A question about the aeroplane on a treadmill. Can you solve this conundrum?

http://blag.xkcd.com/2008/09/09/the-goddamn-airplane-on-the-goddamn-treadmill/

I think the answer is to use a Harrier jump jet – no wheel speed at take off.

Also remember the treadmill will take time to accelerate, so long before it reaches relativistic speeds the plane will have taken off. It may cause a lot of wear to the tyres though.

fyi: Can a plane take off on a giant treadmill?

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It’s not engine thrust that makes a plane fly. It’s lift generated by air passing over the wings. If the plane isn’t physically going anywhere, then there air isn’t passing over the wings and hence, there is no lift.

@robmandu thank you, no one seems to understand planes!

Assuming the treadmill will accelerate instantly as fast as the wheels are going (“assume a spherical cow…”), and a small airplane, the journal friction in the bearings will overpower the engine power quickly.

I imagine what would happen is that long before you hit relativistic speeds is that the metal in the bearings will melt (or the tires explode), causing the wheels to fall off, and the airplane to contact the treadmill, which would instantly stop since the wheels are no longer moving. The plane (if we’re lucky) will then slide across the treadmill and take off.

QED.

I mean, yeah, if the treadmill can accelerate in a way that breaks physics, then yeah, the plane won’t take off, ‘cause physics broke.

The problem lies in one’s definition of “speeding up to match the speed of the wheels”-

does that mean matching how fast the bottom of the wheel is moving (because a treadmill with no motor and good bearings will do that), and takeoff is no problem.

or does it mean it matches the speed of the axle moving across a stable frame of reference (in which case the wheels will just spin twice as fast as the plane’s motion, and takeoff will be no problem (if takeoff happens at stable-frame 100mph, the wheels will be spinning at 200mph, which is totally within design limits).

OR does it mean that the treadmill accelerates to whatever the top of the wheel’s speed is from a stable frame (essentially the axle’s speed across the stable ground + the bottom of the wheel’s speed across the treadmill), which is theoretically what the plane’s speedometer would read if it was reading speed via its wheels. Because that system quickly spirals up to infinite speed for both the wheels and the treadmill, and additionally is a type of treadmill that humans don’t know how to craft.

If this third-possibility treadmill was anchored well enough, and the near-light acceleration happened fast enough, then yeah, you’ve broken physics, and you might even see it rip it’s continental plate orthagonally off the rest of the earth like some kind of wonky gyroscope.

I saw videos supposedly “proving” both points of view.

It seems to me that there has to be friction between the plane’s wheels and the ground for the plane to move forward. The plane has to push hard enough to overcome that friction, otherwise it can’t move forward. However, if the conveyor belt runs backwards fast enough to provide no friction to the wheels, then they won’t move.

However, the rest of the plane will be forcing air backwards, and at some point it will no longer be able to force the air backwards fast enough, thus putting force on the top of the plane, with no force applied to the wheels. The plane will tip the tail end up at it’s speed of acceleration and nose into the conveyor belt. It’ll just look like it’s tipping over. Once the nose hits the conveyor belt, the plane will go backwards at just about the speed of the conveyor belt, although the thrust might cause some movement of the nose forward, thus creating enough friction to burn up the nose of the plane. If the thrust was exactly the opposite of the conveyor belt, the plane would be ground down, nose first, until it got to the engines, which would then cease providing thrust and the remaining part of the plane would be thrown backwards at the speed of the conveyor belt.

I suppose it is possible that the propellers or jet engine could simply push the air backwards. If it is possible to somehow keep the plane level, than the air would be pushed backwards at the top speed of the airplane. The plane wouldn’t take off. The conveyor belt would only have to go as fast as the top speed of the airplane. No sailing off into infinite speed and breaking the speed of light barrier.

Allowing the conveyor belt to go fast enough to cancel any movement of the wheels relative to the ground is similar to tethering the plane in place, except that there is equal force against the entire airplane, not just the part that touches only the ground. It’ll just move air backwards instead of moving the plane forwards.

So the plane will nose into the conveyor belt, and since the nose doesn’t have something to eliminate the friction, depending on the rate of acceleration, the plane will be nose into the conveyor belt at a speed similar to that of the plane flying into the side of a building. It’ll crumple and explode. Except that wouldn’t happen since the plane would nose into the conveyor belt very soon; long before it built up enough speed relative to the air above the conveyor belt to really damage the plane when it hit the ground.

Take the treadmill out of the equation. (It’s obviously too distracting what with friction, acceleration to speed of light, tectonic plate shifting, and gyroscopic motion of the Earth).

Attach the entire plane to a set of cables with some small amount of slack in the y-axis (up and down), but not x- or z-axes (forward and back, left and right). And hoist it off the ground so the wheels aren’t touching.

Now, run up the engines.

Will the plane fly (i.e. generate lift)? – measured by how the aircraft rises up in the harness

No.

Because, just like the treadmill, the plane is not moving… and air isn’t flowing over the wings (if you think the propeller can make that happen, you’re wrong… but for argument’s sake, just use jet engines instead where the exhaust does not flow over any wing surfaces).

@robmandu That’s not the same scenario as the treadmill because the original doesn’t push the body of the plane at all. It just touches the base of the wheels.

What is it about the wheels turning that equates to “pushing the body of the plane”? The entire point of this hypothetical is that the plane is not moving its relative position to the ground (and surrounding air).

@robmandu

Of course, if you keep the plane from moving forward it won’t go up, unless it has vectored thrust.

The original “thought experiment” plays on our mental gap between the way a car moves and the way an airplane moves. The POINT was that the journal friction and the rolling friction of the tires is minimal enough that the plane can still accelerate no matter how fast the treadmill below it is rolling.

The MythBusters did it on one of their shows, but it showed the plane taking off. I don’t think it is that easy, because the wheels on a plane are moved in a different manner then that of a car. In a car, the wheels are connected to the differential, or some power source. But on a plane, they are independent.

@brinibear you’re right. On most planes that I know of, the wheels are not powered, and only move because the plane is moving relative to the ground (or treadmill). If the treadmill keeps the plane from moving relative to the surrounding air, the wings will develop no lift (thanks, @robmandu), and will not take off. The exception to this would be if there was a strong wind passing over and under the wings.

The wheels have to move relative to the position of the plane on the earth, or you can’t get air flowing over the wings to lift the plane. It matters not how you stop the wheels from moving forward relative to the air around the plane. You could move the conveyor belt at exactly the speed necessary to neutralize the movement of the wheel through space, or you could put a concrete block in front of it. Until the plane lifts into the air, and contact with the ground is broken, the plane can’t move forward enough to fly. If contact with the ground is not broken, then, while the rest of the plane tries to go forward, the wheels stay in the same place, and the plane noses into the ground, or, if it stays level, then it’s like an open air wind tunnel or a giant fan.

To restate my early comment from a different angle, I think the key point in the whole argument is the wheels, and how they work, and how the treadmill affects them.

The confusion, I think, comes when one assumes that changing the velocity of the wheels is integral to changing the velocity of the plane, and vice versa.

Imagine, for example, that the plane is tethered to a pole off of the treadmill, and that the treadmill is moving at 50mph. If one looks from a fixed perspective off the treadmill, the plane is not moving. But if one examines the velocity of the wheels, they are spinning at 50mph. The tether is at this point only resisting against the friction inside of the wheel mechanism.

If the ball bearings in the wheels had zero friction, one would not even need to tether the plane- its resting inertia would keep the body of the plane going 0 mph while the wheels moved at 50mph, spinning “in place”

The plane’s engines act on the air around it, not the ground (whatever state the ground is in)- all of its thrust is gained by moving air.

Imagine now that this resting plane with wheels spinning at 50mph powers up its engines to 1mph. The treadmill is still moving at 50mph backwards, the plane pushes the air backwards and begins moving off the treadmill at 1mph, and the wheels go 51mph.

Even if the treadmill was to speed up to 1000mph, if the wheels had no friction, the plane would still go forward at 1mph, just with a lot of wheel-spinning, because the only force affecting the plane is its own thrust.

To put it an even simpler way- since wheels are designed to spin, and as such have little friction compared to, say, rocks, the treadmill can never exert the kind of force on a plane that its thrust cannot overcome, because it can only exert that force through the wheels- making the treadmill spin faster will do little to affect the body of the plane.

No amount of treadmill movement will keep a plane in place if that plane uses engines that move air (all plane engines move air), and the original problem specifies a speed of treadmill movement that is within the bounds of friction/metal fatigue reason.

@daloon you are confusing, I think, what the treadmill can do to the edge of a plane wheel (make it spin really fast), vs what it can do to the position of that wheel’s axle (very little).

The thrust power of the plane would certainly move the axles (since they are attached to the body) relative to the earth, and beyond concerns of friction, the treadmill would do little to the overall plane save speed up the spin of the edge-of-the-wheels.

@Yetanotheruser Except in cases of very, very, sticky wheels (to the point where they are less wheels and more circular protrusions), there is very little the treadmill can do to keep the plane “in place”. It may affect the motion of the outer edges of the wheels, but it can’t do much to the body of the plane itself, upon which the only force of real concern is thrust, a force which the original problem states is entirely based on the air the plane is pulling itself through, not the “speed of the ground”

I love how we’re all in Camp #2 or Camp #3 per @drdoombot‘s link.

@robmandu Or camp #4… this is the Interblags, you know. :-)

I cast my vote for the Harrier.

It makes the most sense and saves the most time.

The treadmill will have little or no effect on the airplane trying to take off. If anything, it’ll be easier for the airplane to lift off, because there will be less wheel friction. Treadmills don’t resist anything. In fact, they have less resistance than ground does. The reason walking on a treadmill gets you nowhere, is because walking moves you forward by pushing with friction between your feet and the ground. An airplane engine, on the other hand, pushes air and moves the plane forward, which the treadmill will not resist at all (or even less than solid ground will).

@robmandu To be clear, do you believe the wheel speedometer to be hooked to a controller powering the treadmill, the airspeed indicator to be hooked up to a controller powering the treadmill, or the impossible Vc=Vw+Vc scenario?

AFAIK there are no controllers that have zero response time. So the only two possible outcomes are a treadmill accelerating to infinity (which would obviously break in real life long before it hit relativistic speeds) or a treadmill that reaches twice the takeoff speed of the aircraft. For practical purposes only one good interpretation, and thus answer, exists: the treadmill accelerates to twice the takeoff speed and the plane takes off.

As a person firmly ensconced in Camp #2, I see this problem as the plane is not moving relative to the ground. It’s only moving relative to the treadmill. The treadmill’s maximum acceleration/speed need only accommodate the maximum performance of the plane and not break any laws (or conventions) of physics. And because the plane itself isn’t really going anywhere, there’s zero air flow over the wings. And therefore, no flight.

Camp #3, as I understand it, seems to think that the treadmill is trying to slow the plane down. And that the plane should be able to accelerate up to rotation speed and take off. Of course, the wheels in that situation must turn fast enough for take off (like normal) plus even faster to make up for the treadmill.

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I think Camp #3 misses the point of the riddle/puzzle/hypothesis (and I’m sure they think I do, too).

You see, I read the phrase “treadmill is supposed to match the speed of the wheels” as a builtin limit or rule of this game. And that if, for example, a plane needs to reach 100mph under normal conditions, that the treadmill under that plane will only be accelerated to that speed exactly as needed to maintain an equilibrium where the plane itself does not actually move. That is, the plane is putting out nominal thrust to achieve normal takeoff/rotation speed and the treadmill is running underneath it just fast enough to keep it in place.

Camp #3 reads that same phrase as a challenge to overcome.

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I think folks who find themselves in Camp #3 are probably more likely to excel under adverse conditions in business, sports, whatever and that I (and my Camp #2 kin) will languish in the same situations under what we perceive to be immutable and unchangeable laws of the universe. ツ

@robmandu So… how does the treadmill keep the airplane in one place?

@grumpyfish, well, how does a treadmill that you’re running on keep you in one place?

@robmandu It keeps a human in place because we are directly attatched to our feet. Our feet do not have ball bearings.

Plane wheels do.

I fail to see how the treadmill can exert the kind of force upon the plane-body to keep it in place, if all it can do is spin the wheels?

@aphilotus, the mode of locomotion does not matter. You’re assuming wheels+bearings == no friction.

If you put the plane on the treadmill and turn the treadmill on, the plane will start moving backwards, right?

So, the plane then fires up the engine(s) and increases thrust just enough to hold in place.

Increase the speed of the treadmill and thrust of the plane together and you can maintain equilibrium such that the plane does not move relative to the ground.

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The thing is, there is friction. That’s why the plane moves backward when you first turn on the treadmill after all. Even though the plane’s wheels turn on bearings, friction is still very much in play. There’s friction in the contact between the treadmill and the rubber tires as well as friction in the workings of the bearings themselves. It’s that friction that the plane’s thrust is being used to overcome.

Sure there’s friction, but the coefs are tiny—the pneumatic tire rolling on concrete is around 0.049, the coef for the bearing friction is roughly 0.0001+/- depending on the quality of the bearing.

Likely, that means that the force required for the accelerate the airplane backwards is likely much larger than the requirement to simply spin the wheels in place. The plane WILL start to move backwards, but we’re talking about inching very very slowly.

The force provided by the engine of the airplane idling is likely more than the frictional losses to the rolling & bearing friction that are being transmitted to the airframe.

@grumpyfish, ever fly a plane?

@robmandu I’m quite sure that a jet engine can easily overcome the negligible friction between the wheels and the axle.
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And if the problem is set up in such a manner that the treadmill increases its speed not to “match the speed of the wheels” but to deliberately abuse just this small wheel-friction as a manner of keeping the plane in place, then its a pretty dumb problem- yeah, the plane will lose.

@robmandu – Both your camps 2 and 3 are wrong. See my answer above. Treadmills don’t slow anything down – they only stop walkers because they offer no resistance (unless someone is holding onto a handle).

Guys, you’re arguing Case #3 as I already described. That’s fine. I’m sure you’re right.

I explained that us Case #2 folks believe that the equilibrium is a rule of this game. If the equilibrium is maintained, it doesn’t matter how fast the treadmill is going.

@robmandu I’m a fish.

The problem with Case #2 is that the math doesn’t add up. Ever see the trick where somebody pulls the tablecloth out from under the glasses on the table?

@robmandu See but the Case 2 that you are citing (from the link at the top of the post) doesn’t ask for equilibrium.

Under its purview, the plane would take off at it’s correct take-off speed (lets say 200mph), the treadmill would be moving backwards to that at 200mph, the wheels would be spinning at 400mph, and the thruster would be exerting enough power that given no friction the plane could be going, lets say, 205mph.

@grumpyfish, you’re of the opinion that the minimum thrust of an idling plane (prop or jet) will overcome the friction of wheels dragging along any treadmill.

That’s obviously wrong as a idling aircraft should be able to taxi on a real road if that were true. And I can tell you that an aircraft at idle will not go anywhere.

@robmandu even with the brakes off?

@grumpyfish, yes!

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And to this riddle/theory/whatever, even that doesn’t matter. The assumption that a Camp #2’er would make is that the plane’s thrust could vector up from zero with infinite gradation to perfectly match the treadmill speed.

Even a magic treadmill with hyper-stupid acceleration created by belief in badly-worded story problems won’t slow down the airplane – they’ll just make the plane’s wheels spin, but un-braked aircraft wheels can spin as much as they need to without significantly slowing down the airplane, which is being pushed by engines and air not involved with the wheels and treadmill.

@grumpyfish @robmandu I think at this point we are just arguing about a matter of degree.

me and @grumpyfish are of the opinion that friction and resting inertia are minor, minor numbers compared to thruster-power, while @robmandu thinks that they are the crux of the whole game.

@robmandu okay then—point granted.

I’m still saying that the wheels would have to be spinning unreasonably fast for the airplane to stay in one place (OK, not with the brakes on). An FA-18 can do an unlimited vertical, meaning it can produce more thrust than the weight of the craft. Based on the rolling friction math, you cannot produce more friction than the weight of the craft.

That is, Friction from rolling = (Normal Force) * (Coefficient of friction) / radius If I have an engine that can produce more thrust than the Normal Force, I can overcome any treadmill.

QED, again :-)

Wow. Now I really understand why Randall Munroe does not allow this debate in the xkcd forums.

I already stipulated that Case #3 works. A real life plane on a real life treadmill certainly can take off. The plane’s thrust likely can overcome the friction of wheels and bearings on a treadmill to achieve its rated takeoff speed relative to the ground (thus generating air flow over the wings and producing lift).

Now, is anyone here willing to acknowledge that for Case #2, a plane with its thrust being used in measure to maintain an equilibrium with the treadmill is not actually moving and hence will not fly?

@robmandu indeed—but it’s a fun discussion!

So, what you’re saying is that if I take a plane, and make it so it CANNOT move no matter how much thrust you produce will not take off (e.g., move)?

In short, if I take a plane and bolt its wings to the ground using infintitely strong bolts, it certainly won’t take off =)

@robmandu But Case #2 doesn’t specify that. At all. It just specifies that the treadmill will move at the same speed as the wheels-are-moving-relative-to-a-distant-observer. There is nothing in case two about the engine playing nice and only providing enough thrust to counteract the rolling friction. It just goes full blast and doesn’t care what the wheels and treadmill are doing.

The point about Case #2 is that how we perceive the question being asked.

“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.”

Robmandu asked, “Now, is anyone here willing to acknowledge that for Case #2, a plane with its thrust being used in measure to maintain an equilibrium with the treadmill is not actually moving and hence will not fly?”

I don’t think so. For one thing, case 2 already says that the treadmill will move at the speed of the axle in the opposite direction. Good for it. It doesn’t stop the airplane, because the wheels spin freely between the plane and the treadmill, just at twice the usual speed during takeoff. The plane is moving relative to the ground as usual, and takes off just fine. The treadmill is just left moving backwards at takeoff speed, and the wheels are spinning faster than usual.

Also from the xkcd post: “The #2 crowd is busy explaining to the #3 crowd that planes aren’t driven by their wheels.”

And now note my quip here.

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I just can’t stand it when we’re all arguing the same frackin’ point!

@robmandu And as you said before, it comes down to what people think the question is asking.

I think the problem is, at some level, is the question asking if a plane on a treadmill will take off, or if a plane that cannot move forwards or backwards will take off.

I think we’ve all proven that a plane on a treadmill (any treadmill) will take off, and we all agree that a plane that cannot move relative to the airstream won’t take off.