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.

815 thoughts on “The Goddamn Airplane on the Goddamn Treadmill

  1. @ the guys thinking prop plane will, jet won’t

    because “wind from prop” – lift generated directly from accelerated air from props is relatively small compared to the lift generated from forward motion. It would help achieve takeoff faster, but by itself, it is nowhere near enough for a plane to take off. At best, you can lift the tail.

  2. I hate to disagree, Patrick. But while you’re right about the wind eventually becoming a major issue (1), it wouldn’t result in the airplane ‘taking off’. Unless we’re assuming a treadmill hundreds of feet wide and half a mile long, the wind simply isn’t going to be consistent enough for the plane to take off, at all. It wouldn’t ‘stall’, really, it would never get to that point.

    However, at some point or another, a threadmill spinning at the absurd speed required would indeed create a gust of wind be strong enough to lift the plane off the ground enough that the super-duper magical wheel friction that everyone is pretending can happen will be lost. Probably on only one wheel. Thus resulting in one side of the airplane being suddenly allowed to move forward, rotating the plane and resulting in it hurtling down the threadmill end over end.

    Not that free-spinning wheels can possibly general enough friction to keep the plane in place, but that’s what we’re apparently assuming!

    1) Although as I point out, we’ve got 250,000 pounds of thrust disappearing into wheel axle friction, which would cause rather serious heating problems well before the wind could do anything at all.

  3. The plane will take off, and in the problem the treadmill is irrelevant. The planes body (including the wheels) accelerate forward relative to the ground (not the treadmill) due to the conservation of momentum.

  4. i really enjoyed to visit this site.it is a very nice site and i book mark ur site.

    This is a nice content with lots of information. It’s a very excellent idea for raising money for charity. I think honest review is more important for authors. I like your whole discussion. Keep it up.
    Keep blogging.ur site is good.and finelly ur site going to in high position

    This is a really good read for me. Must agree that you are one of the coolest blogger I ever saw.
    Thank you for this very interesting infographic ! I’m talking about Google Venice on my blog.

  5. I know it’s late, I just found this. Daniel H. is right, but I don’t think he adequately explains why.

    The question is basically “Can a plane accelerate if the wheels have 0 traction.” This happens all the time, since the wheels on an airplane have no traction while the airplane is in flight, and airplanes accelerate in flight all the time. Slightly more scientifically, the engines (whether it’s a prop or a jet doesn’t matter) use the air around the airplane to create thrust and lift; the runway serves no purpose except to reduce friction from grass and weeds and rocks and small mammals. For a similar real world example, look at sea planes. In fact, the water creates more drag on the pontoons than a runway or a treadmill at any speed could produce on the airplane, and yet sea planes can still take off.

    Airplane engines create thrust, either by creating forward lift using a prop, or by creating a Newtonian action-reaction by the combustion of a fuel. In either case, whatever the plane is resting on has no affect on the plane’s ability to move forward, except where it creates drag (friction) against the plane’s wheels.

    The forward thrust of the engines causes the plane to move forward. The movement of the (relatively) motionless air passing over the lifting surface of the wings creates lift. This works using Bernoulli’s principal. Again, whatever the plane is resting on has no affect on the plane’s ability to create lift.

    An airplane’s wheels are free-spinning, except in some very expensive airplanes that have motors built in to spin the wheels up prior to landing – this reduces wear on the tires. Except for during landing (or braking), these expensive wheels are still free-spinning.

    Now to answer the question about the treadmill.
    1.1: The treadmill is free-moving. Friction between the wheels and the treadmill will cause them both to move forward in tandem as the plane’s engine pulls the plane forward. The plane would take off normally, except there would be no tire wear from rolling down a runway. If we assume that the treadmill runs on a frictionless track, the plane would actually take off faster, since friction between an ordinary plane’s wheels and the runway slows the plane’s acceleration.
    1.2: Vb = Vc. As the plane’s engine pulls it forward, the treadmill starts moving forward (as in 1.1), which causes the plane’s wheels to turn forwards (by some phantasmic unnatural force, or just a motor linked to a speedometer attached to the treadmill). The forward motion of the wheels against the treadmill increases the plane’s rate of acceleration and the plane hopefully takes off, leaving the treadmill behind, before this feedback loop causes it to accelerate to Warp Factor 10.

    2. Vc= Vw. The answer given is correct, the wheels will spin freely, twice as fast as the plane’s ground speed (Vb = 2*Vw), but the plane would take off the same as if it was on a normal runway. There would be a negligible amount of friction in the wheel’s bearings that might slow the plane’s acceleration slightly.

    3. The equation doesn’t match the premise.
    3.1: “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?” The speedometer is linked to the wheel, and not the plane’s air or ground speed indicators, and therefore it’s the same as 1.2: Vb = Vc.

    3.2: Vc = Vb + Vw. In this scenario, as the plane’s engine generates thrust and pulls the plane forward, Vw > 0, the treadmill would start moving backward at Vw. In this first infinitesimally small moment, Vb is still 0. In the very next infinitesimally small moment, the treadmill motion forces the wheels to start turning and Vb = Vw = Vc. Once we move past the infinitesimal, the stated answer is correct that the treadmill would very quickly ramp up to infinite speed. Your wheels would burn off and, if the plane didn’t manage to get airborne before then, the whole plane would crash on an infinitely speedy treadmill. But, depending on how long it took that treadmill to increase in speed, it’s possible the airplane would be able to take off, since, as the wheels are free spinning, the motion of the treadmill has a negligible affect on the plane’s forward motion.

  6. i really enjoyed to visit this site.it is a very nice site and i book mark ur site.

    This is a nice content with lots of information. It’s a very excellent idea for raising money for charity. I think honest review is more important for authors. I like your whole discussion. Keep it up.
    Keep blogging.ur site is good.and finelly ur site going to in high position

  7. Pingback: Ask The Pilot. - Page 616

  8. Here’s something to think about: equal and opposite reaction.
    747 engines pull in air from the front, which pulls the engine forward. The engines also push air out the back, which pushes the engine forward. These 2 forces, the pushing and pulling on the engine and the air cause the engine to move forward. The engine is anchored to the wing (where some of the landing gear is attached) which is affixed to the fuselage to which the other part of the landing gear is attached. The purpose of the landing gear is to separate the plane from the ground.
    From what I can see, there are 2 endings to this scenario, both of which include the plane moving forward
    A) The plane takes off as usual with the landing gear wheels spinning >=2x faster than usual due to the fact that the runway can move. Have a nice flight. I wonder what landing on a treadmill would be like?
    B) The plane accelerates forward. The wheels start to spin and the treadmill engages. The treadmill and wheels respond to each other by going faster and faster until the rotational speed of the wheels far exceeds what the landing gear is capable of. The landing gear would then break appart and the plane would turn into a fiery mess. Sorry group #5.

  9. it’s the speed of the AIR moving over and under the wings that creates lift. Not the speed of the wheels or engines. If the BODY of the plane is motionless, zero LIFT is created. Otherwise, all airports would be verticle and you would launch planes like the space shuttle saving a ton of money and space. The runways are needed not only for landing, but to allow enough room to accelerate the BODY speed of the aircraft to generate lift from air moving PAST the deferentially curved surfaces of the wing.

  10. This is not that hard to solve..
    first let us consider the following
    A. is the plane “indestructible”
    B. Real life Airplane

    If A. then the plane will not take off the ground/treadmill. As long as the total thrust capacity of the engines cannot lift the plane straight up.
    It does not matter that the plane can push forward .. it only takes off once air can push beneath the wings and since the only circulation of air that happens is inside the turbines then i’ts like having just a turbine on wheels that can generate no vertical lift run around a track.. straight line …whatever. .. plane takes off because of both engine and it’s wings.
    If it could take off on engines alone then no wings are needed :)

    If B. then the planes wheels will explode and the plane will be thrown by the threadmill towards the back of the runway into a fiery death.

  11. Omg… yes I think I get it now. Okay then, the plane can not take off. If the wheels are spinning due to the thrust provided by the engines but the treadmill is matching the rotation of the wheels then there’s no way that the wings will be able to supply enough lift to get the plane off the ground (treadmill). I agree wholeheartedly that the wheels do not drive the plane but a plane HAS to develop forward motion before enough lift can be generated BY the wings to support the plane and overcome the weight of the plane. If the treadmill is cancelling out the forward motion produced by the engines then there’s no way the plane can take off

  12. Really a great Insight! Thanks for sharing. The conveyer belt is designed to exactly match the speed of the wheels at any given time, moving in the opposite direction of rotation. The conveyer has complete control system that tracks the plane’s speed and tunes accordingly.

  13. I can’t tell if the blog is giving a definitive answer, but as a pilot, to me the answer is obvious. Taking off and landing are very similar. Simply suppose the plane is not taking off, but landing. It has an airspeed, and when that airspeed crosses the speed needed for flying, it either takes off or lands.
    Now, suppose somebody on the internet is putting a treadmill on the runway, and trying to make it match the speed of the wheels. It can’t be done, because the airplane is moving.
    So the statement of the problem is not physically possible.

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong>