# 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.

## 800 thoughts on “The Goddamn Airplane on the Goddamn Treadmill”

1. There’s one key here that, from what I’ve read so far, has been over looked: the engines don’t directly turn the wheels.

The only reason the wheels turn on a plan normally is because the axle is being pushed forward by the engines, and the bottom of the wheel is being dragged along the ground. The plane would already have to be moving forward for the wheels to turn.

That is really where the paradox is, the conveyor belt and wheels won’t move if the plane is moving, but the plane can’t move if the conveyor belt is matching it’s speed. In real life, the plane would still take off because of friction and the inability to build a conveyor belt to match the speed.

Now, build a conveyor belt which would match the *force* of the engines, and you’ve got a whole different problem.

See, what I initially thought the problem was is..

Things with wings (Gliders, paper airplanes, birds, whatever) hit air friction, causing pressure underneath the wings, and creating lift, causing the flying thing to float, fly, glide or whatever. So, if you put an airplane (Well, a 747 in the post Im reading), will it fly? Or else.. without using its wings, if the airplane’s wheels are going fast enough, will it create lift and take off. The answer I come up with is no, the wings aren’t slicing through the air and creating lift. The air around the plane isn’t being disturbed. If you’re biking at 10 mph, you usually squint your eyes or wear goggles cause the air is hurting your eyes as it rushes past, but if you were on a treadmill, you’d feel nothing.

First off, one could be sarcastic and annoying by imposing that if the engines were working it would compromise the integrity of the conveyor belt, or it would be different considering what kind of tires are on the plane.. but I was assuming we were assuming perfect conditions.

So again, assuming perfect conditions, “If an airplane is going really fast on a treadmill/conveyor belt, can it take off.” I thought this whole ‘problem’ was about whether the wings will create lift, and I don’t see how it would.

But from reading this, i feel generally confused as to what the actual problem is if it isn’t what Im stating above. Am I mistaken that 747s dont use lift to fly? What are the wings for then, surely you can put a jet on top and below if the wings aren’t creating lift and helping. Pppllleeeaaaasssee englighten me! D:

[ML]

3. ***************I made a mistake on my last post. “So, if you put an airplane (Well, a 747 in the post Im reading), will it fly? Or else.. without using its wings, if the airplane’s wheels are going fast enough, will it create lift and take off.”

It should read: “So, if you put an airplane (Well, a 747 in the post Im reading), will it fly? Or else.. without using its engines, if the airplane’s wheels are going fast enough, will it create lift and take off?”

-ML

4. I really don’t understand this problem. The engines push the plane forward, and the wheels only serve as a mechanism to decrease friction between the plane and the surface on which it stands so it can accelerate more easily.

What happens between the wheels and the ground is of no importance as long as the wheels keeps spinning…

Uuuh… right? So whats the point here? A clever way see how people handle this problem? Is there a single solution/ answer to the question?

5. @Pik & other confused posters:
Mr. Munroe already covered it above, but perhaps not in the most accessible way. The problem is the wording of the question, which Mr. Munroe adjusted, I think in order to fit in what he felt were legitimate alternative points of view. In any case, here is a very long and detailed post on the standard phrasings and answers to the question, if you want to work through it.

The tl;dr is:
1) The question is usually phrased to make people think the plane is being held motionless relative to the ground. If that happens, no, it can’t get lift and won’t take off.

2) There is no actual way any treadmill can actually manage that, since the plane is pushing off against the air and no treadmill could ever push back hard enough to stop that without simply breaking off the landing gear &c.

6. @Mr. Munroe: I was doing a sudoku on the subway in Shanghai and the cute girl who sat next to me got interested in it and started a conversation. She gave me her QQ number and we’re going out this weekend. I can’t get my computer to delete all of Tencent’s spyware, though, and can’t tell if it was really worth it.

7. Here is another possibility:

Suppose the airplane is designed to calculate ground speed (velocity relative to earth) either by the rotation of its wheels (like a car speedometer) or by looking down to see how fast it is going (visually or with radar), and subtract this value from its airspeed (aircraft velocity relative to the surrounding atmosphere), to calculate the wind speed (relative to ground, or, in this case, the conveyor belt). On a moving conveyor, the aircraft incorrectly assumes that its speed relative to the conveyor is its speed relative to the ground, and this results in an incorrect wind speed reading. Before the airplane reaches the speed required to takeoff, the conveyor reaches the speed as which the aircraft’s wind speed reading will exceed the maximum wind speed under which it is safe for the plane to takeoff. This triggers an alarm in the cockpit telling the pilot to abort the takeoff and wait for calmer weather.

So no, the plane cannot takeoff.

However, if it measures ground speed with GPS or accelerometers, then there will be no problem.

8. I am surprised no one has yet mentioned Mythbusters PROVED takeoff is possible here (even the pilot thought it wouldn’t work).

9. To me to solve this problem you would have to determine the following. If 2 747′s play tug of war with a theoretical cable that is the same weight as air itself but strong enough not to break and long enough that air currents produced by the two planes wouldn’t affect eachother. And assuming the 2 planes are matched exactly in power and ramp up in thrust at exactly the same rate. Could the planes move the air around the wings enough to create lift.

I would assume yes, otherwise planes would fall out of the air. So if this is possible then yes once this airflow over the wings via the engines is achieved then the wheels would no longer matter because they could be lifted off the conveyor and the plane would then progress forward.

10. I change my opinion above. Because planes don’t fly because the engines push air around the wings but rather because the engines thrust push the wings into the larger block of air in front of it. So no I don’t believe the plane can lift off.

11. What everyone here is missing is static and rolling friction. Since the belt speed will match the plane speed, there won’t be an instance where the wheel start turning, thus the belt would act as a friction less surface for the plane, thus allowing the plane to build forward momentum, hence creating lift over the wings. The only hiccup in all this is at what angle does the plane take off at? There is a point where the plane could literally be stuck thrusting at some positive angle from the horizontal and stay there if it does manage to break the static friction limit, but if it doesn’t it should still take off. It’s worth noting however, if the plane does break the static friction limit, the boundary layer created from a really fast moving belt would probably increase the positive pressure under the wing, thus making it lift off the belt.

12. The problem is much easier than all of these arguments make it out to be. Imagine an airplane on a treadmill with a cable tied from its nosecone to an immovable concrete wall. Regardless the speed of the treadmill, all that would happen is that the airplane’s wheels would spin. Correct?

Now, use a winch to “reel-in” the airplane. No problem, right? Regardless of the speed of the airplane it’s still mounted to the immovable wall. If the wheel bearings of the airplane are “stiff” enough, it would add some resistance, but assuming frictionless (or low friction) bearings, no problem.

The plane’s engines create exactly the same scenario as the winch-cable, they provide a force in the forward direction which is independent of the treadmill.

Therefore, the airplane would be able to take off – this problem is conceptually difficult, because normal and everyday human movement is dependent on applying a force to the ground. An airplane does not have such restrictions.

13. Yeah, I’m one of those that don’t see the other option AT ALL, and I have a simple explanation, too.

I assume the conditions are ideal, so no friction at all.
Now, if you have a plane on the belt, without motors turned on, and the belt runs backwards, nothing happens to the plain, except the wheels start spinning. Now, if there is absolutely no forces affecting the plane, what would prevent the motors from pushing the plane forwards? Nothing!

Actually, if belt was running fast enough, say just under the speed of light, the top of the wheel is moving forwards at the same rate, and if the plane would move forwards, the top of the wheel would have to move faster than the speed of light, which prevents it from happening. So, that is the only scenario actually affecting the plane at all. And since the belt’s speed is the wheels’ speed inversed, that will never happen.

With friction, the belt should be super fast too, to match the force the motors are providing. No change.

14. Mr. Munroe is correct — those of you who are asserting that there is a definitive answer are begging the question of what the treadmill is. The fundamental problem is “what function does the treadmill serve?” The problem has less to do with physics and more to do with paradoxical axiomatic definitions.

Think of it this way. You dive into the deep end of a pool and need to get to the other side. Without touching the bottom of the pool, you can move from stillness to motion towards the other side by pushing the water around you. Aircraft engines have the same effect (sort of). Therefore, the plane-ground interaction is not “necessarily” the determining factor as to whether or not there can be motion in the form of LIFT. As Mr. Munroe describes, “People who subscribe to this interpretation tend to assume the people who disagree with them think airplanes are powered by their wheels.” It’s only part of the problem.

There are slight albeit negligible differences in the type of plane being contemplated. A prop plane achieves lift by increasing the air speed flowing around the wings by using one or more propellers to thrust air over the wing behind it. Therefore, a prop plane, anchored to the ground, can generate enough thrust over the wings by “creating wind” and can lift off without the need for forward momentum. This is why the Mythbusters test worked and why toy gas-powered airplanes can fly on a tether.

Comparatively, a jet liner uses thrust to push an aircraft through a medium of air. One might think that, because the plane must be pushed by the engines, forward momentum is necessary to achieve proper thrust. But imagine, for a moment, that the aircraft is not being pushed by thrust escaping behind it but is being powered, again by wind, traveling over the wings. The jets are creating thrust by vacuuming up the air in front of them and expelling the air behind them. This also creates wind over the wings, because the dense air in front of the wing moves to fill in the vacuum created by the jet engine. The question is whether the volume of air being pulled over the wings by the engines alone is sufficient to generate lift, or if the concentrated thrust is necessary to push the aircraft wing through the static medium of air that is not filling a vacuum. I do not know this answer.

Obviously a jet is taking in as much air as it is putting out. The question is whether the air being taken in is in fact generating lift over the wings or instead is being sucked in from “non-wing” vectors. Certainly some air going into the engine is flowing over the wings, and this will create lift. However, all of the air coming out of the engine creates thrust, and this thrust propels the fuselage and wings through the air. The determining factor is the ratio of lift-capable intake over lift-capable thrust; all of the thrust is lift-capable, but only some of the of the intake (that which traverses the Bernoulli vectors) is lift-capable. A critical question is whether the intake, excluding the thrust, is sufficiently lift-capable to make a jet operate like a prop plane.

Practically speaking, this would depend significantly on the location of the engine. Jets positioned behind the wings would draw air over the wings; jets positioned under (or on top of) the wings would not draw air over the wings. In these configurations, like a 747, the engine is creating lift entirely from its thrust pushing the airfoil through air.

For the sake of argument, though, let’s assume the engine generates enough lift from the intake. This would mean that both aircraft are capable of taking off while being anchored to a stationary object; tether a jet or a prop plane, and (assuming you don’t break the fuselage apart from the force) the plane will be able to generate enough lift over the wings from the engine power to take off and float like a kite. Admittedly, this can be hard to rationalize. The plane appears to be bootstrapping because it shouldn’t be able to generate the speed necessary (from thrust) to push the aircraft through the air at a velocity necessary to cause lift. Again, this depends entirely on whether the lift is created from the thrust pushing the aircraft through the atmosphere or from the engine creating air motion around the wing.

The tricky part is not in whether the jet can take off while tethered (although in practice that is actually a tricky thought exercise, especially regarding the jet). The question is whether an aircraft, untethered but on a treadmill, can still take off. Again, much (but not all) of this depends on what the treadmill does for the plane.

Putting aside the three breakdowns in the example, the plane treadmill seems to be conceptualized as one of two similes: either the treadmill acts like a tether, holding the plane in place as it tries to generate lift, or it acts like a sheet of frictionless ice, freeing the plane from ground interactions.

In the tether example, the treadmill matches the speed of the plane so that the plane does not move relative to the ground. In such an example, the aircraft’s lift capability is entirely dependent on the air movement the engines can proactively generate around the wings — because the airfoil does not have forward movement, no lift is generated from the thrust. As we have seen, a prop plane can generate enough air movement to take off. A jet may or may not, depending on its configuration.

In the ice sheet example, the treadmill matches the wheel speed so as to negate the effects of the wheels upon the ground. This would be equivalent to your floating in the middle of a pool, or being stuck on the middle of a frictionless, frozen lake. Although your body is independent of any ground forces, you can create propulsion by interacting with the water/air medium around you and expelling thrust. Therefore, in this example, the thrust can be taken into the equation because the aircraft is clearly capable of generating enough thrust to push it through the air and create lift regardless of whether or not there is ground. If, for example, you dropped a plane out of a larger plane so that its X motion was zero, the smaller plane could start its engines and generate enough thrust to move the plane in the X direction. Therefore, subscribers to this theory always see the plane as capable of taking off.

The problem is that there is no way to know whether a person means a “tether” treadmill or an “ice lake” treadmill. Both exist as theoretical extremes. One essentially brings the coefficient of friction to 1, so that the plane can exhibit no X-ward movement, whereas the other sets the coefficient of friction at 0, so that the plane has no ground force interaction at all. Any test based on either of the three scenarios that Mr. Munroe presented lies somewhere between these spectra, so both sides will criticize any practical experiment as insufficient or improper.

15. Edit fail: the differences between prop planes and jet planes is not, in fact, negligible. The differences are actually very important with regards to the “tether” treadmill argument. They matter less in regards to the treadmill vs. treadmill argument, but it is a critical issue in whether a plane can take off at all while fixed in the X-direction.

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18. This question is so poorly defined. Why aren’t any of the forces shown? This question shows the importance of a correctly drawn free body diagram. Are you assuming that some torque on the axle is forcing the wheel to turn (back tire of a bicycle) or is the tire freely spinning about the axle, which has a directional force imparted onto it by the body of the vehicle (front tire of bike). The car/treadmill analogy in #3 above is an invalid comparison because the drive systems of cars and planes are fundamentally different.

The actual goal of the problem is to define a v_c such that the velocity of the plane relative to the ground is zero, otherwise it will take off. #3 is correct, if v_w is ever positive there is no solution. I wonder though, can the treadmill exert a force on the wheel that will exactly cancel out the rolling friction? Such as system could not be reactive, it would have to be perfectly timed with the plane’s engines but I believe that v_c=v_b=v_w=0 fulfills the problem’s requirements.

19. Clearly, nobody on the internet is that practical.

What I’d do: Retract the wheels, install tredmills IN the plane to rotate the wheels backwards to provide additional forward propulsion to the plane via the forward moving tredmill the aircraft is now resting its belly on. Use this propulsion in addition to the engines to take off with ease.

IF powering the wheels DOESN’T spin the Runway-Tredmill… then you are going to attempt to take off by dragging the belly of your air craft… hopefully pulling the runway-tredmill with you due to friction rather than just dragging along the ground, and you might be lucky to take off before the end of the runway (or just crasy into a radio tower, whatever).

20. This is not a question of velocities, but a question of forces. For the airplane to stay in place, the force of the conveyor acting on the plane must be equal to the force of the engine on the plane. For the airplane to take off, the force of the engines must be greater than the force of the conveyor acting on the plane.

In a perfect world, the friction of the wheel bearings would be 0 or negligible. The conveyor would move as fast or slow as possible and the airplane would stand still.
Since the world is not perfect, there is friction between the wheel and the conveyor, as well as friction between the wheel and axle. The greater Vc is, the greater amount of force the conveyor would put on the plane.

Now, with the engines off and the conveyor on, the airplane will move backwards, albeit very slowly. It will not get thrown off like a person would, but accelerate very slowly, implying force.

This implies to me that there could exist a Vc where Fc=Fe. This Vc number could be many times greater than the actual takeoff velocity. But it would potentially exist.

A more enterprising person could confirm this by tying a force gauge between a wall and the nose of an R/C aircraft. The R/C aircraft would be placed on a treadmill, held in place by the wire tied to the force gauge. As the treadmill speed increases, the force gauge should provide the actual newtons of force the treadmill is putting on the R/C aircraft, which should increase with Vc.

Another force gauge would be tied between the tail of the aircraft and the wall behind it, measuring Fe as engine throttle increases.

At that point, it would simply be a matter of matching Fc to Fe. The relative success or failure of the problem would come down to how much actual friction exists, and whether or not Vc is realistically attainable.

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22. I didn’t want to comment here. But I see 2 problems here that aren’t being explained properly, but that could also be me

1 – If the conveyor belt matched the speed of the wheels (when perturbed by the force of thrust), the plane would not be able to take off, as this would require an element from problem 2.

2 – Lift! If a plane is kicking out a million lbs of thrust, and the conveyor belt matched the speed of the wheel as they would be on the ‘ground’, no lift is being generated! The plane has to physically move forward to generate lift, the engines help accomplish this >.< This means that the conveyor belt must slow to at least 0.00001% before things could even begin to happen. It would have to slow much more for the engines thrust to get the body moving fast enough to push air under the wings and then you have lift off. This isn't a very complicated conundrum in my eyes.

23. Aw, shucks. Is this still going?

How big are the treadmills? Are they just little things, just large enough to make contact with the portions of the aircraft’s wheels that are in contact with the ground? Or is it one enormous treadmill as long and wide as the aircraft? Because that could probably move non-trivial amounts of air once it really got going, and moving air around the wings could conceivably generate sufficient lift to pick the airplane up off the treadmill entirely. But probably not for long, seeing as how the mass of air moving faster than the airplane’s stall speed is not going to extend very far up.

…was that even worth posting? >_<;

24. Or you could subscribe to the 5th school of thought and say “Well no one said the treadmill was on.”

26. Why is no one responding to those posting that Mythbusters already solved this problem? They solved it. The plane took off.

27. Ok, so here’s my answer:

It depends on two things:

Can the treadmill respond perfectly and instantaneously to the speed of the plane’s wheels?

Do the wheels have friction?

If the above answers are both yes, then I would suggest the plane can’t take off. The reason being that as the engines rev up and start to break the static friction in the bearings and the rolling resistance of the tires, and the wheels start to move, the treadmill instantly responds. This creates kinetic friction in the wheels and bearings. More engine thrust produces more wheel speed, but instantly more treadmill speed, and more friction. Ultimately, this friction is a drag on the forward motion of the plane and it would seem that no matter how much thrust is applied to the plane, as soon as the wheels were about to gain speed, the treadmill would gain extra speed, and produce extra friction, exactly countering the force applied to the plane by the thrust of it’s engines.

If the plane had no friction in it’s wheels, then I would suggest the wheels and treadmill would spin up to infinity, but would have no effect on the airplane, producing no force against it, and therefore wouldn’t stop the engines from pushing it through the air and taking off.

But neither of those scenarios are possible as we have friction and to build such a treadmill would be impossible.

I think if you really tried to build a treadmill for this experiment, that there would be lag between the speed reading of the airplane and the ability of the treadmill to adjust to it. So that basically puts me in the #2 camp, with the exception that maybe before the plane gets airborne, the treadmill is able to spin up fast enough that the gear produces so much friction that it fails.

28. The entire problem with that question is this completely nonsensical sentence:

The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction.

‘the speed of the wheels’ is a nonsense term. Wheels do not move in relation to the thing they are in contact with. The wheel cannot move in one direction and the belt in another (1), the wheel cannot move in one direction and the belt stand still.

The bottom of wheels always sit still in relation to the surface it is one, the top of wheels always move at twice the speed of the difference between the bottom of the wheel and the axis of the wheel, assuming that ‘bottom’ of the wheel is the part in contact with something else.

I suspect they mean ‘The conveyor belt is designed to exactly match the speed of the _airplane_, moving in the opposite direction.’. Although even that’s incorrect, and what is actually meant is ‘The conveyor belt is designed to keep the airplane stationary’.

Stating it that way, it because rather clear that the conveyor belt has no real way to accomplish this feat, barring some sort of amazing wheel/axle friction and wheel/ground friction that can overcome the thrust of airplane engines. Treadmills have no way to meaningfully slow the airplane down, and modern airplanes can create massive amounts of thrush.

Hell, there are lots of fairly lightly-powered aircrafts that can take off from _water_, which I can assure people provides more friction that some silly spinning-backwards wheels.

And because the conveyor belt can’t possible do anything to the airplane’s speed, of course the airplane can take off. (Assuming it is a runway-length treadmill. Otherwise, the airplane would simply fall off the front of it.)

I am, of course, talking about treadmills in the real world. There are no such things as magical treadmills that can spin fast enough to cause meaningful amounts of fiction in airplane wheels, and I have to suggest in such a world where such crazy things exist, those crazy things would require innovations in reducing friction that we could, for example, use in airplane wheels, so ha!

And even if they didn’t, the result wouldn’t be ‘an airplane that couldn’t take off’, the result would be ‘an airplane whose wheels melted and it hit the treadmill and was hurtled backwards at hundreds of miles an hour’.

1) I assume by ‘moving in the opposite direction’, they actually mean ‘rotating in the same direction’. (I’m being very charitable here.)

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30. So basically, under #1 the plane takes off, under #2 the plane takes off, and #3 makes no sense at all. Therefore the plane either takes off, or your problem is invalid in the first place.

31. The problem with the people trying to create a treadmill where this problem makes sense is they ignore all the ‘required secondary powers’ such a system would need. Forget the problem with ‘infinite speed’, we’ve got problems way before that.

First, once the treadmill starts going the speed of sound, the airplane is going to flip over, period. Probably before that, but I will definitely say that the sonic boom wind is going to be disruptive enough that the plane will not remain level. (And considering how much this realizes on friction, if one wheel even slightly leaves the ground, the entire plane is going to spin out.)

And as for those who point at that, at some point slightly before the speed of light, the treadmill and the wheels will gain enough mass to actually stop the plane from taking off, I have to point out that introducing such mass to the system would be, uh, really bad, for both the airplane which would be ripped apart, and the earth, which would also be ripped apart, or at least part of it ripped off. (We need ‘What If?’ about travelling at relativistic speeds on the surface on the earth. It is almost certainly not a good idea.) However, as the plane would never reach that point without flipping over from the treadmill wind, nevermind.

Secondly, the entire premise that the treadmill can slow it down is via friction in the wheels. But friction is, essentially, turning motion into heat. Converting a quarter of million pounds of thrust into heat would completely melt the axle of the landing gear, or wherever this supposed friction is happening. And if somehow it doesn’t, if the plane is somehow melt-proof, now that heat is going to spread the only direction it can…towards the plane, roasting the people inside.

And please note I’m not a physicist and have basically a high school education+internet reading in this, and those two are just the _obvious_ problems that would show up even if you *had* a magical infinite speed treadmill. There are probably more.

In fact, in explaining the first problem up there, I realized another…this ‘friction’ idea basically requires all the wheels to work perfectly in sync. If you have the left side wheel having 0.0001% more friction (And an imbalance is fairly likely), when you ramp up the friction to this amount, you’ll end up with the airplane falling over. I don’t mean ‘steering to the left’, which is what the plane would very slightly do at low speed. This would not be correctable via steering better, this would be a spinout and tilt and flip.

Actually, just having the steering _very slightly_ not-straight would have the same problem. We’re talking about _massive_ friction_ counteracting _massive_ force. Throw a damn piece of paper in one wheel, and you’ve got an imbalance that, before anyone can react, has resulted in the plane cartwheeling down the treadmill.

At this point, considering how many exceptions to the laws of physics we’re having to make for this, we’re officially in ‘If people could walk through walls, what keeps them from falling through the floor’ question territory. I.e., this is a question that does not actually make any sense, physics wise, if you try to assume #3.

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33. Here’s my opinion (disclaimer: I’m neither a physicist nor a pilot, I’m just another opinionated dumb f**k like the rest of you). Its the speed of the wings relative to the air that makes the plane fly, not the speed of the plane relative to the ground (or treadmill in this case). The engines are what’s pushing the plane forward (not the wheels), so the treadmill can do any bloody thing it wants to without affecting the speed of the plane relative to the air. The wheels will spin at whatever RPM the treadmill makes them spin at combined with the forward velocity of the plane. If said treadmill is programmed to keep the wheels at 0 RPM, then the treadmill will just keep its velocity the same as the body of the plane, i.e. if the plane needs to go 200 MPH to take off, then the treadmill will be spinning at 200 MPH at that moment and the wheels will be at 0 RPM. The wings don’t care what speed the wheels are spinning, so off to Zimbabwe we go…

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35. If the airplane was on the water, wouldn’t it be essentially the same problem? We all know airplanes can take off from the water.

36. As a pilot, I have a two part suggestion based on my anecdotal experience. A different perspective perhaps.

Aircraft (almost) always take off into the wind to ensure they have enough room to generate the requisite lift in order to take-off.

A Cessna 172 (small, 4-seater) has a take-off speed of 56 knots. On a windless day, the aircraft will take off right around 56 knots (relative to the ground). But if I take off into a 10 knot headwind, I will reach take off speed at 46 knots (relative to the ground) because the air is passing over the wings at 56 knots and lift is achieved.

If I take off with the wind (not recommended), and I was flying with a 20 knot tailwind, I would have to reach 76 knots, relative to the ground, in order to take off to compensate.

So, on a treadmill, if a Cessna is facing a strong enough headwind, lift can be achieved even if the speed, relative to the ground, is 0.

However, a 747′s takeoff speed (on average) is about 170 knots, and it doesn’t have the small but not meaningless benefit of creating it’s own wind the way a prop plane does. The big girl isn’t going anywhere

37. In addition to the many worthy thoughts in this post , I just have one scenario I thought I might add:

I assume that when the treadmill starts spinning, the friction of the wheels will begin to push the plane backwards.

The jet engines could be adjusted to maintain a static equlibrium . Now pretend that we keep this equlibrium and accelerate the treadmill could the plane take off without actually moving forward reletive to the ground?

My answer (not exactly the original question I know): Probably not at full scale though lift would be generated, and at small scales…well maybe, if we adjusted our shrink ray gun properly.

Lift may be generated for two reasons I can think of:

1. From the air the engines move in order to compensate the wheel friction. This will cause some air to move past the wing and generate (some) lift.

2. The no slip condition between the air and the blelt will actually move air close to the belt along with it as long as the plane is within the boundary layer. This air will also genrate lift if it moves past the wing. At full scale I imagine that the net speed at about 3m height will provide negligible lift but a scaled down version (say we have a shrinking gun) will experience horizantal air displacement within the boundary layer of the conveyer belt. This may generate (some) lift.

So, if the conveyer belt is spun to a speed wher the jets are working at full capacity to keep the plane in place, at the right scale the plane could lift off….right?

38. A conveyor belt can prevent the plane from taking off.

Experiment: place a plane on a conveyor belt. Attach a spring to the nose of the plane, with the other end of the spring attached to a non-moving structure. Spin up the conveyor belt. See how the spring gets stretched?

And here comes the crucial part: the faster you spin the conveyor belt, the more the spring is stretched. This is because the plane’s wheels are on axles, and higher rotation rate means more friction, which means more backward force.

With some experimentation you can find a belt speed that causes a backward force exactly as great as the plane’s engines are able to provide at full thrust. Trivial vector addition: one backward force, plus second equal opposite force, means zero acceleration, results in stationary aircraft.

An even easier way to picture this is a water plane trying to take off upstream against a rapidly flowing river. Surely it is easy to see how the water can push back harder on the pontoons than the plane’s engines can pull forward.

The simple error Mythbusters did was to spin their conveyor belt too slow.

And please let’s not cheat by assuming frictionless axles or infinitely powerful engines; real universe physics please.

39. “And please let’s not cheat by assuming frictionless axles or infinitely powerful engines; real universe physics please.”
Then why are you assuming an infinitely powerful conveyor belt? There’s no conveyor belt in the entire planet capable of the speeds you’re asking for.

40. With some experimentation you can find a belt speed that causes a backward force exactly as great as the plane’s engines are able to provide at full thrust. Trivial vector addition: one backward force, plus second equal opposite force, means zero acceleration, results in stationary aircraft.

You apparently have no idea how much thrust an aircraft engine generates.

And please let’s not cheat by assuming frictionless axles or infinitely powerful engines; real universe physics please.

In real world physics, you just tried to absorb hundreds of thousand pounds of thrust via three free-wheeling axle friction to hold the plane in place. Axles, it must be noted, that are designed, like all wheel axles. to provide as little friction as possible.

Vs. 250,000 pounds of thrust.

This is basically the same concept as trying to stop a runaway train by throwing ping-pong balls at it, or thinking you can fly by fluttering your eyelashes. You’re a half a dozen order of magnitude off here. You’re not even close.

And not only is the entire concept patently absurd, but if you managed it, if you got it going that fast, you just turned 250,000 pounds of thrust into heat.

In three moderately small piece of metal.

So you just melted the axles, the wheels, and probably the entire landing gear in ‘real world physics’. (Which means the airplane will not take off because it was hurtled backwards down the runway.)

It’s the ‘plane would not take off’ side that needs insane physical rules. Under actual physics, it’s just completely impossible to build a treadmill that can generate enough backwards force to do anything. And if you bend the rules to just allow that, you’ve got an impossible amount of heat to deal with.

41. The ‘what if’ post is nice, but the combination of the post with the replies id *glorious*!
Everybody just blatantly ignores the post and plunges into the nonsensical discussion headfirst.

Randall, if you put this ‘what-if’ in your book, you’re morally (and comically) obliged to include the comments!

42. When I’m on the treadmill myself it doesn’t matter how
fast I move my legs, I do not move forward if the treadmill
is matching the speed of my legs. I work like hell but stay in the
same place. Same with the plane. If the plane is not moving
forward, there is no aerodynamic lift being created and thus
the plane does not get airborne.

43. @gary: The thing is, the plane *will* move forwards. Unlike you, an airplane has wheels. These wheels have very little friction, so they will spin freely in place but not transfer much force to the plane (or at any, if we have frictionless physics wheels). If the plane was on skids, you would be correct.

Also, this is the reason why people in the “take off” group think the naysayers think airplanes are powered by their wheels – because some of them do!

44. The plane eventually takes off.

There are three components here, not two. The plane, the treadmill, and the air. The treadmill is tuned to keep the plane’s speed with respect to the ground at zero. But as the treadmill speeds up, which it’s forced to do as the plane’s engines keep propelling it faster and faster along the treadmill itself, drag between the treadmill and the air accelerates the air. Eventually the plane reaches takeoff speed with respect to the air and liftoff is achieved.

At that point, the treadmill is no longer keeping the plane from moving forward with respect to the ground, and forward ground motion begins. If the portion of the treadmill in front of the plane is sufficiently long, the plane may be able to gain sufficient airspeed to stay aloft once it exits from the regime of the local area whose wind speed is controlled by the treadmill. But if the treadmill in front of the plane isn’t very long, then the plane, moving forward at say 200 mph WRT to the artificially sped-up air above the treadmill (but not nearly that fast WRT the ground), will reach the end of the treadmill and quickly encounter less and less headwind. In that case it will reach stall speed and nose into the ground.

A viewer placed, say on a hill several miles away, directly to the side of the cockpit, will hear the plane engines ramping up. They will see the treadmill move faster and faster (we can color it like a rainbow to make it easy to gauge the speed). Then at some point, the plane will slowly lift off the ground, in what appears to the ground observer to be straight up motion. Then forward motion will slowly occur. To the ground observer it will look like the plane took off with zero forward speed, and that it’s drifting forward at 1 mph, 2 mph… But that’s only ground speed. The airspeed will be the normal takeoff speed for that airplane.