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?