Thursday, January 31, 2008

Airplane on a treadmill

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.


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:

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?
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:

-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:

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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Anonymous said...

The problem with this whole controversy is that the original question had the wording of interpretation #2. Any other interpretation is just wrong. The question explicitly states that the plane does NOT move in relationship to the ground, and therefore will obviously not take off, since there is no lift. It does not fly.

I just don't understand how any of the other interpretations came to be. Why the fuck else would you make a giant treadmill and try to have a plane take off on it other than to somehow "cheat" physics and make it take off in a smaller area. Why else would the question be asked? If you build a giant treadmill, you'd still need the treadmill to the be the entire length of a traditional runway, and what would be the point of that? You'd basically just have to build more than twice as much runway, plus the mechanics to turn the treadmill, WHICH IS WHAT THEY DID ON THE MYTHBUSTERS EPISODE on the scale of the smaller plane, that's why the episode is stupid, they interpreted the question incorrectly. You'd still have to move just as much in relation to the ground. It's just as stupid as the pilot only using just enough throttle to stay stationary in relationship to the ground, but THAT is what the question EXPLICITLY required.

Basically. The whole question is just stupid. Wheels spinning has nothing to do with planes taking off, as the writer of this blog has explained.

What would be more interesting to consider is making a plane take off vertically like a helicopter with some sort of wind tunnel that matches your altitude until finally releasing you. I assume your airspeed would still be zero, so you'd still just crash to the ground, unless you were at a high enough altitude to build enough speed and pull out of the dive, but at least it's a more interesting question.

Still such dumb comments down here too. "If the pilot wanted to take off he could." Yeah, assuming he had a treadmill as long as a runway his aircraft would normally require anyways, so what would be the point of the treadmill? It just doesn't affect the plane at all if you replace the runway with a treadmill surface of the same dimensions. Why would this even be the interpretation of the question, though?

Anonymous said...

My airplane is a glider, with an engine driving the wheels. It is designed to drive off a cliff, then glide. (The "car" part stays behind, on the ground.) MY airplane will NOT fly off that treadmill. ;-)

Also: Some planes CAN do vertical takeoff, and do NOT need forward airspeed to generate lift. Examples: Harrier, Osprey.

Anonymous said...

Also: If the treadmill is run really fast, it will generate wind in the direction of the treadmill. (I'm talking REALLY fast!) So, to prevent the plane from taking off, you run the treadmill FORWARD to create a large tail win. The plane will move forward, relative to the ground, but will never get up enough airspeed to take off. (This argument might depend upon the wheel friction being entirely independent of speed, which is wrong. If wheel friction depends upon speed, then you'd need to do a full calculation.)

Anonymous said...

Would this help? In the 1970s, the USAF experimented with an airplane (a C-8 Buffalo specifically) with the wheels removed and what amounts to a hovercraft stuck on the bottom instead.

It was called the Air Cushion Landing System. It existed, I saw it with my own eyes.

So would "NO" people think the treadmill would stop this hover-plane from taking off? Why would it be different from wheels? Friction would only be marginally less.

Romulus said...

It seems to me the critical flaw is the expectation that increasing the throttle of the plane affects its speed relative to the treadmill. However, the wheels on the treadmill will simply spin faster than whatever the treadmill is set to -- i.e., even if the treadmill operator increases the treadmill to attempt to offset that increased wheel spin, the wheels will simply spin even faster.

The wheels aren't doing anything except keeping the plane from hitting the ground, while allowing it to slide parallel to the ground.

The minute the throttle is raised enough to move a stationary plane forward, it will move forward, treadmill or not.

Imagine instead of wheels, the plane was a hovercraft, sitting on a cushion of air. Or sitting on a long trough of well-oiled ball bearings. It doesn't matter. The plane moves forward as if there was no treadmill at all.

The wheels aren't doing work, they are having work done to them. Therefore trying to counteract the wheels with a treadmill is pointless because there is nothing to counteract.

Romulus said...

Given the above, if you wanted to be cute, instead of a treadmill, you put the plane in an wind tunnel, and now the treadmill operator is operating the wind speed of the tunnel. As the plane's propeller spins faster, the wind tunnel operator increases the wind speed (again in the opposite direction).

In that scenario, the plane doesn't move forward, because the force the propeller working against is basically "cooperating" with it -- which is the opposite of what you want for a plane to work.

That being said, once the air speed is high enough, depending on the wing shape, the plane might still actually *rise,* but still not move forward.

matt said...

If there is no slip and the treadmill has no motor, the plane will definitely not fly! The thrust from the engines will accelerate the treadmill until the frictional force opposing the motion increases to match the thrust force. Then the treadmill will move at a constant velocity...and the plane will not move!

Josh said...

I love reading all the comments from people who didn't read all the way through the article. It's really obvious.

That being said, what if the person who originally asked this question was not a pilot? It is reasonable that a non-pilot would be unaware that a plane, when on the ground, does not get its thrust from pushing against the air, but from its wheels which are in contact with the ground.

If that were the case, then the original question is intended to ask whether you can take off by increasing your ground speed without increasing your air speed, and the answer would be "no". While I agree with the article's author that this is a poor question in the context of actual airplane mechanics, it is possibly a more useful or likely question in regards to physics (otherwise you might as well just do away with the airplane confusion and ask about someone standing on a treadmill wearing rollerskates).

Either way, I imagine it to be a treadmill no longer than the plane itself on the edge of a cliff, on an extremely windy day, and I will be flying a Supercub (you said there could be no lift from the treadmill or the propellers, you said nothing about the weather or the plane's wing shape). I intend to take off, waggle my wings, land, take off, and land again just to taunt the treadmill operator.

Josh said...

^ Confusing typo above:

It is reasonable that a non-pilot would be unaware that a plane, when on the ground, does not get its thrust from pushing against the air...

Should read:

"It is reasonable that a non-pilot would be unaware of a plane's mechanics, and assume that a plane, when on the ground, does not get its thrust from pushing against the air..."

Anonymous said...

First I heard of this controversy and apparent paradox. Your explanation is exactly correct. (Though in the case of a jet rather than a prop plane, most of the forward motion comes from thrust (action-reaction) rather than pushing against the air. And in a vacuum, a rocket engine (which contains its own oxygen supply) will also work and the plane will take off.)

Funny that people would think that it's the plane's wheels that cause it to go down the runway. That would be quite a trick.

Anonymous said...

Imagine a plane flying level with its landing gear down. We construct a mile long conveyor built at the same height as the planes wheels. When the plane passes over the conveyor built and its wheels touch, the plane will continue to fly. The planes wheels and conveyor belt will rotate at some speed, but this doesn't matter. The plane will continue to move forward.

Anonymous said...

If a plane were to be propelled forward by its wheels rather than its turbines, it, of course, would not fly. The interesting thing about such a plane, though, is that it almost always would not fly. It could not sustain flight, as it would require contact with the ground to pick up speed. It would essentially become a glorified hang glider. Thus, only special hang gliders fail to take off.

Anonymous said...

If we assume that the plane in question is not on specifically a conveyer belt, but a tredmill, and, moreover, an immovable indestructable tredmill of massive proportions, then it seems the plane could not take off. As it would begin to coast forward along the belt of the tredmill, it would collide with the part in the middle to which the handrails are usually attached and which displays the speed and other unnecessary information. The plane would then either bounce backward, or recieve a good deal of damage, rendering it unfit for takeoff.

Anonymous said...

I hate to poke holes in arguments...but I'm a geek, so here goes. I want to preface this by saying that I agree that, in a REAL scenario the treadmill could not prevent the plane from taking off, or even slow it's progress. However, this is a thought experiment, so...
The entire above analysis is done assuming that there is no acceleration. However, if the treadmill speed were continually increased, the inertia of the wheels would require a force be transmitted to the plane to increase the wheel's angular velocity. Thus, in theory, by continually (and really really rapidly) accelerating the treadmill, the engine thrust could be matched. However, before the end of the take off attempt the bearings probably would be melted/broken, or the treadmill speed exceed the speed of light. However, if we ignore those things (no-one ever said anything about the mass of the wheels or treadmill, or maximum speeds for the treadmill or wheel bearings), then, in theory, the plane could be prevented from taking off.

GVVitali said...
This comment has been removed by the author.
GVVitali said...
This comment has been removed by the author.
GVVitali said...

If the plane moves the plane wheels are turning at a faster speed than the conveyor belt speed. However if you increase the speed of the conveyor belt the speed of the wheels will increase as well unless the conveyor belt is turning so fast that the wheels start to become a drage on the plane therefore preventing the plan from moving. Excluding the plane from skiding if the conveyorbelt and the wheels have teeth to prevent the skiding.

GVVitali said...

If the conveyor belt is a black hole no plane will escape it's sucking power.the wheels will be turning at speeds less than the black hole conveyor speed.

GVVitali said...

I will explain it step by step, tell me the step you don't understand so so I can explain where you are getting it wrong.
Step1: imagine a 2 Kilometer (KL) conveyor belt runway with a plane facing forward in the midpoint of the runway.
Step2: the runway conveyor belt (CB) begins to turn so that the CB moves the plan backwards at 1 KL per hour. The engine of the plane is still off.
Step3: at the end of the hour do you agree the plan will be at the beginning of the runway?
Step4: as the plan is approaching the beginning of the runway the plane's engines are turned on but there is not enough thrust to turn the plane wheels so the plane is still moving backwards with the CB which is still moving at 1 KL per hour.
Step5: do you now agree that if the plane is not to fall off the beginning of the runway the thrust of the plane engine has to accelerate the plan forward so that the wheels turn forward at 1 KL per hour just so the plane will remain on the same spot marked by a pole at the end of the runway off the CB?
Step6: if you agree on all 5 steps before this one do you now agree that to get the plan back to the middle of the runway in 1 hour the thrust of the engines has to accelerate so that the wheels of the plane have to turn at 2 KL per hour.
Step7: let us now imagine that for a plane to take off it has to travel at 100 KL per hour, so now as the plane is moving at 1 KL per hour back to the mid point of the Runway whilst the CB is moving also at 1 KL per hour the wheels of the plane are rotating at 2 KL per hour. Do you agree till now?
Step8: as the plan approaches the middle of the runway the engine accelerates so that the plan moves 100 KL per hour in relation to the ground on which the CB is built upon. However the CB is still moving in the opposite direction at 1 KL per hour so the wheels of the plane have to move at 101 KL per hour on the CB for the plane to move forward at 100 KL per hour.
Step9: therefore if the conveyor belt travels at 101KL per hour and the wheels of the plan also are forced to turn at 101 KL per hour by the thrust of the engine the plane will come to a stand still in relation to the ground on which the CB is built.
Step10: whilst the CB is turning at 101 KL per hour and the engine is roaring the wheels are turning at 101KL per hour on the CB the plane will remain stationary relative to the ground, if now 100 people get on board through the passenger tunnel the plane engines will have to accelerate to keep the plane on the same mark on the ground. Also if the CB is lifted 10 degrees on one side the engine will have to accelerate some more to keep the plane still on the same mark on the ground. However the wheels and the CB will not be moving at a greater speed of 101 KL per hour.

GVVitali said...

So the concussion is for a plane to take off the wheels have to turn 100 KL per hour faster than the speed of the conveyor belt for the plan to take off if we agree it takes a plane to travel at 100 KL per hour relative to the ground to take off.

GVVitali said...

It all depends on the plane wheels, at what speed can they turn before they drag the plane back.

GVVitali said...
This comment has been removed by the author.
GVVitali said...

If the conveyor is moving at the speed of sound and the plane is moving 50 kilometers per hour relative to the ground, the plane wheels will be moving at the speed of sound plus the 50 kilometers per hour relative to the conveyor belt.
If you increase the speed of the conveyor belt 50 kilometers per hour more the plan wheels will also increase 50 kilometers more. Therefore the plane wheels will always be moving faster than the conveyor belt on condition the plane is moving.
However if you increase the conveyor belt were the speed exceeds the ability of the plane wheels to turn any faster then the plane will be dragged back by the wheels. So the faster the conveyor goes the more drag the plane will have and as the conveyor belt accelerate faster and faster a braking effect will be applied to the plane untill the plane comes to a standstill, presuming the whole landing gear does not tear off the plane. The engine will be roaring madly but not able to move forward because the conveyor motors will be more powerful than the plane engine.

GVVitali said...

The riddle says the conveyor belt can go as fast as the wheels so as long as the plane is moving the conveyor belt can move faster till the engine can not accelerate anymore. When the conveyor belt catches up with the plane wheels speed the plane will be at a standstill relating to the ground.


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ahmed mohamed said...

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Unknown said...

sea planes take off with n wheels

Unknown said...

The question is NOT asking you to insert heat, friction against treadmills, etc!
It is asking you to"HYPOTHETICALLY" accept that the "CONVEYOR BELT" is perfect and can maintain the exact speed of the plane's momentum even under changes to forward or backward movement.Then, the plane will NOT take off!

Take the analogy of the wheel brakes being on and irregardless of whether the propeller is at full speed, full pitch or just idling, the plane will NOT take off! (unless the brake is released)

In real life, there is no perfection and "hypothetical". Natural forces come into play and the plane will take off!

uberkluger said...


(Note: any reference to the conveyor speed means the speed of the belt upon which the plane sits. The conveyor itself is
assumed to be stationary relative to the ground upon which it is placed.)

Neither propeller driven nor jet propelled aircraft "push" against the air or push air backwards to generate thrust (unless
you define the expansion of combustion products in a jet engine as "pushing air").

Jet engines create thrust using Newton's third law. The expanding combustion products push on the inside walls of the
combustion chamber in every direction EXCEPT to the rear (the exhaust port = no wall). Thus there is an unbalanced force
pushing the engine (and thus the aircraft) forward. A jet engine would still work in a vacuum provided that the necessary
fuel oxidizing agent were supplied to replace the non-existent air, either separately (e.g. compressed or liquified oxygen)
or as part of the fuel itself (e.g. a controlled burn of (tri)nitroglycerine). The only problem is one of semantics in that
this is not a jet engine anymore, it is a rocket engine.

Propellers only push air backwards as side effect. Propellers are aerofoils and generate thrust exactly the same way the
wing generates lift, by creating an area of lower pressure on the top (i.e. front) of the blade as the air moves across it.
Just as with a wing, the propeller is angled slightly to maximise its efficiency thus creating a fan effect which blows
air backwards (prop-wash) but this air flow has little effect upon the amount of thrust produced. However, since the blade
generates thrust by moving through air, it is true that a propeller would not work in a vacuum.

uberkluger said...

A closer examination of the question suggests that this is actually a trick question. All of the analysis seems to agree that if the plane moves relative to the ground (to which the conveyor is attached) then it can take off. But the question states (again verbatim but with emphasis):

The plane MOVES in one direction, while the conveyor moves in the opposite direction.

So what does "MOVES" mean? It means that the position changes relative to something.
Which something? There are only two possibilities. The existence of the plane and conveyor is stated and it is reasonable to assume the existence of the ground. So the something is either each other (plane..conveyor) or the ground (plane..ground, conveyor..ground), any other interpretation is mixing reference frames. If the something is the ground then the question states that the plane moves relative to the ground (without imposed limit on speed) and can thus take off regardless of conveyor speed. If the something is each other then the conditions reduce to:

The plane and conveyor move in opposite directions relative to each other.
The speeds of the plane and conveyor are equal relative to each other (speed is a scalar, direction already stated to be opposite).

However, these statements are true for EVERY pair of moving objects in the universe and so say nothing about the particulars of this question except that it is assumed that the conveyor is stationary relative to the ground and so also is the plane which therefore does not take-off. For the purpose of logical analysis, I am ignoring any real world
considerations such as "How does the conveyor know to adjust its speed unless the plane's speed is different, which violates the stated condition?" and "How does the conveyor know when to start?". The only answer is that the conveyor is claire voyant and can predict what the plane will do. But that violates the "Real World Considerations" that produce the
conundrums (I doubt anyone other than Douglas Adams would suggest the existence of a psychic conveyor). In reality, these considerations would simplify respectively as "The plane and conveyor speed is constant" and "Both the plane and conveyor speeds remain at 0", making the whole question rather redundant.

Thus, the answer to the the question is yes or no entirely depending upon how you define "moves" and the behaviour of the conveyor is irrelevant (within the stated rules).

Alternatively, the question is a trick question because it's not about aerodynamics but about recognising that a plane accelerates by direct thrust from the engines unlike a (typical) road vehicle that uses a combination of torque in the driving wheels and (usually static) friction with the road surface. (Typical means NOT coyote chasing road-runner in a
rocket car.)

And for those who postulate about conveyors/treadmills travelling at whatever speed is necessary to achieve the desired result (whether no air speed so no fly or no ground speed but still fly), these are interesting ideas (whether correct or not) but they are not relevant to the actual question which stipulates conveyor speed matches plane speed
(however relative speed is defined).

uberkluger said...

Comments about comments:

Comment containing:
"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 (power) dissipated. ... 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."

This adds unfair conditions that alter the problem.

The circumstances of the question are (reproduced verbatim):
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?

Nowhere is it specified that any limit may be applied unequally. Both the plane and conveyor are either bounded by reality (maximum plane thrust/power and maximum conveyor speed) or equally unbounded (unlimited conveyor speed and plane engine thrust/power). The latter is typical for thought experiments. The comment also ignores the possibility that the extra power dissipated in the wheels is provided by the conveyor. A common error is to use power as a measure of effort (deliberately didn't say "work"). If a chair is on a table then the FORCE of gravity is pushing (pulling?) the chair down but the table is equally pushing the chair up. The chair does not move and so no work (energy change) is done. Thus, power = 0. If I hold the chair suspended and stationary, I think it is an effort (I get tired and sweaty) but if the chair does not move, power is still 0. Similarly, if the plane has its brakes applied and the engine thrust is insufficient, the plane does not move
and again power is 0 (except for the power used overcoming friction WITHIN THE ENGINE). Most of the "energy" produced is heat (with a bit of sound). If the force of the wheel friction is constant (as accepted by the commenter) then the engine need only generate an equal amount of opposing thrust in order for the plane to remain stationary. If the plane
is stationary then the power is 0 (+ engine friction). It doesn't matter how fast the wheel is turning. So, in fact, it IS the conveyor supplying the extra power (the belt IS moving and the wheel friction is trying to stop it).

Comment containing:
"However the original quote was still wrong - friction is normally proportional to speed, and sometimes proportional
to speed squared."

Laws of dry friction
The elementary property of sliding (kinetic) friction were discovered by experiment in the 15th to 18th centuries and
were expressed as three empirical laws:

Amontons' First Law: The force of friction is directly proportional to the applied load. (aka normal force)
Amontons' Second Law: The force of friction is independent of the apparent area of contact.
Coulomb's Law of Friction: Kinetic friction is independent of the sliding velocity.
Thus it would seem that this comment (referring to a comment about a comment about a wheelless plane sliding along the runway)
is incorrect but, interestingly, comments about the frictional resistance of the wheels cannot apply the dry friction laws
(unless the maintenance crew forgot to grease the bearings) and so constant friction force is probably not true in this
case (I was only using the commenters' own assertion of constant friction in an earlier critique regarding engine power).

uberkluger said...

Comment containing:
"i know this was a while ago but i wanna point something out to mikeyberman.
you misread darklooshkin's comment about spelling errors. he wasn't criticizing, he was saying that you guys must not be on drugs because of your LACK of spelling errors, which means you must be insane."

However, even though this comment is correctly stating that darklooshkin did not criticise (yes, an s) the comment posters on this page, there was an implied criticism (spell that with a z, i dare ya) of youtube comment posters AND YET there was a spelling error within that very comment so by darklooshkin's own logic, (s)he must be on drugs (and also insane for posting a comment on this page).

Comment containing:
"To look at it another way, how long a treadmill would you need to LAND a plane safely and then stop?
Answer: About as long as any other runway surface with similar friction."

Comments that mention it state that the wheels DO NOT contribute to the thrust during take-off. However, they do contribute to deceleration when landing (i.e. braking). Previous comments (correctly) state that Work (Energy) = Force * Displacement.
For the brakes, displacement is proportional to the number of rotations of the wheel. If the plane were landing on a conveyor moving in the opposite direction to the plane (this is an assumption but why else would you use it) then the wheels will spin faster and appear to travel farther than the ground displacement from touch-down to stopping (it's actually the distance along the belt). Thus more energy is dissipated in the brakes and the plane will stop sooner although, to actually stop, the conveyor would need to stop too.
How much sooner, I don't know. Also, hopefully the wheels aren't skidding all the way, so the surface friction is irrelevant
(provided that it is sufficient).

Comment containing:
"I'm afraid your pompous blog is basically entirely wrong. All you have done is 'alter' the question to fit your answer."

This is basically saying that, even though it is not specifically stated, the question implies that the plane remains stationary. To return to the pilot conspiracy angle, the pilot didn't want to take-off but some weirdo put a conveyor under the plane so the pilot has to throttle up just to stay in place. Apparently, the question is "If the pilot has applied just enough power to compensate for some stupid prank, will the plane take-off?" Put that way, the answer is clearly no. Further, if the question expects the plane to remain stationary (relative to the ground) "then the belt might as well not be there
and so the question is made moot".

Comments containing:
"If that were the case, then the original question is intended to ask whether you can take off by increasing your ground
speed without increasing your air speed, and the answer would be "no".
"It is reasonable that a non-pilot would be unaware of a plane's mechanics, and assume that a plane, when on the ground,
does not get its thrust from pushing against the air..."

Query: which of the plane's mechanics allows a plane to increase its ground speed without increasing its air speed?
Answer: the engines but the ground has to start moving first so the plane can "catch-up" to where it was.
The question states that the conveyor speed is controlled to match the plane speed so the plane would have to move first.
Uh-oh, contradiction.

Anonymous said...

I can't believe no one's asked the most important question: are there snakes on the plane?

Wednesday said...

"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."


It's not enough that you say it - explain why, and prove it.
Is it because there's no such a big conveyer belt? This is thought experiment, so imagine there is conveyer belt powerful enough. After all, only one century ago, airplanes barely existed; so if not today, such a conveyer belt may exist in one century.

To justify this statement, you need to prove that concept defies laws of physics, not that it's impossible to build it today.

You're essentially wrong here.

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Anonymous said...

La comparaison avec une personne sur une planche à roulettes sur un tapis roulant est séduisante, mais ne tient pas la route... en effet... le fait de forcer des biceps pour faire avancer la planche à roulettes ne fera pas avancer le corps, seulement les jambes... une fois les jambes avancées... le corps restera immobile, les mains agrippées à la rambarde du tapis roulant... DONC l'avion ne bougera pas davantage...

La poussée des réacteurs fait avancer l'avion sur un sol immobile... ou sur un sol glissant comme le fait un hydravion... Sauf que les flotteurs d'un hydravion ne sont pas des roues... pareil si l'avion n'a pas de roues...

Ici, on parle de roues libres qui tournent sous la poussée de réacteurs ou la traction d'hélices sur un sol immobile. Si le sol est mobile... la mobilité du sol ( du tapis roulant ) contrecarrera l'effet de la poussée parfaitement... et s'il la contrecarre de manière plus intense l'avion pourrait même reculer à cause de la friction des roues sur le tapis, peu importe la poussée...

La confusion vient du fait que l'on pense intuitivement que la poussée fera avancer l'avion de toute façon... Or, ce n'est pas le cas d'un avion sur roues... Ce qui serait différent pour un hydravion ou pour un avion sans roues... Sans roues, la poussée des réacteurs peut éventuellement contrecarrer la friction de la carlingue sur le sol, mais le fait de disposer de roues... ce n'est n'est pas la même chose...

Suite 1 ici-bas

Anonymous said...

Suite 1
La roue tourne... imaginons que l'avion est attaché par des câbles au sol et que le tapis roulant est actionné à grande vitesse, les roues vont tourner, mais l'avion ne bougera pas... si coupe les câbles... le roulement des roues va se poursuivre et l'avion va rester immobile... jusqu'à ce que la friction s'en mêle et il va reculer peu à peu... jusqu'à débarquer du tapis roulant...

Quand le pilote met les gaz d'un avion sur tapis roulant programmé pour contrecarrer la poussée des réacteurs, l'avion ne pourra pas avancer parce que les roues tournent. Si elles ne tournaient pas... si elles étaient bloquées... ce serait différent... peu importe la vitesse du tapis... la poussée des réacteurs ou la traction de l'hélice pourraient à force contrecarrer la friction des roues bloquées sur le tapis roulant créant un mouvement vers l'arrière, jusqu'à faire éventuellement fondre les pneus, et faire avancer l'avion, comme s'il n'y avait pas de roues...

La confusion vient du fait que les roues tournent... le déplacement de l'avion est entièrement lié au roulement à billes des roues...

Pour reprendre la comparaison du tapis roulant... si une personne est sur une planche à roulettes et est maintenue par une personne qui exerce une poussée dans son dos... peu importe la force de la poussée, la personne restera immobile puisque la vitesse du tapis roulant sera égale à ce que serait la vitesse de rotation des roues sur un sol immobile quand une même poussée est exercée sans que l'avion ne soit sur un tapis roulant. Imaginons donc une piste et un tapis roulant au milieu de la piste... Imaginons un trottoir, et au milieu une bande de 3 m. de tapis roulant de niveau avec la surface lisse du trottoir... à 3m du tapis roulant, si je pousse une personne sur une planche à roulettes sur le trottoir, elle va avancer jusqu'à ce que les 4 roues de la planche accèdent sous ma poussée au tapis roulant qui est programmé pour contrecarrer ma poussée... je ne pourrai plus faire avancer la planche et la personne dessus, peu importe ma poussée... parce que les roues de la planche tournent sous la friction du tapis roulant et si le tapis roulant va plus vite, à cause de la friction, peu importe ma poussée... la planche pourrait même reculer jusqu'à faire débarquer les 2 roues arrières du tapis... ce qui pourrait permettre à un regain de poussée de faire avancer à nouveau la planche pour que les 2 roues arrière soient à nouveau sur le tapis... sauf que... une fois à nouveau sur le tapis... toute poussée aussi puissante soit-elle ne pourra pas faire avancer la planche... Et... si je ne donne pas davantage de poussée... les roues avant vont tourner dans le vide sans déplacement... faute de traction ou propulsion des roues libres... Pareil pour un avion sur roues...

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