical case in which the conveyor belt speed is unlimited (and the wheels are indestructible). . .trying to figure out the math required to compute the conveyor belt acceleration (and the energy input) required. . .
PS-- I'm thinking that a rocket doesn't generate thrust by pushing "air," if you're referring to the oxidizing agent. . .it pushes whatever the byproduct of combustion is, the exhaust (thrust = exhaust mass * rate at which it's expelled). . .
It doesn't have to be air int he 79%N/21%O sense of earth's atmosphere. It is a mass of various gasses, including CO2 and other waste products. As I said in my original post, rockets carry their own oxygen with them for combustion, so in a sense it is still air that they exhaust.
Well, I've enjoyed this blog immensely. I especially liked Grahame's 747 comments. LOL
I've always explained this problem to people who don't get it by using a swimming pool as the example.... If I put a treadmill on the bottom of a pool that rotates along at a speed equal to the speed i swim, will the treadmill be able to keep me from moving forward.... Of course not, because I'm not walking on the treadmill, I'm pushing myself through the water.
I can't wait to hear the "no swim" answers.... :-)
Great blog man!
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To all those who still don't get it - this is a really dull thought experiment to get worked up about. There are far more interesting ones.
Yes! Thank you!
I've used the same skateboard analogy for other people, and it always seems to help... though, I use rollerskates and a rope, rather than a skateboard and handlebars, but the principle's the same.
Incidentally, two different versions of this puzzle came up in a forum I frequent at the same time... one of them was worded "the treadmill spins at the same speed that the plane would move if the treadmill was absent"... that is, a certain amount of thrust is applied to the plane that'd normally make it move at v(t), then the same amount of thrust is applied to the plane on the treadmill, and the treadmill moves at -v(t). I personally think this is the best wording for the puzzle, because it takes out the human factor which you have in your wordings, but still has the "puzzle" interpretation, as the same treadmill setup would hold a person running stationary (and is, indeed, the way that the speed is chosen for most treadmills). Of course, with a plane, it just spins the wheels at twice the speed, and the plane takes off.
The other one was worded ambiguously, but in a way that (from a literal interpretation) meant that the plane had to be stationary (to be specific: it said that the speed of the treadmill at all times matched the speed of the wheels. Since the speed of the wheels is the speed of the treadmill plus the speed of the plane, this implied the speed of the plane was always 0). Clearly this fell into your #3, and the thread quickly split into (a) people saying basically "it's stationary, therefore it won't take off", (b) people trying to explain how the plane could possibly be stationary (with an absurdly-accelerating treadmill, and friction/rotational inertia), and (c) people claiming that this wording of the question is stupid, and trying to drag the conversation over to the sane wording, where the plane takes off. All three groups were pretty much constantly arguing with each other.
I found the discussion of the first wording to be more civil, the solution more satisfying, and the question more deserving of the title of "puzzle"... and would recommend that wording to anyone considering sharing the puzzle.
Yeah but what about the moon landing thing... as if that could really happen. Also what would happen if you put a ship on a giant treadmill, i reckon you could get it to take off.
To stop the plane from flying, the generated force by turning the engines on will have to equal or be less than the force driving the plane backwards... perhaps they should glue the plane to the belt.. although I think that defites the object of having a belt in the first place.
Brilliant. An excellent and clearly reasoned explanation. I hadn't actually heard of the debate before, but it's an interesting concept. Thanks for taking the time to put this together.
Nice try. You your self have done exactly what the you were complaining about. Misinterpreted the question. It does not ask 'if I set a treadmill to a really high speed, can the plane still take off?' it asks, 'if by some complex control system a treadmill runs at the exact same magnitude of velocity, but opposite direction, of the wheels, can the plane take off?'. For the plane to take off, it must still move through the air, and as the plane starts stationary, and does not provide lift, the plane can only move through the air if it moves relative to the ground (it is not like a swimming pool where external lift is provided, the mass of the plane is being supported by the ground, therefore you must move relative to the ground), and therefore the wheels must spin faster. It does not matter where the force comes from to provide this acceleration, it must still spin the wheels faster relative to the runway. As the treadmill is under the control of some system to maintain the exact speed of the wheels (not the force imparted on the wheels by a drive shaft, but the absolute velocity of the wheels) then the plane can never move faster than the treadmill, can never travel forwards relative to the ground or air. This is of course ignoring friction in the barrings and assuming no head or tail wind. These things would make it harder to take off, as the net force will be backwards relative to the plane, and the wings on aircraft rely on the curvature of the airfoil on the front of the wing, its is not reversible.
The internet thanks you for this pleasantly thorough treatment of what has been a highly tedious topic.
One potential improvement: there's no need to take a tangent about cars in a vacuum, just refer to the Moon Rover.
But then again, some of the idiots involved probably don't think that was real either...
First off, on the "no-swim" thing, the problem is that you aren't connected to the treadmill on the bottom of the pool. In this plane scenario, the wheels are connected to the plane. And the problem with the skateboard thing is that yes, you can pull yourself forward on the treadmill, but only if you don't increase the speed of the treadmill. If you leave it at the same speed, you can move up or down.
Think about running on a treadmill. You match the pace of the treadmill. If you increase the speed of the treadmill but keep running at the same speed you were before, you begin moving backwards. That's why you get all those wacky scenes in movies where people start running on a treadmill, it speeds up, they have to run faster and faster to keep up, and then eventually fall off the back.
The wheels do NOT generate any force on a plane, we get it. But the wheels ARE in contact with the ground, and if they aren't rolling across the ground (which they wouldn't be on a treadmill that exactly matches the speed/force/thrust generated by the plane, then the plane isn't moving relative to the ground, and no air is moving over the wings. It's not that hard.
OK...
I think I have an example that will help people understand a bit better that the wheels have very little relevance...
Many people get hung up on the idea that increasing treadmill speed keeps up with the wheels... eg the last comment: "and if they aren't rolling across the ground [...] then the plane isn't moving relative to the ground"
Imagine the plane is on the runway and you turn on the wheel brakes and then hit full thrust... The plane WILL move forward! Thrust from the engines against the air is greater than can be compensated for by wheel brakes - now I don't know if the wheel brakes could cause enough friction to prevent the plane taking off but I'm pretty sure that just friction between un-braked wheels and the bearings / ground can't...
/me <> scientist but I understood the post and am convinced... It's not rocket science ;-)
YOU SIR, ARE AN INTERNET SUPERHERO!
When I heard for the first time that people on the interwebs are actually debating this I could not believe it. I still don't believe it that some people lack the basic physical intuition or education, but now I can just redirect them to this site and end the argument.
Because frankly put, arguing on this topic is not an option. It's not a religion, it's simple facts.
So, THANK YOU for this site!
"In the first, we're asking "can a plane take off with no runway, if I replace the runway with a treadmill?" The answer, as we know now, is no. The plane must move relative to the ground in order to take off."
Not in a good enough headwind!!
not sure if anyone has already commented about this, but this would be in a wind tunnel (or a senario very similar to one), which is already explained in the artical as: "So you could put a plane in a very powerful wind tunnel, blow air over its wings, and have it fly stationary relative to the ground. But that's another question."
and, im probs wrong, but im sure that if there was a strong enough headwind for the plan to take off without the use of its engine (keeping still relative to the ground) surely the airport and a cow or two would be moving through the air aswell?
The simplest way to bust this "myth" is to look at the implications of the original question.
It says that the conveyor belt must match the speed of the plane.
What speed? If the plane stays stationary, there is no speed.
To have speed, the plane must already be moving and thus it cannot stay stationary.
Depends if the speed is its speed to air, or its speed to land. If treadmill ran at 10mph, and the planes air speed was 8.68976242knots (10mph) then yes, it would take off, (I know it needs to go faster than 9knots to take off). If in relative to the *moving* ground, it travelled at 10mph, it wouldnt take off
if that makes sense
While there will be a certain amount of airflow created by the propeller or engines, it is not enough to create flight.
Slight correction: it is not enough to create flight in a conventional aircraft. The Custer Channelwing manages quite nicely, though. But this doesn't affect the answer to the question, nor the validity of the argument.
Alright, so we've got a 747 sitting at rest on a hypothetical uber-treadmill. the engines fire up, and the plane starts to roll forward.
then the treadmill starts rolling backwards at an equal amount to keep the plane geographically stationary.
guess what guys - the whole reason a plane has engines is so it can move the damn wings through the air, as its the movement of the air over the wings that provides lift.
if there's not sufficient wind flowing over the wings..
....guess what...
IT WONT FLY.
Move the PLANE at 5mph, wheels move at 5mph! move the TREADMILL at 5mph also, the wheels will now move at 10mph! the plane is still moving at FIVE MILES PER HOUR relative to the air and though 10mph relative to the ground, which isnt important for a plane to fly
that it all =)
Move the PLANE at 5mph, wheels move at 5mph! move the TREADMILL at 5mph also, the wheels will now move at 10mph! the plane is still moving at FIVE MILES PER HOUR relative to the air and though 10mph relative to the ground, which isnt important for a plane to fly
edit: by "wheels move" i mean "wheels turn"..sorry
Actually, you voiced some of what I had thought, especially with the wheels. Great post; you seem quite informed.
Okay, forgive me if this is repeating something in an earlier comment, but I read about thirty and gave up.
You've made an error in your presentation. In the summary, you say, "In the first, we're asking 'can a plane take off with no runway, if I replace the runway with a treadmill?' The answer, as we know now, is no. The plane must move relative to the ground in order to take off."
The problem with this, and the thing that is probably going to cause a lot of grief, is that the plane is moving relative to the ground,, because the ground is moving! The treadmill is the ground. You really mean to say here that "the plane must move relative to the air to take off. I realize you caveated this out earlier, but the way this is worded in the summary is simply going to fuel the argument).
(I got directed here by the xkcd forums.)
"The wheels do NOT generate any force on a plane, we get it. But the wheels ARE in contact with the ground, and if they aren't rolling across the ground (which they wouldn't be on a treadmill that exactly matches the speed/force/thrust generated by the plane, then the plane isn't moving relative to the ground, and no air is moving over the wings. It's not that hard."
Oh...dear. You're trying to answer Question 3. There's a problem here. IT IS NOT THE CASE that "on a treadmill that exactly matches the speed/force/thrust generated by the plane, then the plane isn't moving relative to the ground." The plane basically ignores the ground when determining its speed. It doesn't much matter (except for friction, which the wheels make fairly negligible) whether the ground is stationary, moving, moving erratically, or moving in changing directions (so long as the wheels can swivel...or so long as you're willing to let the wheels get ripped off by the moving ground). THE WHEELS HAVE NOTHING WHATSOEVER TO DO WITH THE SPEED OF THE PLANE. Put the plane on skids. Move the treadmill at whatever (non-infinite) speed you choose. The plane's jet or prop engines will throw air behind it and move it through the air, essentially ignoring the treadmill. The ground basically has nothing at all to do with the whole thing, except that the goal of the plane is to get off it. This is why I thing the original poster should have used "relative to the air" all the way though. Speed relative to the ground is completely immaterial to the flight of an airplane.
Ha.
so... if we got a really massive fan that could make the air around the plane match the planes speed it would remain on the ground then. however this not being the proposed method i aggree fully with the results shown here.
what if the pilot is determined to make the plane take off, but hes on crack, and is just balled up on the cock pit floor crying? technically, he meets the criteria of wanting to take off, but it's highly doubtful he'll be successful.
Look, you elitist, college-educated, arugula-eating snob, I will say this once and only once: You are WRONG!
For the sake of clarity, let me repeat that: You are WRONG (the opposite of RIGHT)!
Everyone should know by know that if God had meant for us to fly, he would have given us wings. So, logically, any situation where a plane might not fly, it won't. As it says in Psalm 55:6,
"And I said, Oh that I had wings like a dove! for then would I fly away, and be at rest."
It should be clear to everyone that this means that since we do not have wings like a dove, we cannot simultaneously fly and be at rest, as we would be on a treadmill (assuming we also had wheels).
As for the "Mythbusters", everyone who trusts in God's Laws of Biblical Physics knows that the outcomes of their experiments are influenced by Lucifer and, as such, are highly suspect. So, once again, I say, you are WRONG...
http://meditativeentropy.blogspot.com/
"IS NOT THE CASE that "on a treadmill that exactly matches the speed/force/thrust generated by the plane, then the plane isn't moving relative to the ground."
I think the poster you're responding to meant that the wheels are always countered by the treadmill, in which case, yes, it IS the case that the plane would not be capable of moving relative to the ground.
This is a commonly held view of the problem, interpreting it to mean the treadmill ALWAYS counters the speed of the wheels in such as way as to make them spin in place no matter what changes are made to the thrust.
However this condition may be achieved (this is a thought experiment after all) isn't pertinent. In and of itself this premise dictates that the plane cannot go forward relative to the treadmill (and hence the ground).
Period.
There's really nothing else to consider in this (completely valid) interpretation--the will of the pilot, the frictional cooefficient of the treads, the mass of the wheels--all are irrelevent to the central concept of the wheels being matched by the treadmill.
So...in this version, how can the plane move forward relative to the treadmill? It can not.
ANY forward motion relative to the treadmill breaks the premise--the wheels would have to "skid" ahead, which isn't allowed since any rotational acceleration is countered, making skidding impossible.
The plane can't move, no flight takes place. Yes, it is a "Question 3" scenario, but it's a valid interpretation of the problem as it is often stated (ie, whenever the wheels (NOT the plane) is bound to the speed of the treadmill).
The force of friction from the wheels on the planes is constant with speed, but the relevant value to consider is not the force to be overcome, but the dissipated. Energy is force*distance. Power is energy/time, or force*(distance/time). So the power dissipated by the friction in the wheels scales as force*speed. As the speed of the treadmill increases, the power necessary for the plane engine to overcome increases also. Since there is a limit on the power of the plane engine (and not necessarily a limit on the power of the treadmill), then the operator of the treadmill can force the plane to be stationary.
If you disagree (or even agree), please reply! Hopefully I can convince people to see the problem the right way!
I really can't be bothered to read the comments because its always frustrating. Needless to say I always interepreted the oringinal scenario as the plane remaining stationary with respect to the ground (or air if you want to be really pedantic but it we are talking about any comercial or military plane then the windspeed at the groud would rarely ever be sufficient to make much difference). Lift is generated as a fluid medium (air in this case) passes over the wing. If the plane doesn't move forwards then it can never take off.
The interesting bit in reality is of course what happens at the wheels, but the way I see it, the scenario specifies that the plane remains stationary. So any arguments regarding the forces at play here are pointless.
I'm not claiming this to be the accurate real world answer, but I see the whole thing as a theoretical model. Models aren't always accurate and in this case it isn't. Nethertheless, if we follow the model specified in the question then the plane will not take off.
Neither the explanation nor the "internet explosion" accurately take into account whether the plane is necessarily propelled by wheels, jets, or unladen swallows.
Unladen African Swallows FTW.
....QED
Oohhhhhhhhhhhhhhhhhh.
I'd never actually heard of this debate before xkcd today, so I decided your article would be a fun read.
I wasn't quite sure I believed/understood it all until it got to the skateboard/treadmill analogy. Well-written, sir.
"Much to your chagrin"... lololol.
You are wrong. You are assuming that no matter how fast the treadmill turns, it cannot affect the airplane. This seems obvious... but isn't actually true.
How so? Well... once the wheels star moving at relativistic velocities, their effective mass increases. Increase their mass enough, and there is no way the plane can take off. Since the plane has a maximum forward velocity due to air friction, it must also have a maximum lift. If its weight becomes higher than the lift, it cannot leave the treadmill.
Thanks for the light in this darkness, lol.
Also: Ø friction means that the air would not be able to lift the plane??
I'm only a junior in hs who hasn't taken physics, so pardon the potential for stupidity.
Nice site!
My only nitpick would be the factually-correct "friction force does not increase with speed" - The force itself does not change, true, but as the relative speed increases, the Power required to combat that rolling resistance/friction does increase; by P = F*V .
But since the friction of the wheelbearings and rolling resistance is relatively insignificant compared to the power of the aircraft's thrust, we can ignore that (as you did).
:)
I think a lot of the confusion is in the interpretation of the concept of "frictionless" and having "perfect" bearings, as well as the idea of a powerful enough treadmill to run at infinite speeds.
Let's replace the treadmill with magnets in the plane, and opposing electromagnets on the runway. Turn on the electromagnets and you can make the plane levitate, as with a Maglev train.
Now the plane is levitating in a stationary situation relative to the ground, it's easy to see that if the engines are turned on then there will be nothing to stop it moving forward unless a rearward force is actually applied to the plane. In the scenario of a "frictionless" treadmill and "freewheeling" wheels on the plane, then this is why the plane will take off.
If the electromagnetic fields were manipulated in such a way (as in a Maglev train) as to produce a rearward thrust counteracting the forward thrust of the engines, then it could be possible to make the plane stay in the same place, and therefore prevent takeoff. This is not the same as running the treadmill backwards at a higher speed with freewheeling wheels. To achieve the same thing in that treadmill scenario you would have to use an additional drive mechanism to create a rearward force which is deliberately applied to the plane.
"The propellers or jets create thrust that pushes against the surrounding air and causes the plane to move forward. A plane wouldn't move at all in a vacuum chamber."
Then how does a rocket fly in outer space, which is a vacuum? The engines cause the plane to go forward not because they push on the air around them, but by momentum conservation. If fuel is shot out backwards by the engines, you have momentum created behind the plane (say in the -x direction). Since there are no external forces acting on the plane/fuel system, this newly created momentum in the -x direction must be balanced by exactly the same momentum in the +x direction. This means the plane starts moving forward.
When you blog and berate people saying they don't understand simple physics, you should make sure you understand all the physics you talk about.
'"The propellers or jets create thrust that pushes against the surrounding air and causes the plane to move forward. A plane wouldn't move at all in a vacuum chamber."
Then how does a rocket fly in outer space, which is a vacuum? The engines cause the plane to go forward not because they push on the air around them, but by momentum conservation.'
A jet plane throws air behind it to move. True, "pushes against the surrounding air" was shorthand, but it amounts to the same thing. Rockets carry their own reaction mass to throw behind them. Okay, if you were somehow to provide the jet engines with oxygen to combust their fuel, that fuel would provide some small amount of reaction mass, probably moving the plane forward. A prop plane would not move at all in a vacuum chamber.
Okay, let's get off the shole question for a bit.
Look at the wheels. DO you really think that the planes uses them for acceleration?
First, if that were true, as soon as the wheels left the ground, it would turn into something like a hang-glider.
Second, the wheels couldn't PRACTICALLY balance the plane. Even (on the near impossible case) you manage to do so, as soon as the plane accelerates, it's rear is gonna hit the ground, and a lot of sparks will ensue if it continues.
Therefore, the engines have to be the main propulsion system.
So, back to the question. With the facts above, we concluded that the wheels do not propel the plane, but the engines do. The wheels would just be rolling, just doing their thing, having NO effect on the plane, while the plane does its "go fly" performance. So, we "yes" group WINS.
"If the wheels were perfect - that is, there is no friction in the bearings (and no deformation of the wheels as they spin) - then something interesting happens. When we turn on the treadmill, the plane stays stationary on its own. The wheels simply spin along the track, and impart no force to the plane."
I'm not so sure about this. Imagine if you put the wheel on the treadmill without the plane. When you start the treadmill the wheel will roll but it will also move backwards. With the plane attached the whole thing would move, even with frictionless bearings. It would be very slow so the treadmill would have to be unreasonably powerful, but I think this could theoretically put any amount of backwards force on the plane if the treadmill could keep accelerating. The limit in this case would be the friction between the tyre and the treadmill. The plane would have to slide to take off.
The ENGINES cause the forward momentum, not the wheels. The engine is not being affected by the treadmill, because the treadmill only works on things touching it. Sure, the wheels can create a lot of friction, but all that might happen are some sparks, and maybe the plane will catch fire.
Oh yeah, the "yes"ers already know they're right. Look here. --> http://www.youtube.com/watch?v=IbRcg3ji_Pc
Definitive. I like it.
I came to the same conclusion after converting the question slightly. If I put the plane on a dyno (that measures cars), and ran the engines, would it stay on the dyno?
Nope. It would pop right out because the force is pushing on the air, not generated from the tires.
I liked the airplane on ice and car in a vacuum comparison. That's great!
As this is only a thought experiment, then if we consider the belt has infinite acceleration and finite speed (But very large speed). The friction of the planes wheel bearing is non zero, so there is a force acting to stop the wheel from turning. Therefore, given a large speed of the belt, there will be a large force acting against the turning of the wheel, and hence against the movement of the belt. 3rd law of motion means there is then a force acting backwards on the plane, counteracting the thrust.
So the question is then, can a plane take off if it has its brakes on?
If its Qantas it aint going anywhere!
After writing on this subject—and the ridiculous turns that it tends to take—(http://uberregenbogen.livejournal.com/3115.html), i've come to wonder if the original question was meant to be: would a treadmill moving under a stationary aeroplane—instead of it moving down a runway—allow it to lift off? In other words: would the treadmill satisfy the requirements. The point of this scenario is that the vehicle remains stationary to the air, and is NOT asking if it would be ABLE to move. The answer, in this case is no.
I can well imagine this question being asked of students in elementary (or maybe middle) school science class—both to test their knowledge, and to encourage them think.
It may be impossible to ever find out what the original positor was trying to illustrate.
At any rate, it has become a rather interesting study in interpretation and human nature.
"So the question is then, can a plane take off if it has its brakes on?
"September 9, 2008 10:03 PM"
Yep! Here, the thrust of the plane has to overcome not the small friction of the wheel bearings but the larger friction of the rubber of the wheels against the treadmill.
It will take a greater proportion of the plane's total thrust but will eventually be overcome and the static wheels will skid along the runway as the plane accelerates to take off speed.
Excellent article, should hope to kill all those pesky no-flys.
There is one problem though - you say that a frictionless plane on a treadmill (without any propeller) would stay still. This is not true, as there is a force needed to cause the wheels to rotate, which will cause the plane to go backwards.
But if the plane has frictionless wheels with zero mass - then it'll stay still :)
A mistake -
"stationary to the ground" is different than "stationary to the air".
Being stationary or not stationary to the ground has no relation to taking off.
You wont take off if you are "stationary to the air".
What really happens is that the conveyor belt speed gets high enough to create sufficient headwind for the plane to take off with zero ground speed.
You wont take off if you are "stationary to the air".
True, but the plane won't be "stationary to the air", and WILL take off!
What really happens is that the conveyor belt speed gets high enough to create sufficient headwind for the plane to take off with zero ground speed.
I really hope you're trolling...
Man I can't honestly believe this debate still rages on. Folks this is a "thought experiment" and relies heavily on the actual question....
Will a treadmill runway operated at ANY speed keep me from taking off? Nope. I will require a small amount of thrust to overcome wheel bearing and tire to treadmill friction. After I've applied this thrust, you may increase the speed of the treadmill and I will remain stationary. But once I apply takeoff thrust (full throttle) I will take off....
Bonehead who said that the treadmill speed would increase to a point that would create headwind is an idiot. I would love to hear how he thinks the treadmill creates headwind.
ALL a plane needs to take off is air flowing fast enough over the wings. PERIOD. If I am unfortunate and do not have a treadmill to play with, i can wait for a hurricane, or possibly a tropical storm.... if my plane is pointed into the wind, and the windspeed exceeds takeoff airspeed, the plane will fly with the engine off. Mind you it would be a bumpy ride and we'd probably crash in the end, but the plane will fly.
Now this whole treadmill thing is just insane.... The original question never mentioned anything about a vacuum or outerspace or friction in bearings, etc. It simply said if the treadmill matches the speed of the plane, will the thing take off or not? It is a really simple answer: If the speed of the plane is less than takeoff airspeed NO. If the speed of the plane exceeds takeoff airspeed YES.
Can a treadmill runway in ANY way stop my plane from taking off if I as the pilot want to take off? Not a chance. It can not happen. There is no way.
Anyone who thinks a treadmill runway can prevent a plane from taking off is just "plane" stupid. (pun intended)
All of this about wheel bearings and friction is pointless.... I've heard peeps say that if there were an "infinite speed of the treadmill" and similar things..... My wheels won't fall off and the bearings won't fail no matter how fast that treadmill is going because the question stated: the treadmill matches the planes speed. And I promise that my plane will take off in about 45 seconds no matter how fast your treadmill is going, which is long before the bearings will fail or some other such nonsense.
It is ALL about the question.
And dear lord I hope you folks on the "no fly" side of this aren't pilots. And if you are please tell me who you fly for so i may book travel with someone more intelligent flying my plane. I'd like to know that if something goes wrong my pilot has a clue. And if all else fails I'll have to buck up and spend the extra money and fly myself.
Strictly speaking, cosmicnomad, a treadmill travelling near the speed of light could break the plane in many ways. Aside from basic stuff like unimaginable friction with the wheels, due to Special Relativity a treadmill travelling near that fast would have hugely increased mass, possibly resulting in lots more gravitational pull on the plane (depending on the original mass and speed of the treadmill).
But yeah - if the treadmill is going as fast as the plane, then of course it'd take off.
And unfortunately some of those on the "no fly" side ARE pilots (I've seen a few)... And anyway, when they did it on Mythbusters their pilot thought it wouldn't fly....
Mind you, he wasn't a commercial pilot...
They should put this as a question in Physics exams. It'd be too much fun.
I know I'm going to regret wading into this..
I read the post and saw no mention of one important effect. I skimmed the comments, and only one (anonymous) person mentioned it. Duff called him a troll and Cosmicnomad scoffed at him and asked:
"I would love to hear how he thinks the treadmill creates headwind."
It's very simple. There is a no-slip boundary condition between the treadmill surface and the air. The layer of air in contact with the surface moves along with it, at the same speed. The air velocity decreases with height. This is the headwind the anonymous commenter was talking about.
Contrary to the claim of the author, resistance to forward movement is not constant, and it is not simply due to friction in the wheel bearings. The air around the airplane is not stationary wrt to the ground. It is moving and creating wind resistance which varies with treadmill speed. It also creates lift.
But man I love getting responses such as this.... OF COURSE, there may be some minuscule bit of airflow created by the treadmill and OF COURSE there are varying amounts of friction in several components of this system.
It might be interesting to see if the treadmill could create enough headwind to register on the airspeed indicator let alone get the plane to V1 or V2 speeds. But alas my friend. This is not what the question was about.
I'm sure we could analyze this to death considering that no one has mentioned that a treadmill this large would cost more than the plane. Now we have economics in our once simple question that was intended to illustrate a simple thing like thrust is generated by the propeller and a treadmill has relatively no impact on the speed of the plane.
And while I'm at it, Duff was an idiot. His quote was:
"What really happens is that the conveyor belt speed gets high enough to create sufficient headwind for the plane to take off with zero ground speed."
I'd like to see this conveyor belt....
And while I'm at it, Duff was an idiot. His quote was:
Why, exactly, am I an idiot? for calling the headwind poster a troll? It certainly seemed to me that the only reason he posted that ridiculous claim was to stir up the argument again... maybe I should have called him a flamebait instead? Or maybe you think those words you quoted and attributed to me were mine - sorry, you're wrong, I was cleary quoting the poster immediately above me, the one I called a troll.
I quoted what he said, then responded to it. I never said I agreed with him, and I don't - his claim is simply ludicrous.
You should make sure you know what you're reading and responding to before calling people idiots, it makes you look like one yourself.
In fact, I can bet I CAN prevent plane from taking off by controlling the treadmill.
I will simple start it moving in the SAME direction with plane. Since wheels are idealistic, they DO NOT slide - we take it as axiom, just as we did with "frictionless" wheels.
The plane remains static. Simply, there's NO WAY it can overcome the treadmill movement without making its wheels to slide - a sort of event prohibited by rules of conundrum.
Ellegant solution for no-fly, isn't it? Please note - it DOES NOT contradict the conundrum in any of three your original interpretations. In none of them did you address the treadmill's direction issue.
Am I not a smart guy?
I'm sorry to say it, hata-botsad, but you are not a smart guy as you have ignored the crux of the whole argument:
Regardless of what the treadmill does, regardless of what the wheels do, the plane will always be able to push itself forwards through the air.
If the plane can push itself forwards, then it will accelerate forwards, and eventually it will have accelerated enough to take off.
mikeyberman,
The following argument - Regardless of what the treadmill does, regardless of what the wheels do - is merely senseless. What did you mean by this? There IS strong connection between plane and earth - through wheels and treadmill respectively.
In static position, the plane lies on its wheels. Dont forget about gravity.
Please, read once more my argument - the treadmil is accelerated IN THE SAME direction with plane. This changes everything absolutely.
Make an experiment, or imagine one. You hold a free-spinning wheel in your hand. Move it up and down the wall. Now, can you move it "up" the wall, having the wheel spinning "down", without slide effect?
Can plane move forward with its wheels moving backward, without slide - its THAT simple.
Well, I revised my theory and have to admit now, its wrong.
The plane will take off, refardless of treadmill's direction.
Still I dont give up hope of finding another way to prevent such event.
Of course a plane can move forward with its wheels moving backward...
Eventually the plane will catch up to the treadmill's speed, so then it will be stationary in the frame of reference of the treadmill.
Any inertial frame of reference will hold the same laws of physics, so we can think about this from the treadmill's frame of reference (i.e. we are attached to the treadmill moving along with it):
At first the plane and the wind will move backwards. Then the plane will accelerate forwards, until it is stationary (and the wind will continue moving backwards). Then the plane will accelerate more, and it will move forwards, until it has sufficient velocity relative to the wind to take off.
sorry Duff. was looking only at the guy who quoted you.
now for the new thing. a treadmill moving in the SAME direction as the plane will only contribute to increasing the airspeed of the plane, thereby making it easier to take off. This is exactly why aircraft carriers head into the wind to conduct operations...
And this whole business of the wheels having anything at all to do with the plane's velocity is nonsense. The wheels actually only interact with the treadmill. Think about it like this... A seaplane will take off upstream on a river no matter how fast the river is flowing.... (don't get into rapids, rocks, and other idiotic variations....). If the river is flowing at 60mph, I will need to apply power to counteract that, and once i do, i will begin to move forward relative to the shore. Once I'm moving forward to generate sufficient airspeed to fly, the plane will take off. Now I will give it too ya that the drag induced by the pontoons will be prohibitive in a 60mph river, but the original question mentioned a treadmill and there aren't any 60mph rivers that i am aware of.
A treadmill can not prevent a plane from taking off.
hata-botsad, i congratulate you. Very rarely do people change sides in this argument, and very rarely do people admit they have been wrong.
I applaud you :)
mikey you are on now?
i am always on. i am a god.
got an IM prog? I am on as signsystems on yahoo
too lazy :P
i feel like some delicious Family Guy
ah just figured we could chat. no biggee if you don't want too
feel free to add me, mikeyberman AT gmail.com on MSN, but I won't be on for half an hour or so.
is that the same as Windows Live Messenger? things change so much..... *sigh*
haha yes yes it is... crazy confusing technology...
I agree with the results.
But there is one important detail:
"The propellers or jets create thrust that pushes against the surrounding air and causes the plane to move forward."
Jet planes generate a forward force by pushing air backwards out of the engine, which then creates (by Newton) an equal and opposite forward force on the plane. It does NOT push backwards on the air behind it. For more information, look up the rocket principle.
this article is awesome. with the insight granted to me by 34, the (mostly nude) goddess of the internet, i hereby infer that the people arguing about this are a) on drugs and/or b) insane. given the lack of spelling mistakes (aka youtube comments syndrome), i deduct that it is more likely the people doing this fall into category b.
darklooshkin, I find it ironic that you make a spelling mistake in the very sentence in which you criticise people for being unable to spell. You mean to say "deduce", not "deduct".
And before you say I spelt "criticise" incorrectly, the 's' is the British/Australian spelling, whereas it's spelt with a 'z' in America.
But yes - people who argue the plane won't fly are generally insane. And so are some of those who say it will (e.g. the one's who claim it will entirely from the headwind created by the treadmill...)
"Imagine a plane without wheels. The fuselage would sit on the runway, and as you fired up the engines, it would skid spectacularly along the runway, possibly spewing sparks in its wake and doing quite a number on the body of the aircraft. No matter how fast it was going, the frictional force against the airplane would be constant; friction does not depend on speed"
The friction would actually be less as the plane gained speed wouldn't it? The faster the plane goes the more force is pulling it up (even while lift < weight). So the faster the plane goes, the less the normal force is and the less the friction is
Nice thinking Greg :)
However the original quote was still wrong - friction is normally proportional to speed, and sometimes proportional to speed squared.
So I'd imagine friction may first increase at low speeds, then as the plane approaches takeoff speed it might decrease as you said...
cosmicnomad, while the "plane will fly" is quite obvious, your swimming pool anology is not so good, unless you are wearing skates.
You need to be able to swim (with arms only, while your feet remain on the treadmill) with enough force to overcome the friction between your feet and the rubber treadmill - I doubt you can manage that.
Your example is exactly the same as skates, treadmill, handrails, but substituting water for handrails. You still need to be wearing skates.
I first heard about AoaT from the mentioned MythBusters episode. Initially, I was a "no-fly" on the basis that if the plane is going forward at speed X and the treadmill is cycling backwards at speed X, then the plane cannot go forward to achieve lift.
I was, of course, thinking about airplane wheels like car wheels, and when they did the small scale test and mentioned that the wheels rotate independently, it all made sense and then I realized that there would be no speed a treadmill could go at that would prevent the plane from going forward. However, the wheels would be spinning at the planes liftoff speed plus whatever the treadmill was doing.
For the "Yes" people the wheels are a red herring. A plane without wheels would also take off....that about demonstrates the silliness of this question.
In many ways this is a question about the difference between static friction and kinetic friction.
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