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 thoughts on “The Goddamn Airplane on the Goddamn Treadmill

  1. NERD SNIPE! the only reasonable way of interpreting this question, which is notably unreasonable, is that, given no slip, the runway accelerates with the plane. which means the plane doesnt move, or the runway does. I prefer to think of the latter, which would look something like a very large aircraft carrier flying through the sky.


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  3. 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


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  6. Hehe. I brought this up in Theology class at my school, right in first period, by supposing it to a student. Some overheard and started arguing too. Eventually the class split in two and the teacher joined our team, and we spent the first ten minutes of class debating it, with no clear resolution….

    Of course, it appeared in second period, too, and then third and fourth and so on, and everyone was talking about it in lunch, too, and no one even came to a consensus until the NEXT day.


  7. How about the viscous friction in the axis of the wheel? It is proportional to rotation speed and able to create a force which increases with increasing speed. This friction is able to fully compensate the thrust of aircraft engines.

    And of course we must take into account the friction in this problem – otherwise the wheels will not spin, but will slide.


  8. The wheels of a plane only apply torque on the plane’s landing gear (and by extension the plane) via sliding friction between the wheel and its axel (assuming the parking brake is off). Sliding friction is not dependent on speed so the magnitude of the force counteracting the planes take-off would be the same as if it were on the ground. The pilot wouldn’t even notice that conveyor belt was even there unless he looked out the window.

    Also the wheel wouldn’t move infinitely fast, it would have an angular velocity of (2Vw)/r. Twice the wheel/plane velocity (because the difference in speed between the conveyor and the plane is 2 times the velocity) divided by the radius of the wheel (because thats how you get rotational velocity).


  9. sorry for double posting but i read the comment 2 up from me and i must defeat this troll

    I think force of friction of a wheel on its axel still follows the Fnμk formula (force of kinetic friction times force normal) but i could be wrong. I don’t see speed there though.


  10. *RAWR triple post thats supposed to be coefficient of kinetic friction times force normal


  11. We were able to stop arguments over this by requiring anyone who tried to discuss the “takeoff” problem to first answer the question of “what happens when the same airplane lands on the treadmill?” This brings to the forefront the question of “what is the treadmill actually doing?”


  12. Its funny how simple this really is. The plane engines push on the air, they do not power the wheels. The plane will move forward and take off regardless of what the wheels or the tredmill does.

    Myth Busters solved this one already.


  13. Forget the treadmill. A plane preparing to take off in a westerly direction on an east/west runway is already on a treadmill going 1,000 miles an hour against the intended take off direction. Does anyone think an airplane of any type cannot take off to the west?


  14. What this analysis fails to take into account is that a constantly accelerating wheel with non-zero moment of inertia will produce a constant torque acting as a brake on the plane even with frictionless wheels. So since we are working in an ideal world where plane sized runways exist, if the wheels have mass and are frictionless and operate in a vacuum and there are no relativistic effects, the treadmill feedback loop described by interpretation #3 could continuously accelerate the wheels and hold back the airplane.


  15. Plane will not launch because it needs to be pushed by air, which will not happen beause plane is not moving against the air. It doesn’t matter how fast are the wheels rotating.


  16. I just found this site, so admitedly this is a pretty late response.

    But just because there still seems to be no total consenus on the airplane problem, I’ll explain why it certainly does move with yet another example. Forget anything concerning angular momentum, V(c), V(b), and the lot. All that just confuses the issue.

    Consider this: Imagine that you had two mechanical winches (like the ones on the front of some pickup trucks.. only bigger). Now, imagine that those two winches were mounted onto some really strong object in front of the plane, right at wing level, right in front of each wing of the airplane. Next, extend the cables from the winches and hook them onto both wings of the plane, and then turn on the winches…

    Even if there is a treadmill under the plane running backwards at any speed, by God, our plane is still going to be pulled forward by the cables that are attached to the wings. This is NOT different from just turning on the engine of the plane. It simply doesn’t matter what the wheels are doing — the forward force is coming from the engines mounted on the wings, and the engines produce forward momentum regardless of what’s happening down below.


  17. Thinking about the problem this is equivalent to a plane against a frictionless wall – can the plane take off?

    I reckon yes providing the thrusters provide some vertical lift – otherwise no. (As the plane moving is required for lift)


  18. The treadmill is cancelling out any horizontal component caused by the thrusters – hence why I think only vertical components should be considered.


  19. Vc = Vb …condition for pure rolling,,no skidding (so clearly #1 is not incorrect)

    Vb = Vw + Omega(circular velocity of the wheel) * Radious of the wheel

    Energy dissipated by the engine = 1/2*m*Vw^2 + 1/2 I(moment of inertia of wheel)*Omega of wheel^2

    Vw is the horizontal component wrt ground observer.

    and yes the Vw becomes significant wrt the ground observer. Vw will lead to upthrust required to ply the plane


  20. Settle down everyone, I got it all figured out. The solution lies in the first word of the the problem discription; “Imagine”. I can imagine such a plane taking off just fine, even if the the wheels and the conveyor belt can’t have any nonzero speed and the universe borks. Everybody ‘s happy and look at it go!


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  22. It is simple.. replace the wheels for big Ice cubes and repeat the experiment..
    What about now?


  23. Ummm…..Why is everyone assuming that the treadmill rolls backwards?
    As the wheels are attached to the plane then they must move at the same speed as the plane so the phrase “The conveyor belt is designed to exactly match the speed of the wheels” could be interpreted as “The conveyor belt is designed to exactly match the speed of the plane”.
    Imagine square wheels and you get the idea…
    Similarly, if the wheels are round and the speed of the treadmill matches the speed of the plane then (like the square wheel) the rotational speed of the wheel is zero.
    The only thing that makes sense really…


  24. Thank you, thank you, thank you for finally publishing somewhere on the interweb clear descriptions of the two different things that people are arguing when they think they are arguing the same thing (ie. #2 and #3). I have to admit that even in the question of the definition I find myself choosing sides pretty strongly… even, bear with me, though it was purposely intended to be ambiguous. Basically what I’m saying is that if you take the question at face value, as a question instead of a means to provoke an argument, why on earth would the asker say “speed of the wheels” if they were talking about the forward motion of the airplane? To me those who start answering according to interpretation #2 have already made a linguistic error. If we’re talking about the wheels, we must be talking about some property of the wheels that the other parts of the plane don’t have, namely, that the wheels are spinning. And I would say that it is by far most meaningful to interpret this as the speed measured be the wheels.

    Assuming #3, wouldn’t this be a great “What If”? The jet engines thrust, and for just a fraction of a second the 747 moves forward *without* turning the axles. (Let’s assume that the wheel speed is measure via the axle, since this is the last part that you would expect to be destroyed in the process.) The rubber would move against the pavement, but the force will not yet have been imparted through the rims and into the axle. This gives us some leeway, because as the wheels accelerate, the treadmill lags behind a teensy bit. Next the axle starts turning, (in a microsecond?) and the treadmill matches. Now the acceleration of the jet itself becomes secondary, because it is already moving forward and the treadmill is caught in a positive feedback loop with the only delay being how fast it takes the force on the wheel to affect the axle. Within some small period of time, which you will amazingly narrow down, the runway will be moving several hundred miles per hour. At some speed (once again, you come in here) the wheels will begin to rip apart. But how? And how will this affect the rotation of the axle? Will the leading edge of the tire hitting the asphalt crack? As the wheel falls apart, this would presumably dampen the affect of the runway on the axle speed, which would make the feedback delay longer, and at some point the runway might have no affect or a reverse affect on the axle (perhaps it pushes bits of the tire in the way or spins them in some tiny rubber tornado). I would expect that at some point the treadmill spends a relatively very large amount of time (1 second?) totally disconnected from the axle and the plane will be in near free fall before the inner structure of the wheel (probably the rim) contacts the pavement.

    Then the fun begins. Now the wheel is in position to possibly rip into the runway, or does it? What shape is the rim? Assume the runway holds firm. Now we have very little traction between the runway and the wheel, and the jet has had half a second to accelerate while the wheel, just from inertia, has been spinning at a constant speed with the treadmill matching that speed. The bottom of the rim is now moving forward relative to the runway, which means the inertia of the wheel might be just high enough to allow it to slip a bit instead of trying to sync up with the runway surface. BUT the friction isn’t negligible because we have an infinitely powerful instantly responding treadmill. The feedback loop is now MUCH faster. When the edge of the rim accelerates the axle feels it within… attoseconds? Nanoseconds? The runway reacts so quickly that the inertia of the wheel can’t possibly change fast enough, and…what? The wheel and the runway follow this delicate balance where the wheel has almost stopped slipping against the runway for many seconds as the plane takes off? But what speed was the runway moving when it contacted the wheel? 1000 miles per hour? 2000? Does the rim heat up from vibration as it runs along the runway and melt or break off?

    Finally, perhaps, the entire wheel breaks apart and the axle gets ground into the runway and stops spinning. The runway would then act as a normal runway, with a 747 now attempting to take off with no wheels. Does it collapse the landing gear, glide across the runway on its engines and take off? Does it rip apart before it gets off the ground? Is there finally enough friction to keep it from getting up to speed? Does it make it off the ground but with melted passengers and ruined upholstery?

    What happened when or if the runway exceeded the speed of sound? What kind of airflow would this produce during that part of the sequence of events?


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