Comments on Airplane on a Treadmill: Essential fact(s)

What if it's a helicopter on a treadmill?

2012-02-03T17:26:09.643-05:00

What if it's a helicopter on a treadmill?

The plane on a treadmill would take off and fly af...

2011-08-16T19:21:28.958-04:00

The plane on a treadmill would take off and fly after slightly less distance than normal. The treadmill would assist the plane in flying by creating a slight head wind via the no slip boundary condition.

The difference between the flying and not-flying a...

2011-01-30T21:41:50.315-05:00

The difference between the flying and not-flying arguments is that the not-flying arguments assume that the airplane uses it's wheels for movement on the ground, otherwise the treadmill makes no difference, as stated by the flying-argument people. You're all a bunch of idiots.

I'm a no-fly, but I do believe the plane will ...

2011-01-21T14:58:34.845-05:00

I'm a no-fly, but I do believe the plane will move. :)

So if the plane is not turned on, we can admit that the treadmill will move the plane backwards.

Now if we turn the plane on, it will begin pulling against the air. Assuming relatively stationary air, the plane will thus begin moving forward. For all the true no-flies out there, this thought should help you:

Imagine it's not a plane. Imagine you're standing on a treadmill with roller blades. If the treadmill turns on, you will begin moving backward. Now let's assume you have a rope tied to the front of the treadmill. If you were to hold onto that rope, would the treadmill still move you backwards? No. You'd remain stationary. Now imagine you were able to pull yourself along that rope. Would you move forward? Of course, you would.

No matter how fast the treadmill moves, you would be able to move along since your means of propulsion is completely irrelevant to the treadmill. This is the same truth for the airplane.

However, my thing was that the treadmill is always matching the speed of the wheels, so this provides an ever-increasing speed. I believe the tires would explode/melt, and the plane would crash into the ground and be unable to move. :P

I have truly lost my faith in human nature. Here&...

2010-12-16T02:42:09.155-05:00

I have truly lost my faith in human nature.

Here's the problem with the no-fly crowd:

"no flys read: The treadmill matches the ROTATION speed of the wheels."

This is impossible.

The rotational speed of the wheels (by which I assume we mean their circumference times their RPM) is the forward velocity of their axle mount relative to the surface that the wheels are resting on. If, as in this case, the surface is moving backwards, then their speed can be determined by adding the backward velocity of the surface to the forward velocity of the axle.

Thus, the rotational speed in this case is the forward speed of the airplane plus the backward speed of the treadmill.

S_wheel := S_plane + S_treadmill

Saying that the speed of the treadmill "matches the rotational speed of the wheels" (S_treadmill := S_wheel) is meaningless, because the rotational speed of the wheels depends on the treadmill's speed. That would make the speed of the treadmill equal to the speed of the plane plus its own speed, which doesn't make any sense. That's like saying, "Your speed was 10 mph more than your speed." It's nonsense.

As far as understanding it is concerned:

Imagine that the plane is sitting on the treadmill as before, but that it never turns its engines on. Instead, a pair of helicopters hover, tied with long ropes to each wing. The helicopters begin to fly forward (not upward), causing the plane to roll forward. Will turning on the treadmill at any speed stop the plane from moving forward? No. Even if the wheels spin so fast against the treadmill that they melt and collapse, the helicopters will still pull the plane, dragging it forward relative to the surrounding air as the treadmill spins madly underneath the plane's belly.

Now imagine that instead of using ropes, we've strapped the helicopters to the wings in a forward-facing configuration so that they will only generate forward thrust.

Now imagine that these helicopters are small, have no cockpit or tail. That's what an engine is. Just like the helicopters in the original example, the engines have nothing to do with the motion of the wheels.

It's not sufficient that the plane move. It h...

2010-09-14T10:46:46.498-04:00

It's not sufficient that the plane move. It has to move at an airspeed equal to to the speed necessary for the lift of the wings to exceed the weight of the plane. Every plane has a maximum attainable airspeed with the given thrust of the engine. This is based on a number of factors including the drag of the plane, the drop of propeller efficacy due to speed, and, significantly, axle friction. Axle friction is also proportional to weight. For light planes the drag is far more significant than axle friction. But for heavy planes, loaded to maximum capacity, such that the engine can only just achieve the necessary speed, doubling the wheel speed will increase the friction to the point where the critical air speed cannot be achieved and the plane will not take off.

"...you would have to run the treadmill backw...

2010-07-29T23:20:16.019-04:00

"...you would have to run the treadmill backward at a speed of 1,725 mph to keep [Cessna 182] stationary at full power."

Thankyou so much for providing a concrete, tangible proof, but not how you think.

A Cessna 182 will take off at about 70mph. So basically, the plane will have lifted off the conveyor belt long before the rolling friction force is high enough to hold it back at full engine power. At this point, the conveyor belt will be travelling at 70mph in the reverse direction.

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2010-07-09T12:51:25.676-04:00

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The plane won't move, because the treadmill wi...

2010-05-21T18:16:00.752-04:00

The plane won't move, because the treadmill will attempt to match its wheel speed. The treadmill's speed will approach the speed of light extremely fast, increasing the mass of the treadmill until it collapses into a black hole, pulling the plane and the whole Earth in. Then the plane won't be going anywhere anymore, ever.

Wow, jh, that was a new one. Yeah, I think that...

2010-05-21T16:09:07.392-04:00

Wow, jh, that was a new one. Yeah, I think that's one possible answer to the problem, too. Insanely fast moving treadmill pushing the air over the plane's wings.. LOL! Too bad the treadmill can't be made frictionless to prevent it from having any effect on the air, as that would defeat the treadmill's purpose. Maybe one could assume frictionless air, though?

Amusingly the problem has many possible answers, because of the convention in word problems that one can ignore whatever aspects of reality one finds convenient, such as unavailability of treadmills capable of insane speeds, lack of frictionless wheels or unbreakable wheels with friction, unphysicality of infinite (not merely arbitrarily large) speeds, theory of relativity, and so on. But this "controversy" would be distinctly dull without that convention.

The plane will fly. However, it does not have to m...

2010-05-17T05:51:03.973-04:00

The plane will fly. However, it does not have to move forward.
I know it is the airflow on the wings that create lift in an airplane, so let me explain.

Let's imagine a propeller plane starting its engine and going at low rpm. The engine will create a thrust and maybe it will move forward one millimetre. The treadmill will react and start going at a speed to counteract the plane’s movement. This will create a rolling resistance on the plane’s wheels. However, this resistance depends on the tyre, the surface it is rolling against, the bearings and the normal force on the wheels (as we have zero lift for the moment, that means the plane’s weight), but it is little dependant on speed. That means that in order to counteract a small amount of propeller thrust, the treadmill will have to go extremely fast.

Now let’s assume we have a long (let’s say >20 times the plane’s length) and wide enough (at least twice the plane’s span) treadmill, and capable of achieve ridiculously high speeds.

The movement of the treadmill will cause a backwards laminar flow of air near its surface, due to the air’s viscosity. See http://en.wikipedia.org/wiki/Viscosity. The maximum air speed will be at the treadmill’s surface, and will decrease with height. This air speed, (acting for the moment significantly only in the lower regions of the plane), together with the rolling resistance in the wheels, with counteract the propeller’s thrust. If we increase the engine’s rpm the treadmill will have to move faster, pushing air backwards faster and faster. Eventually, the air speed at the wings, produced by the movement of the surface of the treadmill will create enough lift and the plane will take off, without moving in relation with the ground.

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"Unless you convey the air around the plane, ...

2010-04-01T13:42:34.726-04:00

"Unless you convey the air around the plane, it is just going to move forward through the air with the treadmill having NO effect what so ever. Air planes uses air to propel forward, not the ground. Like many have stressed before, IT'S THAT SIMPLE. Subject over."

This was my original comment. When I said the tread mill would have no effect, I only meant that it wouldn't create the effect that everyone was thinking. If one was to increase the speed of the "treadmill" to generate enough resistance to hold the craft in place, not only would it be going extremely fast, I don't think the aircraft would just sit there either. Try to image an aircraft landing with the landing gears locked... this would be similar to the effect caused by this "treadmill" idea... that is, if it was going fast enough and maybe not as severe. If it isn't going fast enough, and it would have to be pretty freak'n fast, then the craft will simply take off with, yes, some resistance being applied but not enough to stop the plane. I would put any amount of money on this. "Airplane on a treadmill" is a fun subject to talk about but that is all it's good for. If it was of any use, we wouldn't be using giant steam driven pistons to launch jet fighters off of aircraft carriers.

You're right of course but the problem is tend...

2010-03-29T17:44:03.887-04:00

You're right of course but the problem is tendency to obscure the core discussion behind categorical claims and otherwise insulting rhetoric which only exacerbates the issue by forcing people to take a side, instead of participating in what could be a meaningful and enlightening discussion on the nature of the problem.

Too many comments at large and in this thread dismiss any arguments made for the opposing view as stupid, pointless or irrelevant and the posters make no effort whatsoever to consider the other camp's arguments. To them it's all about winning some silly contest that exists solely because a question that looks simple is anything but, the discussion itself and even the problem is of no interest to them, just the victory.

This would not be a problem either to many as we can see beyond these twits to those putting forth meaningful answers and try to have a discussion with them about it while shutting out the background noise. And the twits themselves can continue what they do and contribute to the noise. But this is a problem to people who have a brain but a passing interest in the problem and arrive late to the conversation, they may not know enough about the issue and the physics of airplanes taking off to effectively ignore the twit background noise and get to the real conversation around the issue. So he will, at best, move on and not participate or at worse chose a camp and become part of the problem.

This is why it's good to remind everyone that both sides of the conversation have valid points that need to be considered.

This is true of just about any discussion, in any forum internet and otherwise, but it is especially true for this airplane on treadmill problem which seems to attract a higher than usual proportion of close minded blowhards calling everyone stupid or other less flattering names.

In general I try and obey the following rule: never assume someone who disagrees with you is stupid. If he is and you treat him in a civilized manner and consider his opinion carefully then that make you open minded. If he's not and you carefully consider his argument then that makes you wiser. If he's stupid and you dismiss his argument out of hand then that makes you an ass. If he isn't and you dismiss his argument out of hand then that makes you a dumbass. I can think of very very few situations where that rule doesn't apply.

Thanks Frédéric for this very sensible comment. I&...

2010-03-29T09:25:41.781-04:00

Thanks Frédéric for this very sensible comment. I've found myself in either one of both camps at one time in the past and I've come to pretty much the same conclusion.

However, I still very much enjoy the occasional reply to this discussion, just to see how far this person is in thinking about the problem.
The discussion is more fun and intriging than the actual problem is.

Marc

Seems strange to me how categorical both camps are...

2010-03-29T09:07:19.309-04:00

Seems strange to me how categorical both camps are around this issue, and how otherwise intelligent people seems to completely refuse to consider or even hear seemingly valid arguments from the other side.

To those who are convinced the plane will take off:
Do you agree that the wheels impart some rolling resistance to the takeoff speed? If so what if that resistance were to be increased? Using, say, brakes. If we were to apply brakes to the wheels with enough force to keep the plane from moving would the plane take off?

Okay now lets say that instead of brakes we simply increased ground speed to a degree where rolling resistance is as high as it would be if we were braking the wheels. Admittedly this would require insane ground speed, and many many other things could go wrong, tires would probably burst from heat caused by internal deformation friction, or from centrifugal force. Bearings would fly apart, or something else would break. Not to mention the fact that a treadmill moving at a speed of probably a few thousand miles per hour (I do not have typical landing gear rolling resistance numbers handy at the moment) does not exist at the moment I would think... But it could still increase rolling resistance to a point where it would completely offset the thrust from the plane's engines correct?

Read the question again... Could it possibly be interpreted that way? I believe it can.

To the other camp: Can we agree that rolling resistance is pretty small? Airplanes have taken off on skids and pontoons, both of which create much more resistance than a modern landing gear assembly. And that if ground speed increases at double the rate of the airplane's forward movement the effect on rolling resistance is minor?

Can the question be interpreted in this way? I believe it can.

The only person that can settle this is the person asking the question. And he can do this by clarifying one simple point: is the treadmill's speed directly opposite the wheel's forward movement? Or their turning speed? The question is unclear on this point and it is driving the internet insane! Please help!

Unless you convey the air around the plane, it is ...

2010-03-28T15:29:56.789-04:00

Unless you convey the air around the plane, it is just going to move forward through the air with the treadmill having NO effect what so ever. Air planes uses air to propel forward, not the ground. Like many have stressed before, IT'S THAT SIMPLE. Subject over.

Airplanes require the movement of air over the win...

2010-03-27T18:47:27.966-04:00

Airplanes require the movement of air over the wings to provide lift right?

When you run on a treadmill how much wind do you feel blowing by your face?

No, you're wrong. The plane won't move bec...

2010-03-22T14:59:26.427-04:00

No, you're wrong. The plane won't move because the treadmill will continually push it back. Since it doesn't move, it won't take off. You guys need some serious help.

There is absolutely, without a pilot conspiracy...

2010-03-16T21:33:49.743-04:00
There is absolutely, without a pilot conspiracy, no way for the treadmill to keep the plane from taking off.

Sure it can. The problem is you're comparing forces over different distances. In such cases you need to compare work, which takes distance into consideration. Even easier would be to compare power (force*distance/time).

P_e - Power output of airplane's powerplant.

C_r - Rolling resistance of the airplane's wheels.

V_t - Velocity of the treadmill.

Basically:

P_e = - V_t * C_r

Or, if you want to know how fast the treadmill needs to go to keep the aircraft stationary at full power?

V_t = - P_e / C_r

A Cessna 182 has a 230 horsepower engine. It also has a rolling resistance of about 50 pounds. Assuming an ideal prop, you would have to run the treadmill backward at a speed of 1,725 mph to keep it stationary at full power.

Deleted previous comment, must remember to proofre...

2010-02-15T16:18:14.702-05:00
Deleted previous comment, must remember to proofread...

But 'Matching Speed' is the entire point of the exercise. I truly believe that if the problem was worded as 'the treadmill moves at twice the takeoff speed of the plane' or something to that effect there would be no controversy. In that case it's perfectly obvious that the treadmill would have little effect on the plane, as has been demonstrated in real life.

However, things get interesting then you use the term 'matching speed', because even though this could never happen in real life as this is physically impossible, this is about the most widely known infinity jumps in physics. See if you applied but a bit of thrust to an airplane and a treadmill was setup to perfectly counter the wheel's speed, assuming an also physically impossible frictionless axle, it would instantaneously reach infinite speed. See the moment the plane starts to move, if even just a little, the treadmill would start in the opposite direction, which would accelerate the wheels, which would accelerate the treadmill more, accelerating the wheels more... And no friction or upper limit on treadmill speed means means this happens instantaneously, there is no curve unless we also say there is a reaction lag between wheel and treadmill, and if so the plane could very likely take off as the lag would be exploited, and neither the wheel nor the treadmill would reach infinite speed, just very high speed.

Now then, it follows that in order to take off the wheel speed would need to be infinity + takeoff speed. A flat impossibility since any speed increase is immediately counteracted by the treadmill. This is a paradox situation, a variation of Zeno's paradox, which states that you can never reach any point because you must always first traverse half of the distance between your position and your destination. And this is infinitely divisible.

Same principle with the treadmill. But this is interesting to study nonetheless for the same reason infinity jumps are interesting when plotting functions.

As I keep saying though, if there is some friction, any friction at all, on the axle then the wheel assembly would very quickly self destruct under the tremendous friction heat caused by the very high speed of the wheel, which exploits a loophole in the question as it makes no mention of what should happen if the wheel is destroyed, does the treadmill stop? Does it continue, offsetting forward movement of the plane as it attempts a belly take-off? The question doesn't say.

Which exposes the sheer lunacy of the question in the fist place, and brings into rather sharp contrast the fact that this problem is a linguistic one, not a physics problem. IE the question is not correctly framed and allows two or more interpretations, which has resulted in a fiery and rather pointless internet debate between people who think someone might do this, physically, for some reason and others who are merely interested in conducting a thought experiment using unrealistic conditions.

The debate is kinda pointless because the answer is known, and it is demonstrable: it depends. On what is meant by matching opposing speed, the conditions of the experiment and which aspect you're testing. The problem is that this is a poorly worded question.

2010-02-15T16:09:57.912-05:00
This comment has been removed by the author.

I think it's funny how so many people take the...

2010-02-15T15:22:22.007-05:00
I think it's funny how so many people take their incorrect assumption that the plane remains stationary, and then pretend like that was an explicit condition of the original question. It's not simply assumed that the conditions are right for the plane to sit still, because evaluating why those conditions aren't right is a major point of the exercise!

I also think the wording of the question itself leaves something to be desired. The language of "matching speeds" seems to lead some people in the wrong direction. (even though it clearly shouldn't, since most of us don't have problems with it) Just say that the conveyor belt will move backwards at the take-off speed of the plane, or twice that or something. It needs an explicit realistic upper limit, since the spirit of the question obviously isn't whether you could produce enough rearward force with unlimited conveyor speed or acceleration.

Haha to all of the people that think "the whe...

2010-01-11T15:56:53.506-05:00
Haha to all of the people that think "the wheels don't power the plane" is an acceptable counterargument. I propose a second thought experiment:

I have a car with a super rigid frame and a friction plate on top of it. I put a 600,000 tonne press with a counter friction plate on it that's fixed to the ground. Will the car move forward? Nope, the engine can't overcome the friction from my press. Does that mean that normal car motion is caused by the friction on their roofs?

Conversely, the question has little to do with friction. It's just defined two different competing assumptions and asked what the resulting behavior is. The treadmill exactly cancels the motion of the wheels so the plane is stationary. Even in the frictionless bearing case - it's defined that way in the problem. Thus we get a debate. 5+x=4, x>0. Solve for x.

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