The Goddamn Airplane on the Goddamn Treadmill

Posted byRandall 830 Comments on The Goddamn Airplane on the Goddamn Treadmill

Sorry for the forum/blog downtime today. Many things went wrong during davean’s heroic upgrade. (I blame the LHC.)

Feynman used to tell a story about a simple lawn-sprinkler physics problem. The nifty thing about the problem was that the answer was immediately obvious, but to some people it was immediately obvious one way and to some it was immediately obvious the other. (For the record, the answer to Feynman problem, which he never tells you in his book, was that the sprinkler doesn’t move at all. Moreover, he only brought it up to start an argument to act as a diversion while he seduced your mother in the other room.)

The airplane/treadmill problem is similar. It contains a basic ambiguity, and people resolve it one of a couple different ways. The tricky thing is, each group thinks the other is making a very simple physics mistake. So you get two groups each condescendingly explaining basic physics and math to the other. This is why, for example, the airplane/treadmill problem is a banned topic on the xkcd forums (along with argument about whether 0.999… = 1).

The problem is as follows:

Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?

The practical answer is “yes”. A 747’s engines produce a quarter of a million pounds of thrust. That is, each engine is powerful enough to launch a brachiosaurus straight up (see diagram). With that kind of force, no matter what’s happening to the treadmill and wheels, the plane is going to move forward and take off.

But there’s a problem. Let’s take a look at the statement “The conveyor belt is designed to exactly match the speed of the wheels”. What does that mean?

Well, as I see it, there are three possible interpretations.  Let’s consider each one based on this diagram:

1. vB=vC: The belt always moves at the same speed as the bottom of the wheel. This is always true if the wheels aren’t sliding, and could simply describe a treadmill with no motor. I haven’t seen many people subscribe to this interpretation.

2. vC=vW: That is, if the axle is moving forward (relative to the ground, not the treadmill) at 5 m/s, the treadmill moves backward at 5 m/s. This is physically plausible. All it means is that the wheels will spin twice as fast as normal, but that won’t stop the plane from taking off. People who subscribe to this interpretation tend to assume the people who disagree with them think airplanes are powered by their wheels.

3. vC=vW+vB: What if we hook up a speedometer to the wheel, and make the treadmill spin backward as fast as the speedometer says the plane is going forward? Then the “speedometer speed” would be vW+vB — the relative speed of the wheel over the treadmill. This is, for example, how a car-on–a-treadmill setup would work. This is the assumption that most of the ‘stationary plane’ people subscribe to. The problem with this is that it’s an ill-defined system. For non-slip tires, vB=vC. So vC=vW+vC. If we make vW positive, there is no value vC can take to make the equation true. (For those stubbornly clinging to vestiges of reality, in a system where the treadmill responds via a PID controller, the result would be the treadmill quickly spinning up to infinity.) So, in this system, the plane cannot have a nonzero speed. (We’ll call this the “JetBlue” scenario.)

But if we push with the engines, what happens? The terms of the problem tell us that the plane cannot have a nonzero speed, but there’s no physical mechanism that would plausibly make this happen. The treadmill could spin the wheels, but the acceleration would destroy them before it stopped the plane. The problem is basically asking “what happens if you take a plane that can’t move and move it?” It might intrigue literary critics, but it’s a poor physics question.

So, people who go with interpretation #3 notice immediately that the plane cannot move and keep trying to condescendingly explain to the #2 crowd that nothing they say changes the basic facts of the problem. The #2 crowd is busy explaining to the #3 crowd that planes aren’t driven by their wheels. Of course, this being the internet, there’s also a #4 crowd loudly arguing that even if the plane was able to move, it couldn’t have been what hit the Pentagon.

All in all, it’s a lovely recipe for an internet argument, and it’s been had too many times. So let’s see if we can avoid that. I suggest posting stories about something that happened to you recently, and post nice things about other peoples’ stories. If you’re desperate to tell me that I’m wrong on the internet, don’t bother. I’ve snuck onto the plane into first class with the #5 crowd and we’re busy finding out how many cocktails they’ll serve while we’re waiting for the treadmill to start. God help us if, after the fourth round of drinks, someone brings up the two envelopes paradox.

830 replies on “The Goddamn Airplane on the Goddamn Treadmill”

  1. Pingback: meneame.net

  3. it would take off the. the wheels are in a neutral state and its the engine that is pushing it along so even if the engines are supose to be pushing to go 60 mph the wheels would be going 120 mph

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  6. I was in Boston this past weekend, visiting a couple, and thought I wouldn’t have a chance to see a few bloggers whom my husband and I read.

    On Sunday I was led into a shopping/sightseeing trip while some others of our group went to a “retro video game event” featuring panel discussions and chiptune performances. They made peanut butter cookies to bring, leading me to vaguely wonder whether they’d checked ahead about nut allergies.

    My friend walked me into the Boston Common Garden and I found a surprise birthday picnic party, featuring peanut butter cookies, friends, and the bloggers we’d been hoping to meet! The weather was beautiful and we traded stories about a terrible New Year’s Eve, a security initiative gone wrong, and animated spinoffs of live-action sitcoms.

    It was lovely.

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  7. I just want to say that I love XKCD, and all of its fans! and Mythbusters! and the internet! Have a great day everybody!

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  8. Realists quickly realize that interpretation #4 is full of shit because there is no such thing as a treadmill that can spin at infinite speed.

    Now, what about Snakes on a Plane on a conveyor belt?

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  9. I like to think of this “paradox” as an equivalent of a glass filled with water that you want to put into circular motion (water, not glass). Rotating the glass does not automaticly convert to circular motion of water inside (friction is too small). As for the plane’s wheels – it’s not the speed of the conveyor you have to think about, but the rolling resistance of the wheels (which is almost zero), which is basically independent from the speed of the conveyor. So, the conveyor may be rolling backwards with any speed you can think about, but the force with which it pushes the plane back is small and practically constant. Open the thrust on 10% and you will neglect it’s effect, open it some more, and you will have the plane moving forward with increasing speed.

    I thought of another version of this experiment. Think about a plane with enormous pulling propeller (diameter=wing span), that is restrained in such a way that it can only move up and down. Start it’s engine, open the thrust. Will it fly?

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  10. > I’m sick of these motherfuckin’ airplanes on these motherfuckin’ treadmills.

    Greatersteven, you win the comment thread. +5000 internets to you.

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  11. Nick said: “4 basic forces act on a plane as we all know. lift, thrust, drag, gravity.”

    Ahem… may I suggest strong, weak, electromagnetic and gravity?

    So, something that happened to me recently… well, I watched some very excited physicists cheer at some very exciting white dots made by some very excited protons. It was… exciting. And then we had icecream.

    Life is like an atomic nucleus… full of little ups and downs.

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  12. Pingback: oranchak.com » Blog Archive » links for 2008-09-10
  15. Pingback: I Believe Two Things » Blog Archive » PoaT+xkcd+www+blog = Fun!

  16. I used to think that the airplane would not take off…but if you think twice, all the conveyor belt does is to minimise the friction between the airplane and the ground. The airplane does not power the wheels like a car does (a car would appear stationary in the same experiment). The airplane is free to roll on its bearings….So it does not matter how fast the conveyor belt turns….the airplane will pick-up speed and take off and i would love to see mythbusters or anyone else coming up with such a testbed.

    Keep up the good work

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  17. Favorite nerd / geek / math / physics jokes? Here are a couple of my faves:

    -One atom says to the other atom “I think I lost an electron!”
    “Are you sure?”
    “I’m positive!”

    There are 10 kinds of people in the world: those who understand binary, and those who don’t.

    Why did the chicken cross the mobius strip?
    To get to the same side!

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  18. Quick question for the #2 crowd. If the airplane WAS powered by it’s wheels, couldn’t it still take off? It would just be twice as hard (since the wheels have to spin twice as much).
    If the answer is yes, then I think this airplane/treadmill question was meant either for #1 or #3 people.

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  19. The ambiguity reminds me of the millennium controversy. “When did the millennium start?” is an ambiguous question. “When did the two-thousands start?” is well-formed, and so is “When did the 3rd millennium CE start?”, but they’re different questions.

    *My* millennium started in 1957.

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  20. I’m more interested in what’s happening in the airplane bathroom after those first class drinks…

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  21. I finally went to the grocery store. I like having food in the house again. My wife and I had a ton of fun playing “Still Alive” on Rock Band. It was a free download! JoCo rocks. The drums on Blitzkrieg Bop are almost impossible at any difficulty level above medium.

    My motorcycle ride to work was a little cold, but that means the ride home should be pleasant. Unless it’s on a treadmill.

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  23. 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  24. Everyone is always wrong on the internet. It’s true. Even this post is wrong. If you’re arguing on the internet, you are by definition wrong. And probably a nazi.

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  25. Here’s a nice story that happened recently: I bought an MPD32 from guitar center. I bought it on credit so I wouldn’t have to take it out of my savings. So far I love it. I chopped up some nice piano samples that each took up all 16 pads, and the MPD allowed me to seamlessly switch between the four samples. So I can play a full song that way, something you can’t do with a Trigger Finger. It makes me really happy!

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  27. One way to solve this is to replace the 747 with a Harrier. And all those who claim that an aircraft needs to move forward in order to take off is proven wrong.

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  28. @MrBawn lol. But only, if Monty opens one of the two remaining doors with a goat behind it, you should switch your choice.

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  29. Have you ever seen those self propelled sprinklers that move themselves down the field with water jets?

    Thought problem:

    A self propelled sprinkler, set on “suck”, is placed on a treadmill, set to match the speed of the sprinkler. Next to it is an identical sprinkler on an identical treadmill, pointed in the opposite direction. One sprinkler has twice as much money taped to the bottom of it than the other. You can examine the sprinklers, but you must walk to them in short segments, each one half the distance of the previous one. With each step, you must remove a grain of sand from a heap of sand in your pocket, and must stop when it ceases to be a heap. On some day next week (which one you won’t know beforehand) you will be asked which sprinkler had the greater amount of money: the one that moved, or the one which didn’t. Assuming you are a goat, should you switch doors?

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  30. Pingback: links for 2008-09-10 « pootling

  32. Seems the two envelopes conundrum could be solved by approaching the person in charge of giving you one of the envelopes, smacking him or her upside the head, and walking off with both envelopes.

    Or is that too Occam’s 2-by-4 of me? Sorry. 😉

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  35. …in that case the pilot would be Satan and instead of hearing lyrics in reverse, they’d just be in Portuguese.

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  37. My son had his first day of pre-school yesterday and it went really well. He had a good time and his teachers are very good. I’m glad we got him into the school.

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  38. Ahh, foolish Nick. You clearly didn’t read the blag before you came to post in the comments. Arguments, anyone?

    The two basic interpretations I see are, is an infinite speed allowed for the conveyor belt? If so, this clearly defies the laws of the physical universe as we know it. This shows that we are operating in a new universe, and the laws therefore being somewhat fuzzy, the plane will probably take off or not according to your opinion on the matter. 🙂

    However, if we are operating in the physical universe we all know and love (okay, maybe not love, at least not if you’ve ever bumped your head into it) then infinite speeds are not attainable by the conveyor belt. Thus an attempt to set up the treadmill to go at reverse speed proportionate to the velocity of the bottoms of the plane wheels would fail. You could of course make it go at a proportionate speed to the motion of the axes relative to the ground, though.

    In either of these two real world cases, the treadmill (unless it was very very different from what most of us envision with the word “treadmill”) would not create a noticable amount of wind. The plane’s engines would force air backwards sufficiently to overcome any minor wind from the treadmill speed, and the plane would get going. Its wheels might be considerably hotter than in a normal takeoff, but it would, in fact, take off.

    In MY universe, where infinite velocity treadmills are not only permissible but are the norm, the plane would not take off or stay on the ground (rather, treadmill). Instead, it would be sucked into the time-space vortex caused by a loop of matter accelerating to infinite rotational velocity. The treadmill would create at its center something akin to a black hole, with the attraction power of one, but sans gravitational compression drawbacks. The world would be pulled in from both sides of the treadmill-loop simultaneously and would be turned inside out, resulting in an eradication of all planes besides the ones previously exported to the moon.

    Also, it’s my birthday! 18! Woohoo!

    Captcha: Commander Grizzlies

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  39. I have actually performed the sprinkler experiment, using a pump to suck water into a copper S-tube suspended from a piece of surgical rubber hose connecting to a “T” fitting in the centre. Apart from a “kick” when the water speed changes, it does not move. The kick is in the direction opposite to the normal sprinkler’s rotation, IIRC. The non-moving result is due to head loss at the orifice, which exactly balances the momentum change of the water. This is a very odd fact, which may have some deeper significance that escapes me.

    The envelope paradox is just a variant of the Monty Hall problem, and demonstrates a failure to understand conditional probabilities. You can tell this because they are using the same symbol “A” to represent two different amounts of money in disjoint situations as if those situations were the same.

    Like the Monty Hall problem, if you restate the problem it is trivial to solve:

    One envelope contains H, one envelope contains L. Half the time you will get H, half the time you will get L. The expected value of the envelope you get will be (H+L)/2, and the expected value of the envelope you don’t get will be (L+H)/2. The mathematical relationship between H and L is clearly irrelevant.

    Using the same symbol, “A”, for the value in the envelope you choose in both the case when it has H in it and the disjoint case when it has L in it conflates these two disjoint concrete situations. It makes it SEEM like you can treat the problem in terms of total probabilities, but in fact A means completely different things depending on which envelope you choose, and this produces the “paradox” in the incorrect analysis.

    The Monty Hall problem also has two disjoint situations: 1/3 of the time you choose the correct door at the outset, 2/3 of the time you don’t, and in the latter situation Monty has no choice about which door to open, so 2/3 of the time you will win if you switch, 1/3 of the time you will lose. If you use conditional probabilities to combine these disjoint situations, as you should, you’ll get the right answer. If you treat these two disjoint situations as being the same you’ll make a mess of it.

    The socially interesting thing about these problems is that people cling to the belief that there IS a problem long after the solution ought to be obvious, steadfastly refusing to learn anything about that solution, simply repeating their misunderstanding of it endlessly and at times with great force. I’ve come to believe that some people actually enjoy the feeling that the universe is incomprehensible because it absolves them of the responsibility of actually trying to understand it, much like the nutjobs who claim to think the LHC might destroy the world.

    Which raises a question: are there any “paradoxes” like this that have ever been settled to everyone’s satisfaction? Or once a problem gets labelled as a “paradox” does a certain fraction of the population cling to their confusion forevermore?

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  40. No, you don’t understand: I SHOT THE PILOT. I committed a senseless act of violence, seemingly with impunity and without cause. What did I do with the body? Were there any witnesses? Irrelevant.

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  41. Now, if you had two planes on two separate treadmills that matched their exact speeds, as in your example #1, and they are running ad infinitum, and also suppose that there are velociraptors hidden in the first class sections of both planes, also ordering drinks and discussing logic puzzles, and further imagine that one plane contains twice as many velociraptors; how long would it be until the velociraptors begin eviscerating the surrounding passengers and how can they be stopped?

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  42. What about an analogy with a child running with a plastic pinwheel? As the child runs, the pinwheel spins. Like the plane, the child’s “propulsion” is independent of the pinwheel. If there is a wind, the pinwheel will spin faster, but the child still runs forward (assuming that the wind acts on the pinwheel but not the body of the child).

    In order to keep the plane from taking off, the treadmill would need to keep the plane from moving forward. However, provided the wheels can spin freely, this can not happen; the wheels aren’t providing the forward motion and so won’t impact the engine’s progress.

    If this were a seaplane in a fast moving river, it would be a different problem. The drag of the current would provide a direct counter force on the solid form of the plane and thus the equivalent problem would result in a standstill until the engines burned out.

    However, here the wheel/treadmill system is independent of the jets of the airplane.

    If the plane’s alight you can not fight.

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  43. I dont think anyone posted this yet, but if they did, Im sorry. Mythbusters did do an ep on this, and …..drum roll….. the plane took of. HURRAY!

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  44. Been mentioned.

    Also, in the “post a positive story” department:

    There was a minor earthquake in the middle of the night a couple of days ago. I woke up just in time to experience it.

    It was kinda nice.

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  45. Hmph. Interesting, Mr. Munroe, but you assume too much. Does the plane have wings? Does it have engines? Does it have stewardesses to demonstrate safety procedures ’til the classhole stops the treadmill? And wouldn’t the sprinkler soak through both of the envelopes anyway?

    …Sign me up for group #26: making out in the baggage compartments. With cocktails. But no pterosaurs.

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  46. This is pretty similar to the Helicopter on a Turntable problem. I tried to solve that once, but I ran out of test pilots.

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  47. The people who argue 0.99999… versus 1 haven’t looked at a formal definition of the reals. *sigh*

    I argued my entire physics class into believing the plane took off. Working on remembering how.

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  48. I think the main problem is the limited speed of the treadmill but the unlimited speed (well, in a way) of the plane’s thrust.

    Instead of a treadmill going up to infinity speed, it has to reach a limit, due to friction and other forces. This allows the thrust to push the plane forward allowing for increased airflow underneath the wings.

    My theory? Mythbuster’s Plane simply went faster then the treadmill.

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  49. In other news, my captcha phrase is “Serbia huddled” which is pretty great.

    And now I go, on a mission, to buy cream to thicken the squash soup I made yesterday. Hurrah!

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  50. I’m going to make an attempt to even further clarify the airplane/treadmill problem, specifically regarding Mr. Munroe’s “group 3” interpretation. In this case, there are actually a few different answers. It all depends on whether or not you’re dealing with real or ideal bearings/parts in the plane and treadmill. This leaves four possible interpretations for plane/treadmill: real/real, real/ideal, idea/real, and ideal/ideal. In all cases I am assuming that there is negligible wind. Here’s what I think would happen:
    Real/real: The plane has to take off before its wheels fail. The answer depends on many factors, such as the treadmill’s rate of acceleration, maximum rotational velocity of the wheels (before failure), et cetera.
    Real/ideal: The plane’s wheels explode, the rest of the plane follows (jet fuel + friction=?)
    Ideal/real: The plane’s wheels cannot fail, so the plane takes off
    Ideal/ideal: Who knows? The treadmill could travel infinitely fast, but the wheels would spin freely, so the plane could take off, except for the fact that both the treadmill and the wheels are going the speed of light. Time dilation would make it appear as if the treadmill was going 0.999… C, which equals exactly 1 C (I swear that’s unintentional!)
    If my physics- or grammar- is flawed, please correct me, especially the relativity bit at the end.

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