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.

821 thoughts on “The Goddamn Airplane on the Goddamn Treadmill

  1. You’re wrong.
    (And the fun part is making that stick! Chuckle)

    Considering that this is all about an airplane, it’s amazing how EVERYONE badly glosses over air effects. The direct force of the treadmill on the wheels is NOT the only way the treadmill can apply a backwards force preventing the plane from rolling forwards. The moving treadmill will also apply a drag force on the air above it. The larger the treadmill surface, and the longer it moves, and the faster it moves, the more it will pull the air backwards, generating a wind. This wind can and will push backwards on the plane. And significantly, it will also provide a wind across the wings (in addition to the airflow over the wings being produced by the engines thrusting).

    So what this problem REALLY boils down to is, can a plane take off from a runway, without moving forwards, when there is a sufficient headwind blowing at it. And the answer is, unsurprisingly, yes. A plane in a headwind can lift directly upwards, with zero forwards motion. In fact a plane can even take off while traveling BACKWARDS (compared to the ground), if the headwind exceeds the plane’s liftoff velocity.

  2. My thought is: it likely depends on the weight of the plane upon the treadmill. If the pressure provides enough ‘grip’ without ‘locking’ the tread to the surface underneath, then the treadmill will move, not the wheels, and eventually the plane will reach the end and fall into the sea.

    The same also applies if the plane is too heavy and ‘locks’ the tread. It will then simply move off the treadmill.

    But hey, I’m not well versed in physics.

  3. I won a pub quiz recently, it was a fun experience as the team I was on, titled “Super Sexy Awesome Team”, had never won a pub quiz before.

  4. My first reaction:
    @Alsee –> This is a physics problem… obviously it’s a spherical plane in a vacuum.

    My second reaction:
    @Alsee –> I like your explanation, except I doubt there’s significant wind generated by the treadmill. I’m happy to assume that it’s possible, given the eventuality that a hurricane or storm will provide a strong enough headwind for the plane to lift itself directly upwards.

  5. @James… dammit!

    I can’t believe I’m not the only one who linked here to this random old article from today’s what-if xkcd, and commented on it with that Big Bang Theory episode (with the spherical chicken in a vacuum joke) in mind.

    I swear when I posted your comment wasn’t yet on the page, I was just slow, and when I refreshed with my comment, yours showed up.

  6. Sounds like someone who ascribes to interpretation #2 is trying to discredit people who hold interpretation #1 by ascribing them to the easier to argue against #3. Does anyone really argue for #3 or are they just poorly describing their #1 interpretation?

  7. Problem solved. The Mythbusters (who else would bother?) actually did it and, unsurprisingly, it turns out that no matter what you do with a planes wheels it will take off.

    Also note that #3 creates an infinite feedback loop because the speed of the plane is unconnected to the speed of the wheels, so eventually (assuming a magical motor capable of such a thing) the treadmill and wheels will both be moving at relativistic speeds at which the normal rules regarding relations of differing velocities break down and the plane will start to move forward at something like 0.01c (I understand the concept, but I’m not sure I can explain it without writing a 5 page paper), which is well beyond the takeoff speed for any aircraft ever built (in fact it’s probably fast enough to rip most of them to shreds).

  8. I had a friend in High School. She was pretty, unnaturally thin, shy, and rather antisocial. We had a class together, she sat behind me. The teacher who taught that class is on of my favorite people. She has a genius level IQ and a doctorate. She works very hard, is good at what she does, and was a bit like a mother to me. She really took care of us. She assumed that my friend was anorexic, and at one point confronted her about it. I don’t know if it is true, but I think it is. My friend took it poorly.
    Generally i was worried for my friend. She seemed like one of the least happy people i know. I couldn’t intervene, not directly, we were not close friends.
    Now we are in college. The same college, only by chance. After I realized we go to the same school, I decided to get lunch with her. She was the worst conversationalist I have ever spent any time with. She complained incessantly. about everything, about the food, and how she can’t eat it. I wanted to make sure she was okay, she had friends and was happy. She may have had friends. she was not happy. she caught my intention and resented it. she no longer speaks to me.
    I just found out that she is now a model. She is beautiful, she makes an excellent model.
    I just want to say that beauty comes at a cost. It isn’t free. She may only be so thin and gorgeous because she is anorexic.
    I don’t think that makes her any less pretty. I don’t know if the two are worth it together, i doubt it.
    Most importantly, I want to say that we need to evaluate the two separately. Her beauty and her anorexia may be connected, but they are not two sides of the same coin. We need to be able to ignore the anorexia, just for a second, and rejoice in the beauty. We need to ignore the beauty, and deal with the anorexia. We can have both, a healthy mind, and a beautiful body. We can have both, to build art, and feed the poor. Thank you for reading this.

  9. Has anyone tried this with the Brachiosaurus on the treadmill? I’m pretty sure that was the theme of “The Land Before Time XXVIII”…

  10. Spherical plane in a vacuum? Surely it’s an infinite plane of uniform density.

    If an infinite, frictionless plane of uniform density is moving in a direction parallel to itself … can you tell?

  11. Kris? Mythbusters? Really? This is xkcd. We have actual scientists here. We don’t need special effects bods to swing around the room on a camera.

    If the treadmill follows interpretation #3 it will approach lightspeed, perceive time more slowly than the stationary aeroplane, fail to respond as quickly as necessary and lose the race.

    As for “no matter what you do with a plane’s wheels,” there are roleplayers here as well as scientists, and answers like “Try setting them in concrete” or “Attach a two-inch-thick cable made of human hair to each wheel, pass them over large and sturdy pulleys behind the treadmill and use them to support a blue whale” are possible. For a simpler answer, weld the axles and see how that affects take-off.

  12. Mythbusters didn’t use a treadmill, they pulled a ‘conveyor belt’ under the airplane. Their experiment was flawed from the start since the weight of the airplane overcame the speed of the belt, allowing the airplane to move forward. Wheel speed doesn’t matter, airspeed does. And Mythbusters is a terrible show.

    Otherwise, burritios are f#*@ing tasty.

  13. @Kris,
    The Mythbusters carried out their experiment in accordance with interpretation #2; assuming non-sliding wheels, neither of the other 2 interpretations can be implemented without violating a constraint of the problem.
    So… it’s not what you do with a plane’s wheels that is at the heart of this, but rather how you interpret what was meant by “the speed of the wheels”.

  14. Sea based planes use buoyant ski’s in lieu of wheels. The fact that planes can “still” take off in this manner is an empirical argument for #2.

  15. I’ve been thinking about this. First of all, with a plane, the wheels don’t have any engines to make them spin. They would just start rolling because the plane is pushed forwards by those engines attached to it’s wings.
    Next, the belt would have to move at a speed to prevent the wheels from rolling, yet the plane is still pushed forwards. The belt could not stop that. It can only stop the wheels from spinning. And to do so, the belt itself would have to move into the same direction as the plane, just to stop the wheels from spinning. And it would continue to move in the same direction until the plane lifts off.
    Basically, the belt would be like a catapult, keeping the wheels still while the plane moves forward. It doesn’t even need a lot of force, since the plane itself has the proper engines to do the hard work. It’s like a plane with ski’s instead of wheels, or those thingies that allow it to land on water. Planes don’t really need wheels to take off. It’s just that it helps to overcome the resistance of the ground in case you’re trying to fly away without wheels. Because that would still be possible, but would require more powerful engines plus a new paintjob after every take-off…

  16. As an aeronautical engineer and A&P…

    B747s are extremely heavy. You are going to break your treadmill.

  17. I agree with Alsee (June 4). The treadmill should be able to drag the air above it backwards, creating enough of a headwind to keep the plane stationary, and also allow it to take off without moving forwards.

  18. the speed of the wheels is irrelevant, hydroplanes don’t even have wheels, even tho is not that easily done, what myth busters did is enough to make the point that wheels turning at millions of rpm wont matter since the plane is pushing against the air around it, not the ground.

  19. I think that the real heart of the question is that if a plane’s movement in the x-y plane were restricted but it is allowed to freely move on the z axis (z being vertical) would it be able to lift off? I think most prop planes could get enough airflow over their wings to generate the lift to take off. Like what this plane is practically able to do: http://www.youtube.com/watch?v=f7u1jzjFL8s
    A 747 on the other hand has it’s propulsion sources mounted under the wings and were it not able to move the thrust would probably shear the wings off before generating anywhere near the airflow over the wing needed for liftoff.

  20. W T F

    Seriously though, I think the question was always backwards. I think the pertinent question is “If a plane were on a treadmill, assuming the treadmill were strong enough to operate with a plane on it, and, assuming wheels of a realistic mass but frictionless bearings, and a landing gear of infinite strength and rigidity, how fast would the treadmill have to accelerate in order to keep the plane from taking off? At what point, if possible, could the treadmill move enough air over the wings to create lift? And if it could create enough airflow over the wings and the plane lifted off at full throttle, what would happen? All other physics should be accounted for, including possible relativistic effects.”

    Hmm… this sounds like a great “What if?”.

  21. You know, if we introduce rolling friction into our model of the plane’s wheels, (and we assume the treadmill is indestructible and perfect etc etc etc), there should be some constant treadmill speed — albeit a hilariously and impossibly high speed — at which the drag from the treadmill and the wheels should completely counteract the thrust of the engine. We can call this interpretation #2.5 if you like. I usually don’t see anyone subscribing to this interpretation.

    What I like about this is that under this interpretation, the plane is under no net force, but under an overwhelming net torque. I envision a 747 doing a Looney-Tunes-esque faceplant, which is delightful, but a more realistic model would probably have a lot of chaotic bouncing and skittering shenanigans, even if we’re making the very generous assumption that the plane wouldn’t just come apart.

  22. @Wim ten Brink,

    Did you miss the part where it said the treadmill moves in the opposite direction?

  23. Hmm… that brings up an interesting point: considering that the problem states that the treadmill will “exactly match the speed of the wheels, moving in the opposite direction”, interpretation #1 and #3 are erroneous, as follows:
    #1 “The belt always moves at the same speed as the bottom of the wheel” : In this case, the bottom of the wheel (defined as the measuring point of the wheel speed) is moving in the same direction as the belt.
    #3 ” the relative speed of the wheel over the treadmill” : In this case, the source of the wheel speed is technically the bottom of the wheel, since contact with the treadmill/belt is the only force driving the wheel rotation by which wheel speed is being measured. Again, if our measurement is based on speed of the bottom of the wheel, it’s moving in the same direction as the belt, not the opposite direction.

    This leaves as the only valid interpretation of the three above interpretation #2, right? Can you poke holes in this conclusion

  24. The only force preventing the airplane from moving forward is the transmitted through the wheel bearings. If these are ideal bearings, no forces can be applied to prevent the plane from moving forward (Vw).

    If we assume real world conditions, then the bearings have friction, however the conveyor will not be able to spin fast enough to prevent the plane from moving forward.

  25. So IF the treadmill can accelerate infinitely, it will push back enough on the wheels to keep the plane stationary. But it can’t.

  26. Start by thinking: are the wheels relevant? What would happen if a ‘plane with no wheels applied thrust and took off? Answer – if would move forwards and take off. The contact between the ‘plane and the ground is irrelevant.

  27. Have we just ressurected this old chestnut again? (The plane will take off, for the record.) I’m sure we already had all these arguments, up to five years ago, but one mention on a new what-if and… here we go again!

  28. I agree with Ben. The wheels can theoretically exert a negative thrust on the plane (assuming some friction in the bearings), but the thrust vector will not intersect the plane’s center of mass, creating torque and planting the plane nose-first into the treadmill.

  29. There is an unstated assumption here:
    Any forward movement of the plane results in a change in the speed of the rotation of the wheel.
    At high speeds, the friction of the wheel against the treadmill will mean that the wheel may drag along the surface in the same way that wheels drag across the ground without rotating during emergency car stops.
    I’m not sure of the exact cause of this, and it may be that a larger acceleration is needed than the engines can apply, but it wasn’t discussed above.
    If this assumption isn’t made then the model can be simplified to an aircraft with no wheels taking off, which can happen as seen in water based aircraft.

  30. Airplane wheels have no motors in them. A pilot, attempting to take off, does NOT apply the brakes. So the wheels are essentially frictionless whether on a treadmill or on a paved runway.

    The plane is not accelerated or retarded by the wheels. Those four engines on the wings, OTOH? Each GE CF6-80C2 has a static thrust is 59,000 lb (or 1,150 mBr = 1,150 milliBrachiosaurs). 236,000 pounds of force on a 800,000 (loaded) airframe will accelerate it 3.4 feet/sec^2 causing it to reach take-off speed in about one minute (thrust changes with forward speed, so while v=at, a is not constant).

  31. Hi everyone!
    About Ben’s interpretation I must say that the engines torque in common airliners is positive (nose-up).

    But the most important thing is that the treadmill is not able to transmit Power (I’m talking about Watts) for the plane if the bearings friction coefficient is assumed to be zero, because of the non-sliding condition between tires and treadmill. So, the only source of Power are the engines, that will give to the plane the kinetic energy necessary to take off, whatever the speed of the treadmill (please do not apply to relativistic speeds).
    Read Win ten Brink’s commentary for further explanation. I think he has been very clear.
    Greetings from Barcelona!

  32. I’m inclined to go with @Chris, @Ben and @Ted here, simply from knowing the physics.

    You are going to break your treadmill.

    If your treadmill is somehow capable of reaching sufficient speed to get close to fast enough to sufficient speed before breaking, you are going to break your plane as well.

    At miraculous best, the treadmill will simply collapse in the middle under the weight, cause the plane to manage a barrel roll before landing leaving Group #5 startled and alcohol-soaked. (Admittedly, ‘collapse in the middle under the weight, leaving the plane right there’ likely is objectively better, but not as funny.)

    This is assuming, of course, that the structural integrity of the treadmill is not sufficient for it to reach high enough speeds as to cause it to instead send shrapnel flying & the plane has sufficient structural integrity (and good enough pilots) to survive the experience and not land upside-down.

    …I think I shall be part of group #6, the people watching with alcohol from a safe distance.

  33. So guys, they already did this on mythbusters. There is recorded evidence that you can take a plane off on a conveyer belt. Why is this still an argument. Also the physics makes sense if you think about them

  34. While I’d agree the part about the same speed but opposite direction would mean the wheels are sliding, and if we disregard the “opposite direction” part, I think it would be obvious the plane would take off.

    However, I DO think it would be possible to stop the plane from taking off, even without assuming any friction in the wheel bearings. It would seem reasonable to me to assume the wheels have a MASS MOMENT OF INERTIA, which means that whenever the treadmill applies a force to the contact patch at the bottom to accelerate the wheels, the same but opposite force must also be transferred through the axle to the plane, in order to actually generate a torque to accelerate the wheel.

    Assuming a theoretically perfect control loop with infinite gain and zero delay, the plane would have to be stationary, because any movement would mean there was a mismatch in the speed of the treadmill.

    This would mean the treadmill would have to keep accelerating to keep the plane from moving, reaching improbable speeds very quickly. Before the plane could take off, the tires would expand under the influence of centripetal force, which would probably end badly.

  35. The reason you can’t get a consensus on the answer is because the question is flawed. The conveyor belt as specified can’t exist if the plane is moving, the math boils down to a = a + b which can only be true if b (the speed of the plane) is zero.

    If the plane rolls forward 10 feet, the wheels rolled forward 10 feet and the conveyor belt rolled backwards 10 feet which means the plane didn’t move, except the plane did move so the wheels must have rolled forward 20 feet. If the wheels rolled forward 20 feet the conveyor belt rolled backwards 20 feet which means the plane didn’t move, except the plane moved forward 10 feet so the wheels must have rolled forward 30 feet. Lather, Rinse, repeat — forever.

    Since the conveyor belt can’t exist in the same universe as the plane that is moving forward we have one answer:

    The plane can take off if it has a head wind that exceeds its V2 speed (http://en.wikipedia.org/wiki/V_speeds).

    Any answer that involves the plane moving forward causes the conveyor belt to vanish in a puff of logic.

  36. Sparkygsx, You should say “the tires would expand due to insufficient centripetal force”. As the comment stands you are using a correct physics term to mean an incorrect colloquial concept.

  37. So what if the tread mill were able to instantaneously adjust it’s speed to match the forward thrust created by the aeroplane engines.
    Hypothetically the 747 on a normal runway would accelerate to 160 knots, adjust it’s attitude to increase lift and then take off.
    The forward motion created by the aeroplane on the stationary runway would allow it to have sufficient air flow over the wings to create the lift required for take off.
    If the treat mill were to instantaneously adjust its speed to match the forward motion the aeroplane is trying to create then I don’t believe the plane would take off as it wouldn’t have the required air flow over the wings to create lift.
    Two variables that would change all this (and there may be more) are A: the airflow created by the drag of the tread mill (hold a smoke wand in front of a belt sander, now upscale it a shit load); and B: if the aeroplane were to adjust it’s attitude at all the required lift for take off would change and throw out all the other parameters including the amount of forward thrust required for take off.
    The pilot could screw with us by actuating the brakes on the wheels and then taking them off while lifting the nose (think of a wheel chair where you balance on the back wheels) which may help but I think he’d be a little busy trying to not destroy the plane. Plus in that scenario I think Black Hat would be sitting at the start of the tread mill with a front end loader full of caltrops…

  38. Oh no, it’s happening again.

    “Your problem is confusingly worded. Please clarify some of my questions.” is the only answer you need give.

    If that does not satisfy you, then try the most basic approach you can think of, then explain that as your answer. I like breaking it down into forces, the thrust (T), drag (D), and rolling resistance (R). Try to keep it short and sweet.
    Remember to reread the problem every five minutes to make sure you’re not straying off topic and contesting something clearly stated, or making conflicting assumptions.
    Happy poo flinging!

  39. Now, it’s probably just my terrible understanding of physics, but why is scenario #1 incorrect? If the speed of the plane and the speed of the treadmill counteract each other, why would there be enough air flow above the wing to create lift (assuming the treadmill doesn’t cause the wind)?

  40. “So, imagine there’s a treadmill that a plane sits atop of, and the treadmill moves at whatever speed the airplane engines would have moved the plane had it not been on the treadmill. could the plane achieve lift?”

    “so…the horizontal component of whatever this equation is results in zero?”

    “yes”

    “so while the plane is on the treadmill, it doesn’t move horizontally?”

    “YES”

    “….so….it’s in the same situation as your average helicopter?”

    “YE-wait”

    “and helicopters take off all the time”

    “no, wait, it’s…but…yes, it’s different…but…no?”

    “yeah, proof by helicopters, Q.E.D.”

    “fuck”

  41. Today I wore a black t-shirt with XKCD in white letters and three people complimented me on it. Today was a good day.

  42. Okay, here it goes…

    You all missed the true and best explanation.
    the problem states that the treadmill accelerates to march the speed of the wheels…
    the plane pushes forward due to thrust from engines…
    the treadmill spins faster…
    before the plane reaches the end of the treadmill the bearings in the wheels melt due to friction…
    superheated rubber sticks to treadmill, gear snaps…
    by this point the confused pilot has the engines at full power…
    plane at full throttle encounters treadmill moving thousands of miles an hour…
    boom…
    bOOm…
    BooM…
    BOOM…
    Fireballs…
    plane parts are thrown across state lines…
    best interpretation of problem ever…

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