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. Just sayin- xkcd comments are what web comments would be if they had no friction. Or conveyor belts. Or whatever.

    (Best evAR.)


  2. Ok, I’m a literature guy (just as an fyi, it would be a stupid question even from a literary criticism standpoint) so pardon my feeble attempts at physics here, but it looks like the sticking point of the argument is engine thrust vs. lift. From my, admittedly foggy, recollections of high-school physics we were told that planes flew due to the difference in air speed over and under the wing. So is the question appears to be asking if 747 engines have enough raw power to get the plane off the ground without added lift from actually moving forward and that sounds like a measurable thing; what am I missing?


  3. Okay, I am TOTALLY not qualified to think about this problem… but.
    Airplanes go up because air passes over their wings (which are airfoils), producing lift. In order for air to be passing over the wings, the plane has to be moving forward pretty damn fast (it helps if it’s going into a headwind, so the same airspeed can be achieved with a slower groundspeed).
    So IF the plane is not moving forward (its wheels are glued to the ground, or something) there is no possible way it can achieve rotation speed. (Well, there’s this, but that’s an empty plane in a 70mph gale.)
    But planes are powered by their engines, so it shouldn’t matter what the wheels are doing so long as they’re free-spinning; as 2) says, the wheels will just spin faster. But I imagine there would be some speed (probably a lot less than 2*Vr, or at least in that range) at which the wheels/axles will fail, and everything goes to hell.


  4. First, I want to say I think it’s interesting that all the comments happened after your WhatIf link.

    Second, the problem as stated, is different from how I’ve always heard it. How I’ve heard it is:

    Imagine a plane is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to exactly match the minimum takeoff speed, moving in the opposite direction. If the plane is throttled to the minimum power normally required to take off, can the plane still take off?

    I say the answer to this is “no,” because while the wheels are free-spinning, in the real world there will still be enough friction/drag introduced to prevent reaching takeoff velocity (that’s why the “minimum” qualifiers are there).

    P.S.: How does your CAPTCHA know my library card PIN?!?


  5. So what happens in the case of #1 under idealized circumstances (non-slipping tires and frictionless axels)?


  6. The treadmill IS DESIGNED TO match the speed of the wheels. Due to corruption, nepotism and Microsoft, it fails.

    To everyone mentioning Mythbusters, Mythbusters have successfully proven:

    1) You can polish a turd (they polished turds)
    2) You can polish a turd (they get paid to produce that program)

    They’ve also provided some evidence suggesting that camera guys like swinging around the room in big circles, riding on the camera, facing inwards.

    That’s it. That’s all.

    They don’t do science.

    To Adam K: civilian aircraft, maybe. Modern fighters apply the brakes, open the throttles, wait for the engine thrust to overcome the brakes and then release them before thrust’s finished increasing. You couldn’t do either with a Spitfire, though, or the engine would pull it over onto its nose. Even with the brakes off, they took off at 30% thrust because 40% would cause a nose-plant.


  7. This isn’t really a physics paradox at all. The answer is no. A planes lift is created by the air flow around the wings. The engines may be incredibly powerful, but they are only creating forward momentum, which if the treadmill has the same momentum as the plane but in the opposite direction, then the total horizontal momentum is in fact zero, so the plane is effectively stationary. Therefore, there is no airflow around the wings to generate the required lift for it to take off.


  8. Ok, after some further research, I have realised that I was wrong. I was correct in that it wasn’t a paradox, but the answer is yes. The way I was thinking about it was that although I realised the wheels did not power the plane, I was thinking that there was an opposing force that was equal and opposite to the thrust, but this is not the case. The plane will still move forward as usual, the only difference is that the rotational speed of the wheels will be near enough doubled.
    A great analogy is that if you were to stand on a skateboard on a treadmill and hold onto a peace of rope at the end of the treadmill, at any point, regardless of how fast the treadmill is going, you will be able to pull yourself forward with ease. The same applies to the plane, the only difference being that the plane is pushing itself along, not pulling.


  9. It is a badly defined problem.
    There are so many elements that are not defined.

    i) Friction in the wheels/axles?
    ii) Friction between air and treadmill?
    iii) How does the treadmill counteract the engines force?
    iv) Does the treadmill effect the environment other than the plane?

    The most important thing is that an airplane takes off when the velocity difference between it and the air is large enough to generate lift. Rotary wing aircraft do this by spinning their wings, the rest of the aircraft shape and forces just don’t matter at all. Fixed wing aircraft do this by moving air through their engines, adding momentum to it (by expanding the gas using heat). The scheme works pretty well because it is relative to the airspeed, therefore a sudden tail-wind will not affect the engine output or the plane’s lift.

    Back to our problem, my answers are:

    1) No friction: There is no air friction, the treadmill is ideal, and the wheels are ideal (non slipping, but frictionless axles): No the airplane will move, draw the free body diagram: the engines produce a thrust vector. The treadmill does not produce any thrust vector because the axles are frictionless. There is nothing stopping the plane from moving through the air and generating lift.

    2) The plane will fly if there is any friction in the system (wheel is non-slipping, but axles have friction (or else the treadmill cannot put any force on the plane), the treadmill is ideal (can match the momentum of the engines). If the treadmill is in contact with air, then the friction will cause a layer of air to move. As the speed of the treadmill increases to drive the wheels back, it will eventually move enough air over the wings to lift off the plane. When the wheel disengages from treadmill, then there will be no force vector stopping the plane.

    Lets tie the plane down, therefore it won’t fly:
    3) Any other variation on this would just be crippling the system, and taking it further from any sense of reality. Basically, defining the system removing components that would not normally make an airplane fly in the first place:
    a) remove air – Planes don’t fly in a vacuum. Air breathing engines don’t work on a vacuum.
    b) remove air friction – treadmill does not affect anything else in the environment but puts enough force on the wheels to counteract the force from the engine. (No the plane will not fly because it is virtually staked to the ground).
    c) pretend the wheels are real, treadmill is ideal – wheels will explode as the treadmill starts putting a million pounds of “counter”-force on it.
    d) other things that don’t matter.

    The obvious answer is YES.
    The ideal answer is: this is stupid and I wasted too much time on it already.


  10. I just thought it might be interesting to point out that in China, the Doomsday Argument entry on Wiki is blocked, but the Airplane on a conveyor belt, Feynman sprinkler, pop pop boat, 2 envelopes (and, of course, XKCD) etc, problems are quite accessible…. something about Doomsday has scared the Chinese government!!!


  11. Michael, your skateboard analogy was able to end the discussion I was having with my boyfriend over this problem (somehow, we ended up just like the two groups at the beginning of the problem – condescendingly explaining basic physics and math to each other).
    Thanks, that was awesome!


  12. So, I completely agree with and accept as obvious your assessments in scenarios 2 and 3, and I agree with your assessment of scenario 1 based upon what you’ve defined the parameters of the question to mean. But I think you’ve overlooked a slightly different case, 1B, if you will, which is the one my mind jumped to upon reading this problem. In this case, the speed of the treadmill belt always matches that of the bottom of the wheel, as stated in your scenario, but I’d say the salient fact about the problem description now becomes that the “treadmill is designed to exactly MATCH the speed of the wheels,” which I take to mean that there is a motor in the treadmill that actively accelerates the conveyor belt to match the speed of the bottom of the wheel. What this means is that, while the plane’s engines provide a torque to accelerate the wheel, they always do so with absolutely no resistance (other than the wheel’s intrinsic linear inertia and angular moment of inertia). In this case, the wheel isn’t “pushing” the treadmill via friction to accelerate the belt and make it match the wheel’s speed. Instead, the motor in the treadmill actively “matches” the torque (in terms of the angular acceleration generated by that torque, that is) applied to the wheels by the plane’s engine and applies it to the conveyor belt independently. Thus, the torque applied to the wheels never succeeds in making the bottom of the wheel try accelerate relative to to treadmill, there is no relative motion between them, and there is no friction present. Since friction is ultimately the only external force implied by the problem that would provide the plane with linear velocity, the plane can now not accelerate relative to the air, cannot generate lift, and cannot take off.


  13. Actually, I just don’t know how airplanes work. I was assuming that the only external for on the airplane could be friction, but I didn’t realize its linear velocity is actually generated by the creation of thrust and just a result of conservation of momentum. Everything I said is irrelevant, then. Disregard. That’s what I get for trying to do real-world problems when I don’t know the real-world aspect, only the theory.


  14. Based on the wording of the problem, the wheels of the plane would never turn.

    The treadmill would perfectly match the speed of the wheels and the first (and only) direction the wheels would move is forward. Thus, the treadmill will simply move with the plane until the plane lifts off of it.

    If we imagine a plane with pontoons instead of wheels then my assertion may be easier to picture. As the engines move air backwards, the velocity of the pontoons is always in the forward direction. This would be no different with wheels.


  15. For the record, your reference for the lawn-sprinkler physics problem actually says that if the sprinkler has low enough friction in its bearings and the liquid has low enough viscosity, then the sprinkler will turn backwards. That makes sense to me when I think how the pressures on the insides of each sprinkler arm add up, with an absence of pressure at the nozzle opening.


  16. This one’s a doozy. Since my recollection of physics formulae is pretty fuzzy, I’m going to stick with basic ideas of force.

    My understanding of the problem is that the treadmill spins at the same velocity (but opposite) the plane, resting on the treadmill by its wheels. The net effect of the plane’s engines and the treadmill’s spin is that the plane does not move forward (or backward).

    As mentioned in the article, a jet’s engine produces a ridiculous amount of thrust. The treadmill would have to spin fast enough to counter this. Basically, if the plane is staying in place, it means there is an opposite and equal force matching its engines’ acceleration. (If I’m not mistaken, this opposing force takes the form of friction on its tires conveyed by the treadmill going backward relative to the plane.)

    I’m not even going to do the math on friction trying to match a quarter of a million pounds of thrust. We could assume that neither the tires nor the surface of the treadmill burn away from this friction, and in that case we should throw in that the motor and mechanical parts of the treadmill are somehow up to the strain. What this leaves us is air friction.

    Seems to me that a treadmill spinning that fast is gonna get the air moving. If we assume that the treadmill is as it is in the diagram at the top of the page, it’s pretty big, a bit longer than the plane and at least as wide as its wheel base. That’s a pretty good amount of surface area to start moving the air. True, the air above the treadmill will never move quite as fast as the treadmill itself, but it may be enough to lift the wheels off the treadmill.

    Needless to say, once it leaves the treadmill, all bets are off. I imagine there’ll be a bit of a forward lurch once it no longer has to fight a major opposing force, but airplanes are pretty hardy, so it’ll probably be okay. Even if not, it won’t have far to fall. (Just hope it doesn’t land back on the treadmill!)

    I may be completely wrong. It’s been years since I’ve really exerted myself over a physics idea, and I’d love to hear other people’s take on this idea. But this is what my little brain thinks will happen in this scenario.


  17. I love how the thrust of this post is: “Here is a problem, let’s not argue about it anymore, because it’s been done to death.” And the comments section is filled with people arguing about it. hehehe


  18. Wow, how ironic that people started arguing about this after your “What If?” article, and especially after those people most likely read the bit about this being a banned topic in your forums. 🙂 I think that’s really funny.

    To the person who asserted that the Mythbusters don’t do science: Have you tried any of their experiments for yourself? Have you applied the Scientific Method to their theories? No? That’s what I thought.


  19. Okay, because I’m right and everyone else is wrong (;)), here’s my take on this problem:

    The wheels on an airplane are designed to spin freely unless a brake is applied – there’s no motor attached to them, no transmission, nothing other than the intrinsic friction in the bearings and the interface between the rubber and the ground. So if a treadmill is placed under the plane and starts moving in the backward direction, most of the energy exerted on the wheels will go into making them spin, and only a very small amount of that energy will exert a backward force on the plane as a whole. This is a simple dynamics problem.

    Further, it’s likely that the treadmill is causing a small amount of air movement in the backward direction – not very much, but it would likely be measurable. The treadmill would need to be moving much faster than the airplane’s equivalent takeoff speed to generate enough wind up at the wing level to make any sort of significant difference. Because of the wing’s design, if any of that air flows over the wing, it would generate a small amount of lift.

    In any event, the treadmill would mostly just be causing the wheels to spin fast and the plane to slowly move backward if the engines are turned off. With them on, the engines are going to generate far more thrust against the air behind them than the treadmill is exerting against the tires. The plane will accelerate forward, the wheels will spin even faster, and the plane will eventually take off just as normal. It might need a negligible amount of extra thrust due to the wheel friction, but that could also be counteracted by the treadmill’s own wind generation.

    The only likely side effects of the treadmill in this scenario are that the wheel bearings could overheat and burn out due to being spun outside their normal tolerances, and that the plane could lose control if the wheels were to break before it lifted off. (Not to mention the danger posed by damaged wheels when the plane comes in for a landing.)


  20. First, I’m probably not qualified to discuss this. But here’s how it seems to me, and I would love if someone explained how this is wrong, if it is:

    Even if the wheels have no force applied to them at all, the plane could theoretically “taxi” with nothing but the engines if it was put into neutral. The wheels would spin freely and move with the treadmill. The plane moves forward, with its wheels moving twice as fast as normal taxi conditions but with the rest of the plane moving normally, and it takes off. Voilà.
    As Randall says in the post, “no matter what’s happening to the treadmill and wheels, the plane is going to move forward and take off.”


  21. @Timothy: Well, you’ve basically got it, with one small difference: There’s no such thing as “neutral” in an airplane. The wheels are either spinning freely or have the brakes applied.


  22. Yesterday I went shopping, and as I had one too many bags, one of them fell apart before I could jump on the bus. A nice guy helped me pick up my stuff and chatted with me while we waited for the next bus.
    thanks, Random guy!


  23. In the non-frictionless axle case, you also need to take the potential torque about the y axis into account, which would tend towards pitching the nose forward.


  24. Who said the treadmill couldn’t move? If it moved at the same speed as the plane, the wheels wouldn’t spin and therefore make this point completely irrelevant because it wouldn’t affect the plane’s speed at all. Well, as long as the treadmill was moving forward at the same speed as the plane, it’s belt and the wheels could be going at any speed and it would make no difference.


  25. When the RAF test engines for jets they don’t put the planes on a treadmill, they tie the planes down. Ok, that’s because they are sometimes testing Harriers and those are VTOL aircraft but the treadmill argument almost certainly ends with the plane wheels / axles coming apart unless the minimum takeoff speed is less than half the design speed the wheels can go.


  26. Option 4: lock down the brakes, open up the engines, and let’s get this baby off the ground. The treadmill will provide the bearings that allow the plane to move forward and take off.


  27. If we assume the treadmill and wheels can turn infinitely fast and survive, we can simply say that engine trust will move the whole atmosphere (and the exhaust gasses change its composition) if it has enough fuel and time, changing gravitational center – which means that the treadmill MUST BE moving with the plane (relatively) !


  28. This is one of those paradoxes that only exists because it can’t possibly be real. If you tried this experiment in real life, everything would break: the treadmill, the tyres, the wheel struts… the only real argument would be which failed first. There’s a reason these things are called thought experiments.

    But there is still a problem with this thought experiment: in order for it to work at all, even as a thought experiment, we have to posit conditions that fundamentally change the experiment. Each time we make a condition to allow a calculation to be made, it changes the outcome.

    For example:

    Problem 1: In real life, the plane would take off because no treadmill could respond quickly enough to counter the sudden forward thrust of the engines. It would just lengthen take-off time (much as posited in the article above).

    Solution 1: Let’s assume the treadmill’s reaction time is instantaneous.

    Problem 2: With an instantaneous, reactive treadmill, the wheels will accelerate to the point where they disintegrate and the plane will collapse onto the treadmill.

    Solution 2: The wheels are indestructible, smart-ass.

    Problem 3: The heat generated by the friction inside the wheels will melt the axles and the indestructible wheels will detach from the plane. Same result.

    Solution 3: Ok, fine! The wheels are magically indestructible and frictionless! They can turn as fast as they like.

    Problem 4: If the wheels are frictionless and indestructible, they have no maximum speed and are not relevant to the movement of the plane. In other words, they may as well not exist: the plane is to all intents and purposes sitting on a mag-lev rail, which means it takes off.

    Solution 4: Ok, so the rubber on the wheels isn’t frictionless – it still makes contact with the surface of the treadmill, so no matter how fast the plane tries to move forward, the treadmill will move the plane an equal speed in the opposite direction.

    Problem 5: In which case, the tyres will skid forward, leaving a huge rubber mark down the treadmill. We have now transformed the question into another one: Are the engines on a 747 powerful enough to move it forward with its indestructible brakes locked on and its indestructible wheels sliding along the runway without rotating? And the answer is, I suspect, that although there will be forward movement, it won’t be anywhere near enough to achieve minimum lift, so no, the 747 cannot take off. But then this isn’t a paradox any more, it’s a straight force equation: the combined thrust of the engines of a 747 vs the friction of the sliding rubber of its tyres (assuming everything is still indestructible, of course).

    David nearly got this right a couple of comments above, except he forgot that if you lock the brakes, the wheels won’t turn and, therefore, the treadmill won’t move forward because the rules say it has to match the speed of the wheels (i.e. zero). In his example, the same thing happens as I have posited above: the wheels are forced to slide against the treadmill as the plane moves forward, and it won’t take off.

    Now, if the plane had infinite thrust…


  29. It seems to me that regardless of whatever mechanism was implemented to ensure that the plane did not move forward, if the engines were somehow allowed to pull air through themselves at a speed exceeding a few hundred miles an hour, it seems likely that without bolting that plane down in some fashion, it’s going to move in some other direction than in a neat and tidy path along the treadmill.

    So the plane wouldn’t fly, but it would most likely flip off of the treadmill and explode, or some such.


  30. I went to the dentist today for the first time in 14 years, because I thought my wisdom teeth were rotting. Turns out, the black I was seeing on my teeth was actually my tarnished fillings in the molars right in front of my wisdom teeth. Not only are my wisdom teeth cavity free but so is my entire mouth! Woot!


  31. Randall, initially I hated you for leaving this open for comments. Then I realised that everyone posting here was not posting elsewhere on the web. Thank you for making the rest of the internet fractionally, if temporarily, more intelligent.


  32. I’ve seen Mythbusters solve this one to a degree, it involved a truck pulling a conveyor belt opposite of the plane’s movement, and a Cessna airplane.
    the result was that the Cessna required a much shorter run up before it took off.
    the two of them concluded that it was due to the movement of air over the wings from the propellor that helped create lift in such a short distance which is a fair observation since the propellor created enough vacuum to move the plane forward, and thus also enough air over the wings.

    the scene with the treadmill can be seen in another light, imagine one would remove the treadmill and instead graple the back of the plane against the radio tower (or anything strong). what inevitably will happen is the engines would either
    1: tear themselves and/or the wings off the plane
    2: the plane will lift off the ground and swing like a fly on a string around its anchor point. this is not going to be pleasent for any passenger.

    my conclusion is that the plane will take off, regardless of what the conveyor belt is doing on the wheels.


  33. My solution to the two envelopes paradox is “They both have money in them? And I can take either one, and keep the money?”

    And then I pick one envelope at random and walk away with your money.


  34. I find this a very interesting problem, as the definition of the relationship between the wheel speed and the conveyor belt speed causes a kind of binding situation. I think.
    Here’s what I mean:
    Given the conveyor belt must *always* match the speed of the wheels in the opposite direction, the plane could not move forward, as this would require an increase in the wheel speed relative to the conveyor belt. To counter-act this increase, the conveyor belt would speed up instantaneously and the relationship between wheel speed and conveyor belt speed would remain equal.
    So the problem, as I see it is this: how do you create forward movement with the thrust of the engines, and yet not violate the equilibrium relationship between the belt and the wheels?
    For this problem, I have two answers.
    1) put the conveyor belt on wheels, and move the whole system with the plane until it takes off in due course. I vote we call this the “Trolley Launch”
    2) use a helicopter.


  35. Unless you bolt the plane down to the ground with something solid, taking off isn’t much of a problem. A 747 will even happily take off *without* wheels, they’re just there to keep the thing on the ground while being able to move freely.

    If the runway was made out of ice (#1, an endless frictionless treadmill) and the jet was on skids, taking off is actually quite easy.

    Landing, that’s where it gets really interesting. Unless you’re a sucker for touch-and-go.


  36. This honestly reminds me of a lot of those math problems you find in fourth and fifth grade text books. You’re given a couple of fairly simple and stated fact. (in this case, Plane’s weight equals, coefficient of drag equals, and thrust equals) and literally everything you need to solve the problem is right there, then they throw in Nancy and her five apples, and that’s where people get confused. (true story when trying to help my mother help her nephew solve a math problem about george and betty’s apples, she kept asking about nancy and finally I said “who the hell is nancy? she is irrelevant to the problem!”)

    you can replicate this experiment with a real treadmill and an RC plane (drone as we now ominously call them) and see that, yes, in fact, the craft will get airborne and the wheels and the ground/treadmill have fark-all to do with whether or not that will happen. (unless they are causing drag against the whole thrust/drag argument)

    Meanwhile, I want to be in that fifth group and see how many martinis will be served while the ground crew sets up the treadmill, because kari is hawt and I want to see how long I can watch that while drinking nice drinks and sitting in comfy chairs.


  37. I’m into the PID controller idea: the belt doesn’t match the speed of the wheels exactly, (for that I agree with theory 1,) what happens is that the belt accelerates when the plane is in front of the starting location, and decelerates when the plane is behind the starting location. Once the engines get the plane away from the starting location, the conveyer belt rapidly accelerates to very high speeds. This doesn’t change much as far the plane’s position, but the friction in the tires cause them to heat up and eventually bad things happen to the tires/wheels and they rapidly end up behind the plane. At this point the conveyer belt is basically just a giant belt sander, which grinds away any part of the plane that happens to come in contact with it. Assuming a big enough motor, (and strong enough belt,) the entire landing gear is ground away and then the belt starts grinding away at the lower parts of the airplane, which includes the engines on a 747, (an A-10 would have a lot more time.) There is of course some friction between the plane and the belt and at some point the turbines are reduced to fractions of themselves which causes thrust to go down below that amount of the force from that friction, at which point the plane starts moving backwards and the belt slows down, (possibly with some dramatic oscillation depending on the PID controller/momentum of the belt&strength of motor) and the plane ends up at the starting location again and stops.

    Of course, if the belt takes a while to accelerate and can’t do better than 500 mph or so, the plane is probably going to take off before before the belt grinds up the engines, (unless of course the belt breaks in the process and the plane gets hit with large portions of flying belt…)


  38. I think the best thing about this article is the comments. The whole and entire point of the article is that there is no solution the problem because THE PROBLEM MAKES NO SENSE.

    And now we have almost a THOUSAND comments now from people offering their own misguided pet theories as to why or why not the plane will ever take off.

    Priceless. I love you, Internet.


  39. A simple way to think about it is given in Randall’s point (1): the treadmill may be simulated by not attaching a motor to the treadmill and by further assuming that the treadmill’s bearings are perfectly frictionless.

    As the engine thrust increases, (assuming there will be no slippage between the wheel and the treadmill) the wheel rotational velocity will remain zero and the treadmill will cycle in reverse because the airplane’s wheels are experiencing a translational force, not a rotational one. The system of forces is not intuitive: the thrust acts as an external force either pulling or pushing (depending on how you prefer to visualize it) the airplane’s “system” in relation to its position in the atmosphere, independent of the treadmill.

    Even if we do implement a negative velocity to the surface of the treadmill — although the wheels will begin to spin — the airplane will continue to “surf” forward at an unchanged rate.


  40. When met with a difficult problem, I find the best method to figure out a solution is to step back and find ways to make the problem simpler. It turns out, this is a very simple problem to solve.

    If I were silly, I would suggest simply replacing the tires on the plane; the old tires can be left on the ground, and since they aren’t spinning, the treadmill won’t move. The plane takes off normally. Alternately, place the old wheels on an axle, and as the plane takes off, crank the wheels to create a treadmill-assisted takeoff.

    However, that isn’t the easiest solution. Instead, simply define positive velocity of the wheels in the opposite direction of the plane. As the plane takes off, the wheels spin (with a negative velocity), and the treadmill rotates the opposite direction (the same direction the plane is moving). To maintain equilibrium, the treadmill and the wheels spin at half the speed of the plane.

    Before anyone says, “Hey, you can’t do that!” I would like to point out that the problem does not state the reference frame; and, even if it did, it would be simple enough to swap the wheels on the left and right side of the plane, which gives the same result.


  41. It’s amazing that even after explaining the answer, and clearly describing the basic reason why people disagree about the outcome, and asking people not to debate the question, you still see people debating the question.

    And perhaps not as amazingly, Randall was right. Most of the explanations here match up exactly with the different understandings of the treadmill that he described.

    If you still don’t understand why the plane will take off, there are plenty of thought experiments in these comments to help elucidate it further. I think my favorite so far is the one that says “imagine the plane is on skis instead of wheels”.


  42. So first of all I would like to thank the writer for giving me a problem which allowed me to spend an hour and a half of my AP physics class watching people argue about an unsolvable problem.

    Personally, I agree that the problem is stupid and has no solution.

    Statement A: if the treadmill moves in the opposite direction of the wheels in the opposite direction, the wheels cannot move in reference to the treadmill.

    Statement B: the engines generate an enormous force in the opposite direction of the plane, therefor, due to the law of equal and opposite reaction, the plane has an equally enormous force pushing it forwards.

    Statement C: in a free body diagram of a frictionless system, there is no force opposing the force of the engines

    Combining these statements shows that it is impossible for the plane to move in relation to the treadmill; however, there is no force stopping it from doing so.
    These statements are contradictory and therefor the problem is void.

    But just for fun.
    If you apply the law of conservation of energy, you will see that A: the kinetic energy of the plane cannot change. B: the potential energy of the plane is not changing. C: the engines input roughly 140 mega-joules of energy into the system every second.

    According to this analysis, the only source to absorb all of this incoming energy is the wheels.

    I’m gonna do some math here based off of basic info I have found online. I have no idea how accurate information is because if came from answers without citations but here is what I found: a 747 has 18 tires, the tires have a mass of 110 kg each, the tires have a radius of .5 meter.

    Using this information, we can determine the rotational speed of the tires in terms of time.
    *(P) is power (140,000,000 watts) = Energy input(j) x time(s)
    *(J) is be energy input
    *(m) is the mass of one tire
    *(r) so the tire’s radius
    *(w) is rotational speed (radians/s)
    *(KEr) is rotational kinetic energy of one tire
    *(I) is the moment of inertia of the tires = (1/2)(mr^2) (I am assuming the tires are 2 dimensional disks of even mass distribution)
    Energy input= 18KEr

    P =18(1/2)Iw^2
    P = (9/2) (mr^2) (w^2)
    (2/9)(J)(t) / (mr^2)) = (w^2)
    Sqrt[(2JT) / (9mr^2)] = w

    Plugging in t=60 (1 minute) gives us a w of 8239 (radians/second)
    Dividing this by 2pi gives us 1311 (rotations/second) or 78,676 rpm.
    Plugging in t=600 (10 minutes) give an answer of
    A pretty standard cross ocean flight time for a 747 is 10 hours so let’s try that.
    Plugging in t=36000 (ten hours) gives us a w of 201801 (radians/second). Dividing by 2pi gives us 32.118 (rotations/second) or 1,927,058 rpm

    Just for more fun.

    If the treadmill suddenly stopped after running the thrusters for 10 hours, all 18 wheels would break free from the plane. If they transfer all rotational kinetic energy to linear kinetic energy they would move across a flat frictionless surface at 71,351 meters per second (Mach 208).
    KE = 36000P/18
    (1/2)mv^2= 2000P
    V= sqrt[4000P/m]
    If it hit a ramp which redirected it vertically upwards, it would be able to escape earth’s gravity well and still have a speed of 60150 meters per second and a kinetic energy of 199 gigajoules (over four times the energy yield of the MOAB).

    The lesson to learn from this is simple: if aliens ever invade, forget about missiles. All we need is boeing’s fleet, a few treadmills, and some well placed ramps.


  43. my first thought was “dear gods and goddeses, ~where~ can i buy a treadmill that SIZE, please?!?!”

    and then the Heinlein story “The Roads Must Roll” came to mind, folowed closely by the whole mental image of people flying off the terminus points like water droplets from the tip of a cracked whip that always went through my mind when i’d read it.

    who needs airplanes, anyhow, hmmm?


  44. Ok, here comes THE solution. At least sort of.

    The treadmill is large enough to influence the air at that area, so the faster the (stationary) plane is going, the more wind we build up. Once there is enough wind relative to the stationary plane, it can take off that silly runway and navigate to an airspace with less wind and it’s gone.


  45. One can easily recreate the experiment with a model plane (or a toy car) on a gym threadmill. Pushing the plane with your hand is a good substitute for the push given by a jet engine. One can clearly see how, in absence of limitations on the wheels rotation speed, the threadmill can NEVER prevent yoy from moving the toy plane forward, no matter how fast it can go. If its speed is so high that air starts getting dragged backwards, it could even help the plane take off earlier!


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