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. I greatly enjoy reading many of your interpretations, including all the Looney-Toons-esque scenarios.

    I think I will join group #6, but I will forgo the alcohol if you give me a nice burrito.

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  2. First of all, I have to disagree with David Thomas, above. Planes’ movements are retarded by the wheels; that’s why chocks work.

    It seems to me the diagram is the problem. (I apologize in advance for not knowing how to do subscripts.) Vb = Vc only if “opposite direction” means that the wheel and the conveyor belt rotate in opposite directions– one going clockwise, the other counterclockwise. But that would make the question basically pointless. What I think the question is trying to suppose is that Vb = -Vc. In other words, the point of the tire that’s in contact with the treadmill wants to rotate from right to left at 40 m^s, and in synch with that, the point of the treadmill in contact with the tire is trying to move left to right at 40 m^s. The forces counteract each other, keeping the plane from moving. (If the tire can’t rotate, the landing gear can’t move forward, and if the landing gear can’t move forward, neither can the plane it’s attached to.) Vb and Vc are both zero, making Vw also zero.

    At least until the motor of the conveyor belt blows out and stops working, at which point the plane takes off normally, or the landing struts break and you have all sorts of interesting discussions about the friction coefficient of a fuselage, and the nightly news comes out to the airport… Well, no. The thing that’s going to fail first in that system is the traction between the tire and the treadmill. The plane will slide along like it’s on skis and take off to the smell of burning rubber.

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  3. I agree with Lizzie and I will join her in group #6, however I will not forgo the alcohol. In truth, Lizzie won’t either, she’s just being coquettish because we’re traveling. Come on you minx, let’s get to first class before the rabble get their dirt on us.

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  4. If we consider this from a force perspective: the force of static friction between the treadmill and the wheels would opposite that applied by the motor. However, if we then rotate the treadmill so that it would create a frictional force opposite to that, we would create a non-rotational wheel. Of course, this gives a few issues in that it would introduce slipping in any real situation, so if we assume that the maximum possible static friction between the wheels and the treadmill is infinity–note: obviously abandoning reality–then simply no motion occurs for either the treadmill or the plane. The motors quickly destroy themselves, and all is well.

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  5. If we consider this from a force perspective, assuming that static friction can go up to infinity, if we consider the treadmill rotating in the same direction as the plane motor applies its force, then the static friction between the wheels and the treadmill would be zero–and neither the wheels nor treadmill would rotate anymore. This would destroy the motors of both the plane and the treadmill and all would be well. Therefore, it’s possible to conceive of a set-up that would give a non-moving plane, but one couldn’t create it, I think, and it’s not the set-up proposed by Mr. Munroe.

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  6. Since planes do not get their propulsion from their wheels they do so from the engine the treadmill would have no effect. if you want to balance out the effect of the engines, place it in a wind tunnel, the wheels only contain brakes. The engines of a plane exert force against the air.

    MythBusters solved this years ago, even before this post was made, it was done in their first season, somewhere like 2003

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  7. Here’s a simple solution: set the speed of the goddamn treadmill to zero (it is just a runway). Replace the wheel and axles with blocks of lead such that as the goddamn airplane moves down the runway, they won’t spin, just skid.

    The terms of the problem are satisfied. The goddamn airplane takes off, possibly melting or breaking off its landing gear, not that they were functional to begin with.

    Argument over. Interpreting the question as ‘the treadmill somehow exerts so much backwards force on the plane’s wheels that the plane stays still’ is nonsensical.

    “What I think the question is trying to suppose is that Vb = -Vc. In other words, the point of the tire that’s in contact with the treadmill wants to rotate from right to left at 40 m^s, and in synch with that, the point of the treadmill in contact with the tire is trying to move left to right at 40 m^s. The forces counteract each other, keeping the plane from moving. (If the tire can’t rotate, the landing gear can’t move forward, and if the landing gear can’t move forward, neither can the plane it’s attached to.) Vb and Vc are both zero, making Vw also zero. ”

    No. If you stopped the airplane’s wheels from spinning ‘forward,’ then they would just spin ‘backwards’ while the plane continued to move forwards. You can simulate this with a toy car and a sheet of paper – set the car on the paper, push the car forward with your finger slowly. See the tires rotating forward? Now press the car down as you move it forward and quickly yank the paper out from underneath it, in the direction of the car’s motion. The wheels spin ‘backwards’ but the car continues moving forwards, because the wheels are free-spinning and simply reduce the friction of the toy car against the ground.

    Similarly, plane wheels have brakes but this is simply to INCREASE the friction. The brakes could not actually stop the plane if the engines were running at the same level they are running during takeoff.

    You also, apparently, didn’t realize that making the wheels spin in the wrong direction would mean the treadmill was moving FORWARD.

    And anyone actually reading this, take this comment as evidence that it’s not just condescension: some people REALLY DO BELIEVE that a plane cannot move forward if its wheels aren’t spinning the right way.

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  8. “The brakes could not actually stop the plane if the engines were running at the same level they are running during takeoff.”

    Not actually true… aircraft brakes are almost always capable of holding the plane at a standstill while the engines run at takeoff power. You don’t necessarily want to rely on it for long stretches of time, but often short runway ops call for pilots in big planes to set the parking brake, set the throttles for take off, wait for the turbines to spool all the way up (which can take four or five seconds), and only then release the brakes for the takeoff roll in order to get maximum thrust from as near to the beginning of the runway as possible.

    To the bigger question I’m of the mind that the plane will take off, because the vector of the axle over the ground is independent of the direction the tires are spinning. You can try this at home… get a wheel and axle of some variety, set it on that treadmill that’s been sitting in your basement since the late 80s. Turn the treadmill on and hold the axle in place. The tire spins. Now move the tire back and forth along the treadmill. The axle moves independently of the direction of the treadmill/tire combination, with the speed of the tire relative to the axle (but not relative to the treadmill) changing to compensate. Your hand is the force from the aircraft engine. It’s the same thing.

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  9. I’m afraid you may have a sample bias problem in you judgement regarding the lawn-sprinkler – I _highly_ doubt that the answer would be immediately obvious for most people in general: the immediate intuitive response would be to imagine the reverse on a normal sprinkler (spinning backward) which turns out to have nothing to do with reality on the first level (no net force on the sprinkler in an ideal system, even if it does rotate backwards because of the initial impulse) and only starts resembling it again if we grandfather non-zero viscosity back into the system (while in the real world the sprinkler won’t rotate much at all due to friction). I’d say most people would just assume some sort of anti-reactive force being applied and therefore get it plain wrong.

    Also, regarding the airplane problem: I’d rather treat it as an airplane with _skis_ instead of wheels – a non-crucial distinction making things simpler to imagine and discuss by removing irrelevant complexity – which I think we can agree is equivalent to the wheel situation in that it’s only relevant as an unpowered but either frictionless or non-frictionless interface to the treadmill.

    If there is no friction, the treadmill is clearly unable to exert _any_ force on the plane no matter how fast it tries to spin its wheels backwards / drag on the skis, so the plane merrily takes off propelled by its thrusters as if the treadmill weren’t even there (=plane on ice). If there is friction, it’s simple to argue that the treadmill _can_ exert _some_ force on the plane by dragging it backwards equal to the friction involved. If the trusters are weaker than that, the treadmill can hold the plane down by dragging it backwards until it falls off at the end (or just by staying still, for sufficient static friction). If the thrusters are stronger, it will simply take off anyway after speeding up sufficiently, regardless of how fast the treadmill is trying to drag it back – the friction force is not supposed to increase with speed (if it would, the treadmill could indeed keep the plane down, motionless relative to the ground, at _some_ speed with friction drag-back force equaling thrust – except Coulomb seems to disagree with that).

    I think the ambiguity is caused by not clearly explainig which of the above assumptions are made in an argument, considering the versions I’ve heard were never much concerned in postulating any magic ability to match some arbitrary speed on either the plane or the treadmill, simply trying to figure out what would an actual treadmill do to a plane under ideal / real conditions. I think once we let go of the arbitrary “speed clause” and just examine the possible scenarios it’s apparent all cases are more or less imaginable.

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  10. The plane on a treadmill problem is clearly intended as scenario 2 by people who haven’t noticed that it is more easily composed as either “what happens if you try to take off in a jumbo jet into a 580mph wind?” or “What happens if you leave a jumbo jet sitting on a runway?”. The answer being, “it both takes off and stays where it is” or “nothing”, respectively. Interpretations 1 and 3 are from awkward buggers who just don’t want to accept that it’s a stupid question.

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  11. Bear in mind that when a treadmill is applying a force to the wheels of a plane, a large chunk of the force goes towards making the wheels spin (exactly how much depends on the moment of inertia of the wheel and the mass of the rest of the plane) to achieve a state where the wheels are not skidding and their hubs are moving at the same speed as the rest of the plane. Once the points of contact of wheel and treadmill are not moving relative to each other, the treadmill stops applying a force to the plane.

    This means that to continue applying a force, the treadmill must have constant acceleration (not just a constant speed). This quickly sends us into the crazy lands where the wheels explode or the treadmill catches fire.

    Camp number 3 is the camp of horrible death.

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  12. In either explanation, why would the plane take off? I am stuck on the idea that a plane needs lift to fly. Lift is generated by the flow of air over the wings. If the plane is stationary on the treadmill (regardless of how its propulsion is delivered) then wouldn’t the plane not take off since there would be virtually no airflow over the wings, even at full thrust?

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  13. VB is a rather unpotable brand of beer, nearly as watery as American beer. Hence, solution #5 may be suboptimal in this example as the VB arrow is pointing towards the front.

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  14. I bought some new running shoes yesterday, intending to go for a run this morning before work. In the end I was too comfortable in bed so I didn’t get up until about 8.30am, and only then because the cat leapt on my bladder.

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  15. @mpetersheim, it says: The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction.
    Opposite direction of what? The plane? The wheels? The bird crap that falls out the engines every time it sucked up another eagle?
    The wheels have almost no influence on the plane itself. What matters is the amount of drag the plane will get from the ground while the engines pushes it forwards, Wheels are just there to reduce the drag.
    Using a belt to spin the wheels in any direction or even to keep them stationary just has no influence on the plane itself, lifting off. You could put a bunch of weasels under the plane instead and make them just run with the same speed as the plane and it will still lift off.
    Each wheel has an axle. And like the axle’s of toy cars, it can rotate forwards or backwards at any speed you desire. But while doing so, it won’t affect the way you can move this toy with your hand over the ground, your Superman figurine, your cat’s tail or your girlfriends leg. Of course, the drag of the conveyer belt will have some influence due to friction and stuff, but not enough to stop those brontosaurus-moving engines. This friction is mostly overcome because those wheels can just turn in any desired direction. Their movement matters not, just their ability to reduce friction.

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  16. Or to put it in simple terms: as long as the wheels kill most of the friction the plane has with the ground, no conveyer belt will be able to keep the plane stationary. They can keep the wheels stationary, but not the plane. Pull the breaks on the wheel and then the belt will have a chance…

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  17. The airplane will travel forward while the wheels stay motionless on the treadmill. The position of the wheels on the treadmill will remain the same, however the location of that position will move FORWARD due to the airplane thrust. much like if it wasn’t a wheel but merely a post. The wheel has no reason to roll as the treadmill has effectively taken on that role (pun intended).

    A person standing off of the treadmill looking at the plan sitting on the numbers will see the plane, and numbers move down the length of the treadmill until the plane takes off.

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  18. I think my pick would also be group 6, although i would rather have my alcohol in Worst Class with a bunch of drunk swedes that i don’t understand, being extremely drunk and seeing what happens if you smash the window and vomit on the conveyor belt. ( It shoots of? Does some of it stay stuck on? (maybe that depends on how much sticky stuff i’ve been eating?) What does the conveyor belt operator say when he gets covered in it? )

    ( the lonely parenthesis

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  19. Can we shoot a plane, accelerating at full speed, backwards out of a large cannon/slingshot?

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  20. I just don’t like the picture of the poor brachiosaurus above, or the T Rex being fed to the Sarlaac on the What If pages.

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  21. @Clark:
    “In either explanation, why would the plane take off? I am stuck on the idea that a plane needs lift to fly. Lift is generated by the flow of air over the wings. If the plane is stationary on the treadmill (regardless of how its propulsion is delivered) then wouldn’t the plane not take off since there would be virtually no airflow over the wings, even at full thrust?”

    Yes, IF the plane were stationary on the treadmill. It is not. The treadmill makes the wheels spin faster, but has no appreciable effect on the plane’s speed (relative to the ground).

    The scenario you’re imagining is more along the lines of the plane being bolted to the ground and then its engines turned on full blast. In this scenario, yeah, the plane doesn’t take off, though the engines might rip it and/or the moorings to bits.

    @derangedlemur:
    “The plane on a treadmill problem is clearly intended as scenario 2 by people who haven’t noticed that it is more easily composed as either “what happens if you try to take off in a jumbo jet into a 580mph wind?””

    Nice job attempting to be condescending, but that is a VERY different scenario, because airspeed produces lift, and groundspeed does not. In that scenario you could turn the engines OFF and (assuming that’s the necessary airspeed, idk) the plane would rise into the air, albeit going ‘backward’ We can all agree that if you turned the engines off and put the plane on a treadmill, it would not take off!

    Depending on the specific speeds, amount of lift required, etc, the plane can either take off while not moving relative to the ground, take off while accelerating backwards, take off while moving at a much slower ‘forward’ speed, or stay on the ground while hurtling forward at a rapid speed.

    The only conundrum there is if you don’t realize that the plane’s speed relative to the air around it governs whether it takes off, and (if we’re altering the speed of the wind at will) this is independent to the plane’s motion relative to the ground (whether it’s moving ‘forward’ or ‘backward’)

    And in case you’re now wondering about a 580 mph TAILWIND – it depends. Do remember that the wind IS pushing the plane forwards, and will accelerate the plane up to a point (either 580 mph or less, i am not an aerospace engineer) even without the engines on. At that point, when you turn the engines on, it’s effectively the same as starting the engines if the plane were stationary relative to the ground and there were no tailwind.

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  22. @Alastair Ward: to paraphrase Vaya con Dios: “Girls don’t cry for T-Rex / Raptors wouldn’t cry for you / When you share the skies with ‘dactyls /
    You better… RUN FOR YOUR LIFE, MAAAN!”

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  23. Curious, someone actually agrees with me that the landing gear would effectively turn into skis and still thinks I’m an idiot.

    “And anyone actually reading this, take this comment as evidence that it’s not just condescension: some people REALLY DO BELIEVE that a plane cannot move forward if its wheels aren’t spinning the right way.”

    Try nailing an airplane’s tires to the tarmac and see how well it takes off. We’re not talking about free-spinning frictionless tires; we’re talking about things that are gripping the ground. Without friction, the question doesn’t make any sense. The clear intent of the question is to posit a scenario where the plane couldn’t roll forward. So yes, some people REALLY DO BELIEVE that a plane can’t take off if its landing gear are immobilized.

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  24. There seems to be a new argument coming up that has the treadmill rotating in the same direction that the plane is trying to move. And those arguing in this vein say that this will stop the wheel from rotating and have it skid forward.

    But think about it from a standstill. The plane starts trying to move… the treadmill sees this, and starts to rotate forward so that the wheel will not move. And as the plane moves forward faster through the air, the treadmill will have to move faster forward to make sure it is still moving so that the wheels don’t move. So now you have a treadmill that is accelerating forward with your plane at the same speed that the plane wants to go.

    This will greatly increase the life of airplane’s wheels (in fact… it doesn’t need wheels anymore…), and it will still take off. So I’m sure maintenance mechanics will greatly appreciate your invention.

    Bottom line is… a plane moves through the air. The reason it has wheels is so that it doesn’t have to drag its belly on the ground. I had a hard time comprehending this too… until I actually watched it happen on Mythbusters. And then everything clicked. So go watch the video (www.youtube.com/watch?v=YORCk1BN7QY) and see if that resets your brain.

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  25. Perhaps one way to get past the silly 3rd interpretation is to replace the wheels and conveyor with gears to avoid confusions around slip. Since the gears/wheels are free-spinning, the main friction the engines have to overcome is not actually friction with the “conveyor” but resistance placed on the axle. If there is no resistance on the axle (i.e. brakes are off), the thrust is going to push the axle forward relative to the ground no matter how fast the gears are spinning.

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  26. I’m in group 25 from the future and I came to tell you that after an expensive amount of testing we decided that to go and give up on life. Also 999999999999999999999999999999999999999999 internet points if you send me a burrito in the mail!!!!

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  27. First off, thank you, I haven’t laughed so hard in a long time.

    Second, when is someone going to launch (bad choice of words?) a Kickstarter to settle this? Where are all the doers and the putter-uppers around here?

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  28. If you read this problem and wonder what the weight-and-balance shows for the 747–Is the airplane full? Empty? Is group # 5 drunkenly staggering forwards and backwards down the main aisle, negating any weight-and-balance work you did previously?

    Also, if the spool-up time for the treadmill is slow enough, you could trick it into working as a catapult by starting off with reverse thrust, then switching into full forward thrust and using soft-field take-off methods to minimize the weight on the undercarriage. I think you could stagger into the air and build up speed in ground effect…anyone want to program Xplane to represent this situation?

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  29. If the brakes are not engaged, the force transfered by wheel spin to the aircraft is so small it can be regarded as zero. The wheels can be spun in any direction you want and it will have very little effect on the aircraft. This can be seen on landing just as the wheels touch down. The wheels will spin up very rapidly, usually with some smoke on large, fast airplanes, but the airplane does not slow measurably just because the wheels touched down. That doesn’t happen until brakes are applied.

    The engine force is entirely directed against the air, or the Earth/air reference frame if you want to look at it that way. Run up the engines and the airplane moves relative to the air and takes off. The treadmill makes no difference.

    This problem is commonly given by instructors in basic flight training to get students to understand airplanes are driven by thrust and are not cars driven by wheels.

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  30. Take-off in an airplane is determined by airflow speed over the wing. The wheel speed things seems to be a red-herring. If the jet can suck enough air over the wing to lift the plane (which seems unlikely), then the plane should take off. This works with a propeller plane, but those work differently as they actively blow air over the wing at speed. Take a look at Channel Wing Airplanes (http://en.wikipedia.org/wiki/Channel_wing) for a good example of the airspeed-over-wing issue.

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  31. Very funny conversation. Found myself thinking of the song “Contact” by Phish the whole time I was reading.
    I also find myself stuck on the drag and lift part of the problem, not on the wheels at all but I never took a physics class so I have to admit to an unfortunate level of ignorance on the matter. Perhaps I will look up the Mythbusters thing and see if that speaks to my confusion about why the wheels are really an issue at all.
    Thank you for the giggles…. I may not understand all the math but somehow I know it’s funny anyway! (Which is truly magic. Well done!)

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  32. As much as I enjoy physics, I enjoy people more. Watching a hundred people respond with great fervor using the same arguments Randall just requested that everybody shut up about is like viewing a solar eclipse before the advent of calendars: you can’t make it happen. You just have to be in the right place at the right time to bask in its glory.

    Just don’t stare too long.

    (also why is the captcha a photo of my home address- oh shi…*)

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  33. Come on!

    I’m still trying to get the speeding car up the ramp into the back of the airplane taking off ……

    Now, you have the 2-wheel drive car with front wheel drive… But that can only accelerate at the right speed until the front wheels leave the runway/treadmill and hit the ramp – at which time all Hades breaks loose.

    Then you have the 2-wheel drive car with a rear-wheel drive ….. Which is fine until the car’s rear wheels try to climb the ramp, lose speed and traction, and drop the car back off the back of the ramp again ….

    Then you have the 4-wheel drive car which can’t do either because the front wheels and back wheels need to be about 160 mph (take off speed) different at the same time ….

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  34. Really? Still arguing years later?! Are you people that thick when confronted by evidence? This “problem” was definitively solved in 2008:

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  35. “Without friction, the question doesn’t make any sense”

    No matter what, the question doesn’t make any sense! It is s stupid question, made arguable only by wilful misinterpretation by internet maniacs.

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  36. the engine pushes (or pulls) the plane. Unless the pilot engages the break, the plane will move forward. If the belt moves backward as the plane moves forward, the wheels will spin twice as fast, while the plane moves forward basically as fast as it normally would.

    The engines apply force to the airplanes body. The conveyor belt applies force to the wheels. The belt MIGHT effect the air pressure under the airplane, but that really isn’t the point of the original problem.

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  37. I like how Randall tried to lay this to rest in 2008 only to have his creation backfire on him in 2013. How does it feel Dr. Frankenstein?

    I think the people in the camp of “it can’t take off on a treadmill” should consider the following: if your treadmill can keep the plane from moving when it has its engines on, it should be capable of launching the airplane into the air if run in reverse and the plane’s engines off. No one seems to want to argue that point, though…

    For anyone who has taken physics in college, you’ll remember that the first thing you learn is that your natural intuition is almost always wrong. This is why we have math!

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  38. I’ve only ever thought of this problem as interpretation 1, and honestly can’t imagine why anyone would think of it any other way.

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  39. It is rather obvious that the people trying to answer this question are focussing on the wrong thing. The forward momentum of the plane while it is on the ground is irrelevant. What actually matters is how much air is flowing over the wings.

    Let’s pretend that instead of a conveyor belt the problem involves an unbreakable tether tied to something behind the plane. Now we need to look at the plane itself. 747s are Jets, they’ll never take off because the plane moving forward is how the air gets to moving over the wings to cause lift. However if your plane is propeller driven then it is a different story. Even without moving forward air will still be moving over the wings and if your engine was moving enough air it is quite possible to get lift even without thrust. A propeller driven plane could lift off, but the tether would hold it back still; if you transitioned back to the conveyor belt you will still get the lift but then there would be nothing to keep the plane from getting the thrust it wants.

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  40. Also the wheels aren’t connected to anything so their max rpm is whatever the failure point of the rubber is. Planes wouldn’t really care how fast the conveyor belt was.

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  41. Then again if the conveyor speed was based on the rpm of the tires, which is itself based on the difference in velocity between the plane and the conveyor….as the plane moved faster it would make the tires spin faster which would make the conveyor move faster which would make the tires spin faster which would make the conveyor move faster etc. etc. until the tires meet their critical limit and fly apart, at which point the plane would fall to the conveyor and there would be a major crash.

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  42. Take a child’s toy train (not powered) and put it on a running treadmill.

    Tilt that treadmill at 5 degrees (along the major axis smarty pants)… The little train will start rolling down the hill due to gravity, it will accelerate and happily roll off the end.

    Start the treadmill at a nice sedate 5km/h and what does our little train do? It happily rolls down the spinning treadmill and trundles off the end. The joint friction in our spinning wheels doesn’t counteract gravity at 5 km/h.. crap.

    What if we make the treadmill go faster, how about a nice sedate 50 km/h?
    Again it rolls down the spinning treadmill and trundles off the end. The joint friction in our spinning wheels doesn’t counteract gravity at 50 km/h.. Oh double crap, joint friction doesn’t increase with velocity.

    The only way to make our little train not slide down the treadmill is to keep on accelerating the treadmill so that the kinetic energy ends up in the spinning wheels, not body of the train.

    Now replace the little train with 747, and replace gravity with jet engine… and we have a crushed treadmill… that simple really.

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  43. The plane will not take off, mainly because its a normal 747, while the conveyor belt is magical. The conveyor belt as presented by the problem is required to move backward at the same speed the wheels go forward, even though at first impression you would think that since the wheels are moving freely this wouldn’t make a difference, the problem here is that the conveyor belt will output a speed that counteracts any force applied to the wheels, included the force of the 4 airplane engines. The conveyor belt then has to go at enough speed so the friction on the wheel axis is enough to counteract the energy of the 4 engines (if the plane moved anywhere the conveyor belt would not be going at the same speed as the wheels) though the wheels would be destroyed before it could reach max speed.

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