Plane on a Treadmill - SOLUTION

Cecil's Columns/Staff Reports
61

Oh, sure. I agree with that.

Sure you can. That’s why I quoted the section from the first column. It just depends on what your interpretation of “the speed of the wheels” is. If it’s “the speed of the outside of the tire with respect to the hub” or “the speed registered by a speedometer connected to the wheels” then that leads to BR2. Other interpretations are possible, of course, and there’s nothing wrong with that.

Or perhaps (CLEARLY) to make people stop and think that airspeed, and not groundspeed, is critical for liftoff. Or, more likely, the phrasing has become muddled over time, and it’s not really possible to say with certainty what the original intent was.

62

Yes, that would likely be the case. :slight_smile:
I think I’ll give this problem to high school classes as an interesting constructivist activity. I’m thinking water guns at fifty paces might be needed… :stuck_out_tongue:

63

OMG, it’s back. :eek:

64

Well…as I mentioned when this made the big hit, if the treadmill moves fast enough, it can create enough friction at the axle of the wheels so that the propulsion, rather than moving the plane forward, will simply keep the plane sitting in place.

If we are to assume that the treadmill can and will try to speed up to keep the plane stationary, friction does allow it to.

Muahahahah!

65

I think I can offer some info for those looking at a realistic approach to this problem. Aircraft have max rated wheel speeds, on the ground obviously, above which the tires and bearings may overheat. So, even if it was at twice normal wheel speed, the tires and bearings will likely fail before the aircraft could reach take off airspeed, assuming a large aircraft. There is a surprising amount of friction on aircraft wheel assemblies even when not loaded. Even on jacks a wheel takes some effort to rotate. Just my 2c

66

Now that this thread has been resurrected for a while, let me repost something that I’ve posted a few times before:

There are plenty of answers to this question, because the key to the question is the wording and your interpretation and what you assume from the beginning. And these answers can all be correct, but the assumptions are the key. Let’s start off at the top:

A. Suppose we actually built a treadmill and put a 747 on it, and had the treadmill match the speed of the plane. Would the 747 take off? If the treadmill matches the plane fuselage speed, then yes. The treadmill simply accelerates in the opposite direction that the plane does. The wheels wind up rotating twice as fast as they normally would, but the plane will take off, leaving a treadmill behind that’s rotating in the opposite direction.

B. Let’s reword the question. Suppose we actually built a treadmill and put a 747 on it, and had the treadmill match the speed of the wheels. Would the 747 take off? Depends. If “exactly matching the speed of the wheels” means that the treadmill matches the hub speed of the wheels (the speed of the wheel center, which is the same as the fuselage speed), then yes. Just like in the last scenario, the treadmill accelerates in the opposite direction that the plane does, the wheels rotate twice as fast as they normally would, and the plane takes off.

C. But that problem is trivial. Let’s assume that “exactly matching the speed of the wheels” means “matching the outer diameter surface velocity”–the velocity with respect to the hub, or the “speedometer” speed. Would the 747 take off? Almost certainly it would, but only because we can’t build a treadmill capable of keeping up with the thrust transmitted to the plane by the engines–in other words, we violate the spirit of the question, because the treadmill isn’t matching the wheel velocity.

D. OK, that’s stupid. It’s a thought experiment. Posit a magic treadmill that can accelerate as fast as desired. And it doesn’t break. I imagine the wheels will skid on the treadmill, because the friction won’t be able to transmit the necessary force. In that case, we again violate the spirit of the question, and–

E. It’s a thought experiment, smart guy. Assume there’s enough friction to rotate the tires. All right. When the engine lights off, the treadmill will accelerate until the force transmitted through the wheel hub to the plane exactly balances the thrust. The plane would stay stationary as the thrust power was dissipated in the wheel bearings (as friction), tires (hysteresis), and in accelerating the wheel to ever-increasing speeds. Since all the power is dissipated in the wheels, eventually either the bearings would overheat, the tires would blow, or the wheel would rip itself apart due to inertial forces. After that, the plane crashes and burns. Then you’ve destroyed a rather expensive magic treadmill.

F. Thought experiment, I said! Let’s posit ultra-strong and heat resistant tires. All right. It turns out the real world is rather complicated. If the treadmill is a long, runway-sized treadmill, it will eventually, running thousands of miles an hour, pull in air at high enough velocity that the plane will lift off at zero ground speed (but substantial air speed). However, now you’re running into trans-sonic compressibility effects…

G. No speed of sound effects! And assume magic air that doesn’t become entrained with the treadmill motion. And don’t throw in any other crazy stuff, either. In that case, the treadmill speeds up (still balancing the plane’s thrust force) and the plane stays in place until the engines run out of fuel. I imagine the treadmill goes pretty fast at that point. The plane stays put until the fuel’s gone, at which point the magic treadmill whips it backwards.

H. Backwards, shmackwards. Now we’re getting somewhere. What if we had infinite fuel? Then the wheels keep going until they’re running near light speed, and relativistic effects take over. The wheels get smaller, I suppose…

I. None of that! No relativity-- Hey, wait a minute. Back up. Suppose we have zero friction bearings and tires. That doesn’t seem so unreasonable for a thought experiment. Well, zero friction tires would mean they just skid on the runway, since nothing turns them. So the plane will take off, tires motionless, and the treadmill won’t move.

J. Hey! Quit it! I already said the tires don’t skid! Sorry. Just friction on the tire/treadmill interface, then, but none in the bearing or sidewall. With zero friction in the bearing, you lose the friction coupling between the treadmill and the jet. But you still have inertial coupling. The wheels accelerate, and that acceleration takes force. Now you have the same case as you do with friction. The jet stays stationary as the wheel accelerates; the wheel just accelerates faster.

K. Well, how about the other way around? Massless wheels, but you still have friction? Here it starts to get complex. As you accelerate the wheels, the bearings will change shape and heat up and so forth, so it’s reasonable to guess that the “friction coefficient” goes up with increasing speed. If that’s the case, then when the engines start, the treadmill accelerates up to whatever speed will give enough friction to balance the thrust. The plane stays stationary, wheels rotating at some reasonably constant (but large) velocity, dissapating the engine power through friction.

L. But I want massless wheels and a constant coefficient of friction. Indestructable wheels, remember? None of this hand-waving “it’s gonna get bigger” crap. OK. It is a thought experiment. With a limited “friction coefficient,” only a limited amount of energy can be absorbed by the friction. When the engine lights off, the treadmill instantly accelerates to infinite speed. It’s never able to counteract the thrust force, and thus plane takes off, leaving the infinite-speed treadmill behind.

M. Ah. OK, one last step. What if we had no bearing friction and massless tires? What happens then? Pretty much the same thing. There’s now no energy losses in the wheels and tires, no coupling between the treadmill and the plane–no bearing friction, no inertial effects, no air resistance, and no way for the treadmill to affect the plane’s motion. The same thing would happen as above, with the plane taking off, leaving the infinite-speed treadmill behind. However, there’s one added interesting thing: This is now an unstable runaway system. There’s no resistance to treadmill motion, and a positive feedback circuit. Imagine the poor mechanic who bumps a wheel, setting it in motion. A very slight roll by the tire is sensed, and the treadmill luches forward. The tire goes faster, the treadmill goes faster, the tire goes faster… Since we’ve posited an instantly-accelerating treadmill and no relativity and no air resistance and no wheel inertia, the treadmill goes from zero to infinity in no time flat. Try to keep your balance on that.

Pick your scenario–they’re all correct.

67

Excellent. There is one additional scenario. If the magic treadmill has zero friction in the direction opposite the plane’s moement then it presumably has zero friction in the smae direction the plane is moving. In that case as the plane moves forward the whelels will not move (they have intertial resisitance) and the treadmill will move with the wheels. Since the wheels are not rotating they have no speed according to your scenario C, and the magic treadmill will not provide any force to resist them. In this case the airplane takes off and the wheels do not rotate.

68

That should be scenario I, I believe.

I might be wrong about this, but I seem to recall someone posting that landing speeds (which the tires naturally have to withstand) are over twice takeoff speeds, so even at double-speed, the tires should be under their maximum rating. Nonetheless., a good point.

69

This wouldn’t be true in general. Landing is almost always done at a lower weight than takeoff (due to fuel burned), so minimum controllable airspeed will be lower (the speed ratio will be the square root of the weight ratio). There might be sound reasons to land at a bit higher than minimum controllable airspeed, but certainly nothing like twice as fast.

Nonetheless, unless the treadmill is allowed to accelerate without limit, put me in the “plane takes off” camp.

70

This is the one that messes me up. I am not intending to disagree, I’m just having trouble wrapping my head around this.

I don’t understand why the wheels wind up rotating twice as fast as they normally would. Lets say the plane needed air traveling over the wing at 2 miles per hour in order to take off.

Normal “real world” scenario. Plane’s engine is pushing the plane forward at 1 mile per hour. Plane rolls onto a treadmill, and the treadmill comes up to a speed of 1 mile per hour. Plane stops moving forward. Plane’s wheels are moving at the same 1 mile per hour.

Plane accelerates to compensate… such that were it not on a treadmill, it would be traveling at 2 miles per hour. Treadmill increases speed at the same rate to 2 miles per hour to compensate. Plane never moves forward. Plane’s wheels now moving at the same 2 miles per hour that the treadmill is moving backward. No air is moving over the wing, plane remains stationary on the ground.

I don’t see where anything in the treadmill needs to be able to be capable of infinite acceleration (thus taking this out of the real world, into the realm of ideal physics) or the wheels move at twice as fast as they normally would.

Can someone please explain to me the part of this mechanism that’s like a rope, tied to a wall beyond the treadmill that the plane is pulling on to move forward?

I don’t think I’m moving from the real world (BR1) to the ideal/theoretical (BR2) unless you account for the treadmill’s ability to compensate smoothly and perfectly matching the plane’s acceleration .

I am seeing where there might be an issue if we were talking about a prop driven plane where the engine is pushing air over the wing surfaces, but i am not even sure that these forces would have enough effect.

(Yea, sorry folks, it got linked on BoingBoing to article to an article in the Times, which lead back here… the discussion is coming back… and the hamsters are gonna get a work out.)

71

What is the sound of one thread awakening?

72

The issue is how the problem is worded and the interpretation of the wording. In this particular case, the wording is, “the treadmill matches the speed of the plane.” There are different interpretations of just what the “speed of the plane” means. Let’s take a look at how you’re imaging the scenario:

OK, there’s nothing wrong with your interpretation of the problem. However, take a look at what you just wrote. You said “the treadmill comes up to a speed of 1 mile per hour” and then the “plane stops moving forward.”

So. How can the speed of the treadmill match the speed of the plane if the treadmill is moving and the plane isn’t?

It depends, of course, on the interpretation. In your scenario, the speed of the plane with respect to the treadmill matches the speed of the treadmill with respect to the ground. Nothing wrong with that. However, other people interpret the question as requiring the speed of the plane with respect to the ground to match the speed of the treadmill with respect to the ground.

In that case, the plane’s speed with respect to the treadmill is twice its speed with respect to the ground. Since (we assume) the plane needs to hit a certain speed with respect to the ground to take off, the wheels (which are running against the treadmill) must be going double-speed.

Two different scenarios here. The second (the double-speed wheels) I covered above. The first (infinite accelerations) occurs when one assumes (as you did) that the treadmill will compensate speed to keep the plane stationary with respect to the ground.

In this case, ask yourself this: what limits how fast the plane is trying to go? You arbitrarily cut off your example when the treadmill goes two miles an hour. However, unlike a plane on a non-moving runway, resistance to forward motion doesn’t increase with increasing speed. So, with a particular amount of engine thrust, nothing prevents the plane from trying to go faster and faster and faster.

Not me. It’s a superficially useful but ultimately misleading analogy. I don’t think using it does anyone any favors.

Uh oh. I notice they linked to Cecil’s stupid answer, too.

73

See, that’s just the mental leap that keeps whooshing me.

If the plane has sufficient speed relative to the ground (or more accurately the speed of air over the wings), then it has to be able to take off period. If the plane’s take off velocity is air moving over the wings at 2 miles per hour, it doesn’t matter if the wheels are moving 2 miles an hour or 4 miles an hour, it’s going to take off. The only way I can see the “problem” being any sort of a puzzle at all is if the plane is kept stationary relative to the ground by the speed of the treadmill.

And I think that’s the scenario you were explaining when we got to…

Well, I would limit that to the speed the plane’s engine is capable of. It’s not like a 747’s engines can keep accelerating into infinity. Sooner or later they are going to run out of fuel or we’re going to get a mechanical failure from some part of the system. The only reason I didn’t continue going up the speed scale is I didn’t see how doing so enhanced my illustration.

Umm… ok, you have just confused me… Resistance to forward motion does not increase with increasing speed means what in this discussion? The only things increasing in speed in my scenario is the wheels and the treadmill. The plane’s engines are certainly increasing their levels of thrust, but isn’t that limited by the horsepower of the engine?

To elaborate, it seems like you’re telling me that even without increasing the thrust, the plane will continue to accelerate and the treadmill will have to go faster and faster and that’s not making sense to me.

OK… I just re-read that last quote. I’m pretty sure you’re discussing an element of fluid dynamics that comes into effect if the plane is actually moving relative to the ground, but can’t then follow that to a point that supports any supposition we’ve mentioned.

74

zut is awesome. I hope, at this point, we all agree that when the belt matches the plane’s air speed/relative ground speed, the plane takes off. But I think zut has made the “belt matches wheel speed” answer far too complex. And Cecil? Well he’s just plain wrong.

Think of a toy airplane with free-spinning wheels, just as a real airplane has free spinning wheels (like rollerblades, but unlike a car, a concept we all should agree on at this point). If I were to hold the plane in my hand and set it down on an actual treadmill traveling at 10 mph, the treadmill itself would push the wheels at 10 mph in the opposite direction, with zero thrust applied. And the plane, being pinched steady in my hand, would have an airspeed of 0 mph. Increase the speed of the treadmill to 20 mph, and the plane’s wheels would travel in the opposite direction at 20 mph, and the plane would still be held steady in my hand with an airspeed of 0 mph. Therefore, the paradox does not exist as Cecil suggests. There is no requirement whatsoever that the belt speed would force the wheel speed to double. Now, of course, I could simply move my hand forward independently of the wheels and overcome the speed of the treadmill — the way thrust would work, the way pulling the rope would work if I were wearing rollerblades on a treadmill — and the plane would move forward, attain lift and take off. And in the real world, that’s exactly what would happen, and the plane would fly. A-ha? I got myself? No. The instant my thrust overcomes the friction and the speed of the treadmill, the wheels would be moving forward faster than the treadmill is moving backward, thus violating the spirit and rules of the question that the treadmill and wheels are always moving at identical, opposite speeds. In order to adhere to the rules of that question, thrust would never be allowed to overcome the friction, the wheels would never spin faster than the treadmill, and any actual forward motion is disallowed by the rules themselves. Therefore, so long as the treadmill can INSTANTLY adjust for any change in wheel speed and always match that wheel speed, the plane will continue to have an airspeed of 0 mph. 300 mph belt, 300 mph wheels, 0 mph airspeed. No airspeed, no lift, no flight. It may be overly strict, and I agree it’s a poorly worded, unintentional variation of the REAL question where the belt matches the plane’s speed. It’s no longer about physics, it’s about semantics. The question itself dictates that forward motion — any airspeed or relative ground speed greater than 0 mph — simply cannot occur, because that would require that the wheels move faster than the belt. Forget all the other complexities outside that simple argument. The plane cannot, and will not, take off under these rules.

75

Well, sure, I happen to agree. But most versions of the problem I’ve seen could be interpreted either way, and most of the arguments about this “puzzle” I’ve seen are between people who interpret the plane’s motion one way or another, and never check to see what the other person’s assumptions are. In any case, it strikes me as more difficult and less enlightening to argue what the interpretation ought to be than to just acknowledge that there are multiple interpretations and move on.

Well, now wait. You’re confusing yourself here, I think, with what you’re calling “speed.” (Or I’ve confused you. Whatever.) Let’s assume the treadmill tries to keep the plane stationalry with respect to the ground, right?

When the plane’s engines light off, they produce thrust that pushes the plane forward. In order for the plane to remain stationary, the treadmill must do something to counter that thrust force.

About the only thing the treadmill can do to counteract the thrust force is to accelerate the wheels, thereby transmitting a force.

How else would the treadmill hold the plane stationary?

Are they? Why would you think so? I mean, they could be, but they don’t have to be.

Exactly. That is, assuming you mean “the plane will continue to produce thrust, and the treadmill will continue accelerating to make the plane stay stationary with respect to the ground.” Why wouldn’t the treadmill go faster and faster? What would cause the treadmill to suddenly balance the thrust force from the plane?
(Note that there could be many different answers here, because it’s possible friction increases with bearing speed, and tires will ultimately fail, and so forth and so on. But this is a thought experiment, and it’s up to you to outline your assumptions.)

76

Ok… Let me back up and check my assumptions again.

A place, given a certain level of thrust (x food pounds) will travel along a normal runway at speed (y miles per hour). If, instead of a normal runway, that plane is traveling along a treadmill that is moving in the opposite direction of the plane’s travel also at speed y miles per hour, the whole thing will remain static, the plane will not move relative to the ground. At the same level of thrust, the plane will remain in the same place.

As long as any change in thrust on the part of the plane is matched immediately by a corresponding change in speed of the treadmill, the plane will remain stationary relative to the ground.

There is no speed at which this system breaks down if the maximum speed of the treadmill is greater than or equal to the maximum speed that the plane’s maximum thrust can provide.

77

Really, I did have lunch already…

foot pounds

78

There’s the problem.

If a plane is moving down a runway at a particular speed (y miles per hour in your example), much of the thrust force is doing one of two things: either accelerating the airplane to go faster, or overcoming wind resistance.

Since, in the treadmill example, you’re assuming that the plane doesn’t move with respect to the ground (or air), then it doesn’t accelerate, and there’s no wind resistance to overcome.

A plane moving y miles per hour on a runway and a stationary plane on treadmill wich moves at y miles per hour are two different things. It’s that difference that makes the problem more complex.

79

I am becoming increasingly confused, but now the confusion is more of the “where’s the difficult part of this puzzle” nature.

Ok, yes there is a difference is the plane on the treadmill vs a plane that is moving relative to the ground because you will get wind resistance if the plane is moving relative to the ground. I can see that.

But the only different I see there is at a lower level of thrust will be required to maintain the same “speed”. The example still seems to hold that as long a the treadmill keeps the plane motionless relative to the ground, and thus no air moves over the plane’s wing surfaces, that airplane will not leave the ground.

The guy in the cube next to me likes the realistic view… the plane rolls onto the treadmill, the treadmill is crushed by the weight of the plane, the plane takes off.

80

Let me paste in my email to BB.

In summary, if the wheels have no friction and the engine can be turned on, then BAM paradox. The belt cannot fulfill its definition, so the whole problem is nonsense, unless we have slippy-slidyness.

If the wheels have friction, then the friction from the belt on the wheels provides backward force to counteract any thrust, and the velocity changes so that these balance out, keeping the plane stationary.

In the first case, the universe explodes in a puff of logic. In the second case, the plane cannot move. End of story.