In any realistic situation, the plane will not remain stationary. Repeat after me: The plane will not remain stationary. It will move.

If you intend to reply on any forum thread, this blog, or in a conversation about the AOAT condundrum, ask yourself this question first: Does your no-fly argument depend on an assumption that the plane will remain stationary? If the answer is yes, then you are incorrect. Erroneous. Fallacious. Double-plus untrue. You are wrong.

I can't stress this enough. THE PLANE MOVES! Read the original post below for clarity.

## 85 comments:

The MythBusters deny any comments to the contrary... watch and be enlightened:

http://www.youtube.com/watch?v=IbRcg3ji_Pc&feature=related

Wow....people just don't get it do they? And yet it's so simple. Of course it would fly, WTF is wrong with these no flyers?

As posted to the XKCD forum:

Gravity is acting on the plane pulling it down into contact with the treadmill. Through this contact energy is lost via friction. Only by overcoming the friction of this contact can the plane move forward. If the bearings of the wheels have zero friction, the plane would sit stationary while the treadmill ran at any speed from zero to infinity. if the bearings have ... i dunno... 100% friction, then the plane must always match the treadmill's speed to keep stationary. on a treadmill capable of spinning at infinite rpm's, the plane can never take off in the 100% friction scenario, and can take off at any time in the 0% friction scenario. anywhere between 0 and 100, the plane must generate enough thrust to overcome the friction of its contact with the treadmill, but there is some amount of thrust that will let it take off.

I think one particular point might help the no flyers. It's not the wheels that pull the plane forwards, it's the propellers/turbine engines. If the plane isn't moving (or barely) relative to the ground, there is still one hell of an amount of air being sucked past it by the engines. That's why it can take off while barely moving compared to the ground. It's not a propellerless glider that would need to be hurtling along the ground at high speed before it could get any lift.

Even if you have frictionless bearings and a jet-powered plane, there is a way for the treadmill operator to stop the plane taking off. "All" he/she has to do is move the treadmill backwards at relativistic velocities. The mass/energy of our ideal wheels can be made arbitrarily large. If the weight of the wheels is larger than the planes maximum lift, then it can't take off.

Real bearings are not frictionless, and real wheels can't take relativistic stresses, so happens in the "real" world? The trick is to look at the friction. The approximation that you use is that friction is independent of velocity. This works rather well in normal experience. However, you can show through dimensional analysis that if the velocity approaches the speed of sound in the material that makes up the wheel, then friction will no longer be independent of velocity.

Once your assumption that the friction is velocity independent breaks down, then your result also fails. "All" that the treadmill operator needs to do is exert a friction force greater than the thrust by having a sufficiently fast treadmill to have a sufficiently large friction.

Unfortunately, the real world is messier than this. The heat deposited by the friction would melt the wheels... and then the rest of the plane. If the plane is a melted pool of aluminium then it isn't taking off since its engines don't exist any more. ;-)

since the connection between the plane and the treadmill is frictionless, the treadmill really has no affect on the plane at all. The plane thus takes off just as it normally would.

To make it even more obvious imagine a treadmill with a surface of ice, and give the plane skis. Same situation, same result.

I know what the problem is. It's the ambiguity in the definition of the conveyor belt's speed. The no-flys assume (and sometimes the conumdrum is stated as such by no-flys, but not on this case) that the belt matches the rotational speed of the wheels. Me and all the other flys assume the belt is moving at the same linear speed as the plane itself. I figure landing gears are designed with a margin large enough to roll at twice the intended take-off speed. No-flys assume the treadmill will spin infinitely fast until the tires are burnt.

To put it short:

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

Therefore, the wheels turn like mad and the plane doesn't move.

flys read: The treadmill matches the plane's LINEAR speed.

Therefore the wheels at all time turn at twice the speed they would if the plane were on a runway, but the plane can still take off. If the plane were not moving, the treadmill would be still as well. In this cenario there is no treadmill running under a static airplane.

There is no point in arguing if you are not answering the same question.

Maruo, and others. You still don't get the point here. There is NO way that the treadmill can make the plan not take off. If the treadmill was going at some fraction of the speed of light it would only be inducing a force vector upon the wheels. In all cases except with the pilot conspiring with the treadmill operator the wheels will spin at Speed of the treadmill + Speed at which the plane is moving forward under it's own power.

There is absolutely, without a pilot conspiracy, no way for the treadmill to keep the plane from taking off.

As an exercise try thinking about standing on skateboard sitting on a treadmill and how much force you would have to use to keep yourself from moving backwards. I bet if you experiment it will only be = to the friction being generated by ball bearings which will be almost meaningless and the force vector due to friction will not increase or decrease no matter the speed you turn the treadmill up to.

All that being summed up the force due to friction being constant, any extra force in excess of the required force to overcome friction will cause the plane to move forward, causing airflow, causing lift off.

Seriously, these responses almost make me cry. I've lost hope in humanity. THE SPEED OF THE TREADMILL MEANS NOTHING - THE TREADMILL CANNOT STOP THE PLANE TAKING OFF.

If the treadmill is moving at close to the speed of light, then the wheels will be spinning such that their outer rims will be moving close to the speed of light. Their rotational energy will be comparable to their mass via E=mc^2. If you make the wheels spin fast enough, then their weight increases via special relativistic effects to be larger than the maximum take-off weight of the plane. The plane simply becomes to heavy to fly.

The force you've forgotten to take into account is gravity, not friction. Friction in the wheel-bearings can be zero, and this effect still occurs.

Of course this situation is completely ridiculous, but it shows how counter-intuitive extreme physics is. :-)

Unfortunately, it is unlikely the Mythbusters producers will be able to obtain a treadmill capable of moving that fast, but it's interesting to imagine.

To explain a little more clearly... the situation described is:

The plane moves forward at a velocity 'v', thrusting backwards, and thus experiencing a force forwards; 't'

The plane experiences a lift force upwards; 'l'. If you have a lift force, aerodynamics says you also must have a drag force 'd' acting opposite in direction to the velocity.

The plane has a weight force m*g acting downwards.

The solution has: t=-d and m*g = -l. The forces add to zero and thus the plane doesn't accelerate. However, it does have a velocity... so you'll need to wrap the treadmill around the Earth.

The reason this works is that we've managed to increase the effecting mass 'm' via the weirdness of special relativity. Also note that the velocity of the treadmill is u!=v, which is something close to light speed.

There should be some Godwin's law equivalent to relativistic physics in classical mechanics problems :).

Alright Kaizo, I am a flyer as well because I agree that a treadmill capable of spinning so fast that it would make the planes' tires explode/burn is not realizable and therefore the plane does fly. But the main reason why the discussion stretches to infinity is the semantics problem, and this is what I try to address here.

I am confused. Since when do planes use their wheels for takeoff? The wheels are for taxi-ing around the runways. Big engines make plane move forward. Those shouldn't be effected by a treadmill?

I think everyone is missing the point. This is a thought experiment. This is like Schrodingers Cat, it's not a realistic scenario, it's a hypothetical situation that's assuming things impossible in the real world.

If we break the experiment down what we're looking at is:

Can an object gain lift if another object is pushing it in the opposite direction with the same amount of force.

Clearly the answer is no, because this is not a real world scenario, and in this fake world we DO have an object that can match the planes thrust exactly.

The problem is that we tried to apply this to a real world scenario, if we did this with Schrodingers cat we'd have animal rights activists knocking on our door.

It's a thought experiment, and as such should be taken in context.

Chris, I was a true no-fly believer, and translated the question like "pope" above. Thanks to you I now see the light.

Thank you for explaining it to all of us morons.

(anyone knows if the mythbuster episode aired in Europe already?)

Exactly. A plane could take off on skis if there were snow on the ground.

Island hopper planes take off from floating water skis all of the time.

The point is that the wheels are not pushing the plane, the plane is simply using them to reduce friction and move faster due to airflow around the wings.

Whether the treadmill matches the plane's x-velocity or its wheels' rotational velocity, you're just going to cause an amount of friction in the wheels as they spin like mad when the plane drags them along the conveyor belt. The worst that can happen is that the friction somehow melts the axle in the short time it takes to take off, anyway. In this case the plane is broken, and falls onto the conveyor.

On another note, why don't you just get an r/c plane and put it on a treadmill? Just as a 747 doesn't use its wheels for forward motion, neither does the r/c plane. It would be a perfectly good small-scale test for the real thing.

The plane will fly. But there's a flaw in your argument regarding friction being the only force acting on the plane. Even if the bearings of the plane are frictionless, if one assumes that there is no slippage between the wheels of the plane and the surface of the runway, then the whole plane would initially experience a backward force when the runway treadmill is turned on. This is because the wheels experience a force applied at their bottoms which acts to cause both an angular acceleration and a linear acceleration. Granted, this backwards force is no greater than the force the plane typically sees when accelerating the wheels during take-off and usuallyin that scenario the plane has to overcome an increasing headwind as it spins the wheels up. In the treadmill scenario the plane doesn't even have to overcome a headwind to remain stationary on the track.

The problem and the reason this debate is so heated is in how the question is framed. The essential facts post is a very clear example of how this is a semantic interpretation problem, not a physics problem. The problem, as I've read it, states that the treadmill moves in exact oppositional velocity to the wheel's speed. This is physically impossible assuming there's no friction at all between the wheel and the axle as the treadmill would quickly ramp up to infinite speed. If there is friction then the treadmill would rapidly gain speed until the friction from the wheels perfectly counterbalances the thrust from the engine and the plane remains still, and no take off. Note that I haven't done the math here but I would not be surprised if that point was very very high, most airplane engines have high trust values and wheels have very low friction coefficients, so if we stay true to the letter of the problem the treadmill could have to move at thousands of miles an hour to keep the plane from moving. And I'm well aware that such a setup would likely cause enormous damage to the landing gear in very short order, but we are discussing theoretical physics here not real world problems (this problem is simply a variation of zeno's paradox).

And really this is the strut of the problem: there is friction but it is very small compared to the thrust of the engine, so even on a treadmill moving at hundreds of miles an hour the effect on plane speed would be minimal.

But the plane must move though in order to take off. the engine itself does not create enough lift on a normal fixed wing airplane to get it airborne.

If the problem is applied to a real world situation then we must assume that the treadmill is moving at some given velocity, and unless that velocity is extremely high, the plane is not likely to be affected.

Worded another way the problem could read : If the ground under the plane moves at the exact same velocity as the plane in the opposite direction can it take off given that the air around the plane remains within the airplane's frame of reference? The answer is no since we have basically said that if we make it impossible for the plane to gather any speed can it take off? And then it become obvious that no matter how powerful an engine, if the plane is not moving it will not take off.

But most people who think the plane will take off assume the treadmill will move at some given speed, and then they are right of course since the speed at which the ground must move to counter thrust is colossal.

Put another way,if the engine provides, say, 1000lbs of thrust (would be a very small engine) how fast must the wheels move in order to provide equivalent friction. Well, if no brakes are involved then the technical answer is freakin' fast. So much so that I suspect the wheel would fly apart before providing the necessary friction. But this is a theoretical possibility nonetheless.

Question: If I add wings to my car, and an airplane engine that will start

only once the wheels leave the ground, and I drive said car very quickly on a treadmill (so that, without the treadmill, it would take off), willthattake off? This is probably what was intended by the creators of the question.No I don't think it was their intention to ask whether a plane standing still can take off, it can't. Those that think it can are debating another problem entirely. And I don't think it was intended as a riddle, just a physics problem, only like so many of these debated issues it's not a physics problem but a language problem.

I say again: if the question is this: if an airplane is on a treadmill that spins in opposite the engine thrust at a velocity exactly opposite the *wheel's* speed can it take off? the answer must be no since you are saying, right there in the question, that whatever speed the wheels spin at is directly opposed, and for the plane to move the wheels must spin at least marginally faster than the treadmill, and i'm explicitly *telling* you that this cannot happen in the way I'm framing my question than we see that this is not a questions at all but some sort of cruel joke. Physics has nothing to do with it it's all semantics. The airplane cannot move because I have said so in the question. Wheel friction is irrelevant because if I say there's no friction then we have a problem of opposing infinites, IE in very short order both wheel and treadmill will be spinning infinitely fast, and opposing infinites mean one can never ourdo the other and both cancel out, hence the reference to the zeno paradoxes. It's like me telling you I will take from you one apple for everyone I give you, and no one else will give you apples, how many apples will you have? The answer is none since I take away everything I give to you. It's all about how the question is asked not the question itself.

If, then, the question is: 'if an airplane is on a treadmill that spins in opposite the engine thrust at a velocity exactly opposite the *airplane's* speed can it take off? This is a completely different question and a fair physics question, but not a very hard one at all as at these low speeds wheel friction differential between spinning at 150 or 300 mph is quite small compared with the thrust of the engine, and the plane will almost certainly take off.

Most people dig much much too deep into this, an altogether simple, problem. it's a linguistic trap much like the one about where you bury the survivors of an airplane crash exactly on a border. The problem is simple because of a few basic axioms:

a plane takes off when air around it's wings moves past them at a certain speed. How this is accomplished is irrelevant, IE ground speed is not important it's wind speed.

Airplane engines *cannot* be responsible alone for this air movement on a normal plane, though designing one with engines that move air correctly onto the wings may be possible if not particularly efficient.

Physics problems need not be practically realizable in the world. Such problems are common and serve to explore or explain the physical laws around us, meaning that if a question states that a treadmill has infinite maximum speed and wheels have no friction then such are the conditions under which the problem must be resolved, to reject these conditions is to reject the question and renders the answer invalid. You are answering *a* question to be sure but not this one.

The debate around this is all about how the question is interpreted, not about physics. And when this happens then we have a communication issue not a physics issue. And there's really no way around this. To say the plane *will* means you are assuming the treadmill has some given maximum velocity, it is an answer an engineer is likely to give because in the real world treadmills have maximum velocities and wheels can spin much faster than treadmills and therefore the plane will take off.

A mathematician on the other hand is used to dealing with pure theoretical abstract problems and is likely to say no it can't take off since a precondition of the problem is that the treadmill directly opposes the wheels, since the wheels must spin faster than the treadmill for the plane to move, and we have said this is not the case, the plane won't move and is not likely to take off (unless it's some special airplane that can do vertical takeoff).

Neither is right or wrong it's a simple question of what do you want to know? what do you think the question is?

On Oct 21, 2008, at 10:19 PM, Winslow wrote:

The plane does take off b/c the wheels of a plane free-rotate and are not connected to a drivetrain and forward force of the plane (either jet or prop) is generated by the engine pulling against the air, not pushing against the ground. Thus, all the force the plane needs to overcome is the basic friction created from the weight of the plane sitting on the wheel system, which is, relatively speaking, a minimal amount of friction. Additionally, friction is a unique force in that it does not vary relative to the movement or speed of the object [it only varies when pressure between the two surfaces increases (e.g. more weight) or the physical properties of the surfaces change (e.g. get stickier such as melting rubber or get rougher)]. An easy conceptual version of this static property of friction is to imagine a skateboard sitting on a treadmill. Imagine that you turn on the tread mill to 1 mph and that it takes 1 pound of pressure for you to hold the skateboard in place (with your hand pushing the skateboard from behind, or directly against the force of the treadmill). Now, if you increase the speed of the treadmill to 2 mph, or 5 mph, or even 100 mph, you will still only need to apply 1 pound of pressure to keep the skateboard from moving. In fact, as long as you do not add weight to the top of the skateboard or the wheels don't start to melt and become sticky, the force of friction will not change and only 1 pound of force will be needed to overcome it. Now, if you are holding the skateboard in place with the treadmill going 30 mph and you want to move the skateboard to the front of the treadmill, all you have to do is increase the force you are applying, say to 2 mph, and the skateboard would easily move forward on the treadmill.

Now back to the plane. Because the engines apply force in relation to the air and not the ground, and because the amount of friction would not change (at least not until lift is created in which case friction is reduced), then the engines need only generate enough force to overcome the friction. Let say that it takes 10,000 lbs. of thrust for a 747 to overcome this friction, then once the engines produce 10,000 lbs of thrust it would not matter how fast the treadmill was moving underneath the plane, the plane would sit still. Now, if you increase the thrust to 11,000 lbs, the plane would move forward, irrespective of how fast the treadmill under the plane was going. If it takes 250,000 lbs. of thrust for a plane to move forward fast enough to generate lift and take-off, then the 747 on a treadmill would only need to generate 260,000 lbs. of thrust in order to take off. Easy-sleazy for a modern jet airplane.

This is the same reason that if you filled a 747 with enough helium balloons that it had neutral buoyancy in the atmosphere and floated 10 feet above the runway it would still take-off without trouble (or why planes on skis or skids can take off on low friction surfaces such as ice, snow, and water). The force of the engines is applied to the air and not the ground. NOW, if you put the plane on a runway that was enclosed in a giant vacuum and fired up the propellor, it would not move forward (let alone take-off) no matter how fast the propellor was moving. In this situation you would have to have a drivetrain apply enough force to the wheels to get the plane going fast enough (say 170 knots) to generate lift. Unfortunately, as soon as the wheels left the ground the plane would slow down, lose lift, and come back down to earth. That must be why they use propellors and jet engines on planes instead of twin 454's with positraction.

I never understood the problem with this question.

the wheels on the ground serve no purpose other than providing a platform for the wings and prop. The prop forces the air backwards providing momentum, and over the wings providing lift.

I stumbled upon this site, and honestly I'm surprised this is even a topic for debate.

"In any realistic situation, the plane will not remain stationary. Repeat after me: The plane will not remain stationary. It will move."

So let me get this straight. You are talking about a REALISTIC situation in which we've taken a full sized airplane and placed it upon a giant treadmill that is able to perfectly and instantly match its speed. Yeah, that seems realistic.

Not trying to hijack the blog, I enjoy these thought experiments and like to offer the thinkers here one of my own.

Do the windshield wipers on a car have to remove more volume of rain from the windshield while it is traveling at 60mph than it does if it was sitting still?

No, it won't fly. There is a giant elephant on the plane.

http://rancemuhamitz.googlepages.com/ele.gif>I never understood the problem with this question. [...] The prop forces the air backwards providing momentum, and over the wings providing lift.

You just showed your misunderstanding. Lift is not produced from the air forced backwards by the prop (for general prop/plane design). It is generated by the air flowing over the wings due to the planes movement through the air.

Just imagine a plane being held stationary on a runway. Do you really thing that the props can be spun fast enough that the plane will do a vertical takeoff?

i found this, to me it perfectly explains it but you to have to contextualize it...

Dear Cecil:

I have a question about the inability of matter to exceed the speed of light.

Suppose I am in a spaceship traveling at the speed of light minus five miles per hour. What would happen if I fired a gun in the direction that the ship is moving? Or tried to run forward at six miles per hour?

Would I prove Einstein wrong?

— John B., Niles, Illinois

Dear John:

There are two ways we can go about this.

First there is the Way of the Wimp, wherein I simply tell you no, you wouldn't prove Einstein wrong, and we leave it at that. This avoids distracting complications, but leaves something to be desired from the standpoint of intellectual rigor.

Then there is the Way of Righteousness, which requires mental tenacity and moxie. Fortify yourself and we'll give it a shot.

You probably have the idea that if you are standing in a bus moving at speed u, and you walk forward at speed v, your total forward speed w is expressed by the straightforward sum u + v. Alas, this is a cruel illusion.

In reality, what we might call "addition of velocities" is governed by the awe-inspiring equation

w = (u + v)/(1 + uv/c^2)

where c^2 is the speed of light squared. (This may give you pause next time you hike to the can on a Greyhound.)

At so-called Newtonian (i.e., slow) speeds, the term uv/c^2 is pretty close to 0, and the equation reduces down to the familiar w = u + v.

However, if we are traveling at, say, 0.9c (nine-tenths the speed of light), and we shoot a bullet forward also at 0.9c, we discover via the above formula that the slug does not attain an overall speed of 1.8c (i.e., more than the speed of light), but rather a modest

(0.9c + 0.9c)/(1 + [0.9]^2) = 0.994c

(roughly).

Does this mean the bullet just dribbles out of the gun like a freaking gumdrop, for Chrissake? Not at all--to you, the space traveler, everything looks normal.

However, a stationary observer would note that you were suffering from the unique effects of the Fitzgerald contraction--which is to say, (1) time would slow down for you (although you wouldn't realize it), and (2) you and your spaceship would get compressed like an accordion along your axis of travel.

The following poem may help to illustrate this:

There once was a racer named Fisk

Who took a considerable risk

When his dragster got traction

The Fitzgerald contraction

Reduced his wazoo to a disc

Sorry, couldn't resist. Anyway, if you ponder this matter awhile, you will inevitably come to the following conclusion: the faster you go, the slower you go. Ergo, the speed of light cannot be exceeded.

And you wonder why physics is my favorite subject.

— Cecil Adams

the planes relativity would not be to the ground, it would be to the conveyer belt/treadmill. the engine will function, it will pick up speed enough to propel itself into the air. IT WILL FLY

***i meant IT WILL MOVE

sorry lol

The OP is needlessly complex, all that's needed to illustrate the problem is a treadmill and wheelbarrow.

Imagine you have your feet on the floor and are holding a wheelbarrow with its wheel on a treadmill. You start the treadmill and the wheel is turning with it. You then push forward on the wheelbarrow, what happens? Even if the treadmill immediately matches pace, something has to push back on you with the same force to keep the wheelbarrow from moving. Imagine pushing the wheelbarrow with all your might and not being able to move it, it just doesn't make sense no matter how perfect the treadmill or wheels.

Substitute you pushing a wheelbarrow for an engine pushing a jet and the answer is clear.

The plane definitely will take off (if engine force is able to overcome friction).

I don't understand, what is wrong with that task?

Why some people rely on "obviousity", "common sense" etc., when pretty straightforward method of solving dynamical problems works here?

Here is a classical method of solving dynamical tasks, that children are taught at school at the age of 14. Let's apply it to that problem:

Applying second law of Newton to the plane.

We need only horizontal projection, directing axis to the direction of the plane and having 0 point somewhere on the ground.

What forces affect the plane:

1) Gravity. Projection = 0 (perpendicular to the axis).

2) Support reaction from treadmill. Projection = 0 (perpendicular to the axis).

3) Engine force. Let it be Fj. Projection equal to the force itself. It depends only on internal engine characteristics. I don't care whether it is constant or what it depends on, the only thing I need to know is that it does not depend on threadmill or wheels characteristics, that's enough.

4) Friction force from treadmill.

Projection equal to the force itself (but is directed opposite the axis). Specially for Anonymous from "September 10, 2008 11:15 AM" and the others that think the same: I don't care of exact treadmill adaptation algorithm, I don't care what acceleration treadmill has, I don't care how it works at all. The only thing I need to know is that it affects the plane using friction force.

Let friction force be Ff.

However, friction of rolling wheels is limited to sliding friction (worst case - wheels blocked) which is in term limited by N*k (support reaction, multiplied by friction coefficient). Sorry, not sure of proper terms in English here.

Ff <= Fsf <= k*N

Total force affecting the plane is equal to:

F = Fj - Ff

If Fj > k*N, that means that Fj - Ff >= Fj - k*N > 0

To summarize: F>0 if Fj>max friction force.

In that case, acceleration (=F/M, where M is the mass of the plane) is >0 as well, plane starts moving relative to the ground. As well and reaching the velocity necessary to take off is just the question of time.

Result: if jet engine force is over k*N (maximum friction force) at any time, plane will take off and the algorithm of treadmill does not matter.

I'd appreciate any comments or errors found.

Let me make this simple. No big words or long ass formulas that most people won't understand.

An airplane isn't a car. The thrust is from the engines. The only difference between this scenario and a regular takeoff is that the wheels spin twice as fast. The only thing friction will do is heat up the wheels a bit more. It takes off.

It's crazy that this is still being argued. the fact is, is that a treadmill cant stop the plane from moving forward because the force that moves the plane forward is thrust. If I drop a car onto a treadmill that is moving 66 mph backwards and I have the car pegged at 66 mph on the speedometer.....the car will remain in place on the treadmill....everyone agrees with that, whether you are a fly or a no fly. If I strap a jet engine on top of this same car, put the car in nuetral , and fire up the engine to a thrust level that would propel the car along at 66 mph on regular pavement and somnehow "drop" this car onto my backwards at 66 mph treadmill......Guess what happens.....the car will move forward, eventually reaching 66 mph! If you don't understand this.......then you are hopeless. It's just like walking through the airport with a bag on wheels, if I come upon a people mover moving in the opposite direction and I decide to walk outside of the people mover treadmill but reach over a place my wheeled bag on the people mover and pull it along in the opposite direction....what happens? I pull the bag along with the same effort,because my bag is ON WHEELS...JUST LIKE AN AIRPLANE...doesnt matter if the treadmill is moving 5000mph in the opposite direction, I can still pull my bag along it in the opposite direction just fine. It ain't rocket science kids.

wouldnt the most basic question really be if a plane is not able to physicaly move forward could the thrust alone cause the plane to take off?

The spirit of the actual question is supposd to be a conveyor that keep the plane from moving forward. whether or not you make the tirs frictionles is irrelevant if you do not nitpick the question apart.

A fairer repesentation would be, if you placed a plane on a giant sized dynamometer, the thrust fromt he jets causing the wheels to spin the rollers under them rather than moving the plane forward. In this scenario would the plane somehow generate lift even though it is not traveling forward, does not reach escape velocity and is not causing air to flow over the wings.

wings provide lift people.

the wings!

jets and propelle do pull the plane forward but they do not provid lift! excepting VTOL and helicopters.

the people who say propellers and jets push air from them over the wings are absolute idiots.

if this was the case, ultrlights, rear engine jets and rear propller planes would all not function.

the jets and propellers provide thrust, the wings and air flow provide lift, a jet in the back of your plane blows no air over the wings. sheesh.

all who say the wheels have nothing to do with th takeoff are also disturbed. if that was the case, then a plane with no wheels or with wheel blocks blocking the wheels would take off from stading still if you goosed the engines hard enough. if that were the case aircraft carriers and runways would not be needed for take offs. and a fast car upon reaching escape velocity would become airborne on its own. you need lift fromthe wings to achieve flight. if you didnt then flying would be extremely easy for almost anyone.

people are supposing thatthe plne moves forwar on the treadmill, when the question is supposed to make you assume the plane does not actually move forward due to the treadmills motion. as in when you exercise on a treadmill, turnit up so you canbarley stay on it at max speed, you have no forward velocity, Velcoity is cancelled out by the treadmills motion.

heres a simple fact people

a plane that does not move forward cannot fly. ( excepting as i said VTOL aircraft.

the orignal question is not supposed to lead you to fricitonless wheels or treadmills. the original question as far as can be found stated "a treadmill like" body . meaning something that keeop the aircrafts forward movement in check. whether you call it rollers perhaps , like a dynamometer. ( or do you think a race car withthe engine doing 200mph on a dynomometer actually goes forward?

people would rather pick apart the question than answer the question .

Protip: A racecar that has a jet engine attached to it will jump out of a dyno and through the wall of the shop.

The wheelbarrow and the airport people mover are my favorite comparisons for people that don't get it.

I have dual thoughts about this: if the treadmill is frictionless, I am a "fly".

If the treadmill has friction, it can be set to spin so fast to burn the wheels (as it cannot stop the plane from accelerating, since friction is constant). After the wheels have burnt, I doubt the plane will be able to take-off, unless it had infinite thrust. I don't even think there is need to bother relativistic mechanics.

So, while I agree with the points a) b) and c) in the "Airplane on a treadmill", I think there should be a

d) the conveyor belt cannot jeopardize the aircraft structural integrity

to let the plane take-off ;)

it wouldn't take off you fucking 4chan reject moron, theres no airflow over the fucking wings

Consider a wind tunnel and a model airplane. A wind tunnel is simply a tube with a huge fan at one end. Turn the fan on: It pushes air past the model airplane at 200 mph or whatever you want.

It is this moving wind that causes the plane to have lift, and the reason for lift is due to airflow over the wings. It is the whole reason we use wind tunnels to help design airplanes.

Treadmills don't create airflow, sorry. If you "flys" are seriously not trolling, then damn son, America is doomed. We don't deserve to hold the scientific throne anymore.

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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.

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'tright 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.

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.

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.

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.

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?

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.

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!

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

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.

"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.

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.

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 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.

"...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.

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.

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.

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

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.

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.

What if it's a helicopter on a treadmill?

Chris, the plane will move,upward as the wind lifts it. The plane does not have to move horizontally as long as there is wind for lift.

It's sad so many "hurr is gonna fly cause mythbusters" idiots (idiots because you can't understand the difference between a treadmill and dragging a tarp on the ground) seem to think in this thought experiment, in which the treadmill INFINITELY matches the forward momentum of the plane is "impossible because of wheels bearings durp a hur".

Friction exists. Otherwise the plane would take off the moment the engines were on. With infinite frictionless wheels, the plane REMAINS stationary due to the fact that by default neither can overcome the other as they infinitely match each other.

Which means the flyers are akin to idiots saying that 1 ton of bricks is heavier than 1 ton of feathers.

Ok so we have several interpretations

1) Speed(Plane)wrt(Ground)=-Speed(Belt)wrt(Ground)

2) Speed(Plane)Wrt(Belt)=-Speed(Belt)wrt(Ground)

3) Speed(wheel rotation)wrt(Ground)=-Speed(Belt)wrt(Ground)

4) Speed(wheel rotation)wrt(Belt)=-Speed(Belt)wrt(Ground)

In 1) The belt has very little effect on the plane, as per the Cessna example, the belt can stop the plane moving at 1725mph but the plane takes off at 70.

Summary: Plane Moves+Lifts off

In 2) The plane doesn’t move, this interpretation defines it as such.

In 3)+4) because rotation isn’t affected by reference frame this condition is met whether the plane is moving forwards backwards or standing still in any reference frame and whether the conveyor is moving forwards or backwards, as long as the wheels are rolling.

So 2) is the controversial interpretation. As shown by Robocop there are real world conditions where the friction and rolling resistance of the plane can be high enough to prevent motion without generating either enough wind to generate lift or enough relativistic mass to create undue gravitational effects.

The question then becomes what is a reasonable interpretation of the question? If you read that a car was on a treadmill that was going backwards as fast as the car went forwards and that the treadmill was going at -60mph would you assume the cars speedo read 60 and thus the car was stationary wrt ground? Or that it read 120 and thus was going at 60 wrt ground?

If you take interpretation 1, the question is asking, given a plane can move freely forward can it take off? There’s nothing interesting there, its not even asking the brain teaser that most people instinctively get wrong “will a treadmill rolling under a plane have a significant effect on the planes speed?”

If you take interpretation 2, the question is asking, given a stationary plane, can its engines generate enough lift by forcing wind over the wings to lift the plane?

It seems, all in all, more reasonable to me to suppose that interpretation 2 was the original intended interpretation.

The interesting questions here are

A) Under what conditions can a treadmill exert enough force on a plane to counteract its engines?

B) Can a plane’s engines generate enough wind movement to generate lift without forward motion wrt the air?

These are both physics questions, with interesting mathematical answers which have so far mostly been ignored in this debate.

If the treadmill is rolling the plane back at 100 MPH and the plane engines were providing enough power to roll the plane forward at 100 MPH then the groundspeed (relative to the treadmill surface) would be 100 MPH but the airspeed (relative to the air surrounding the plane and the treadmill) would be 0. To an observer, the plane is stationary.Therefore, no lift, no take-off. Case closed.

If there is no slip and the treadmill has no motor, the plane will definitely not fly! The thrust from the engines will accelerate the treadmill until the frictional force opposing the motion increases to match the thrust force. Then the treadmill will move at a constant velocity...and the plane will not move!

@matt

If you were right, then acceleration would be impossible.

Imagine a guy pushing a rock, as he starts it may not even move at first, but then it gives way, and then he accelerates because the momentum of the rock moving is adding to his force.

If friction increased to match, then it would cancel out momentum.

Inertia would then be impossible in anything other than a vacuum.

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