Sorry for the forum/blog downtime today. Many things went wrong during davean’s heroic upgrade. (I blame the LHC.)
Feynman used to tell a story about a simple lawn-sprinkler physics problem. The nifty thing about the problem was that the answer was immediately obvious, but to some people it was immediately obvious one way and to some it was immediately obvious the other. (For the record, the answer to Feynman problem, which he never tells you in his book, was that the sprinkler doesn’t move at all. Moreover, he only brought it up to start an argument to act as a diversion while he seduced your mother in the other room.)
The airplane/treadmill problem is similar. It contains a basic ambiguity, and people resolve it one of a couple different ways. The tricky thing is, each group thinks the other is making a very simple physics mistake. So you get two groups each condescendingly explaining basic physics and math to the other. This is why, for example, the airplane/treadmill problem is a banned topic on the xkcd forums (along with argument about whether 0.999… = 1).
The problem is as follows:
Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?
The practical answer is “yes”. A 747’s engines produce a quarter of a million pounds of thrust. That is, each engine is powerful enough to launch a brachiosaurus straight up (see diagram). With that kind of force, no matter what’s happening to the treadmill and wheels, the plane is going to move forward and take off.
But there’s a problem. Let’s take a look at the statement “The conveyor belt is designed to exactly match the speed of the wheels”. What does that mean?
Well, as I see it, there are three possible interpretations. Let’s consider each one based on this diagram:
1. vB=vC: The belt always moves at the same speed as the bottom of the wheel. This is always true if the wheels aren’t sliding, and could simply describe a treadmill with no motor. I haven’t seen many people subscribe to this interpretation.
2. vC=vW: That is, if the axle is moving forward (relative to the ground, not the treadmill) at 5 m/s, the treadmill moves backward at 5 m/s. This is physically plausible. All it means is that the wheels will spin twice as fast as normal, but that won’t stop the plane from taking off. People who subscribe to this interpretation tend to assume the people who disagree with them think airplanes are powered by their wheels.
3. vC=vW+vB: What if we hook up a speedometer to the wheel, and make the treadmill spin backward as fast as the speedometer says the plane is going forward? Then the “speedometer speed” would be vW+vB — the relative speed of the wheel over the treadmill. This is, for example, how a car-on–a-treadmill setup would work. This is the assumption that most of the ‘stationary plane’ people subscribe to. The problem with this is that it’s an ill-defined system. For non-slip tires, vB=vC. So vC=vW+vC. If we make vW positive, there is no value vC can take to make the equation true. (For those stubbornly clinging to vestiges of reality, in a system where the treadmill responds via a PID controller, the result would be the treadmill quickly spinning up to infinity.) So, in this system, the plane cannot have a nonzero speed. (We’ll call this the “JetBlue” scenario.)
But if we push with the engines, what happens? The terms of the problem tell us that the plane cannot have a nonzero speed, but there’s no physical mechanism that would plausibly make this happen. The treadmill could spin the wheels, but the acceleration would destroy them before it stopped the plane. The problem is basically asking “what happens if you take a plane that can’t move and move it?” It might intrigue literary critics, but it’s a poor physics question.
So, people who go with interpretation #3 notice immediately that the plane cannot move and keep trying to condescendingly explain to the #2 crowd that nothing they say changes the basic facts of the problem. The #2 crowd is busy explaining to the #3 crowd that planes aren’t driven by their wheels. Of course, this being the internet, there’s also a #4 crowd loudly arguing that even if the plane was able to move, it couldn’t have been what hit the Pentagon.
All in all, it’s a lovely recipe for an internet argument, and it’s been had too many times. So let’s see if we can avoid that. I suggest posting stories about something that happened to you recently, and post nice things about other peoples’ stories. If you’re desperate to tell me that I’m wrong on the internet, don’t bother. I’ve snuck onto the plane into first class with the #5 crowd and we’re busy finding out how many cocktails they’ll serve while we’re waiting for the treadmill to start. God help us if, after the fourth round of drinks, someone brings up the two envelopes paradox.
xkcd 
Actually, I’m a pilot. Have flown many types of aircraft including high performance jets. So I’ll tell flying stories here in the spirit of the discussion. It doesn’t matter what the wheels or engines are doing at all only how much lift the wings produce due to the difference in pressure/angle of attack etc. My initial training was in gliders – no engines at all and only a single wheel on the aircraft body and two wing wheels – it flew just fine (no treadmill though thankfully – just a HUGE catapult winch or tow plane). Many a time, due to air density at high altitude or over loading, aircraft have not achieved flight despite fully functioning engines
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What is with lift, as the plane doesnt move, thereby no lift is created, so it is impossible to get an airplane fly that way.
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David:
you’re wrong. the treadmill will exert an opposite force on the plane, cancelling out the force from the engines.
Chris:
this is one of those physics problems where we assume that everything is under ideal conditions. our airplane wheels won’t break.
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The wheels start at 0, the tread mill matches.the plane begins to roll forward from the thrust of the engines, the treadmill adjusts speed. The engines will keep pushing the plane, not the wheels, forward, causing the wheels and treadmill to continue to accelerate, but the plane will drag the rapidly spinning wheels across the rapidly moving conveyor.
The plane itself will launch, with shredded tires and a broken conveyor in its wake.
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The wording is a bit too ambiguous to come up with an acceptable solution, but I think we can break it down into two different scenarios, after setting some ground rules.
First, assumptions: All components are indestructible. The tires won’t shred, and the treadmill won’t rip. There is zero friction between the treadmill and the plane, 100% friction between the wheel and treadmill, and zero friction in the wheel assembly* (that last one is redundant, but illustrates how there can be zero friction between the plane and treadmill). It is 75 and sunny (that’s Fahrenheit), and there is zero wind). The plane will take off at exactly 150kts indicated air speed (IAS).
The two scenarios depend on how the entire contraption is rigged. For the first scenario we will state that the treadmill speed is exactly the opposite of the plane’s IAS. I choose this measure because it is directly related to when the plane will take off. The treadmill is going to start as soon as any forward movement is detected.
In this scenario, the plane will start its engine and move forward. As soon as it does, the treadmill will kick in, going against the plane. Because there is no friction between the plane and treadmill (due to the wheels), and because the force from the plane is generated by a jet (or propeller) and acts on the wind, the plane will accelerate as usual. The plane will eventually take off at 150kts, with the treadmill speed at 150kts in the opposite direction, and the wheels spinning as fast as they would if the plane was traveling 300kts (IAS + treadmill).
This is more obvious if you think of it on a smaller scale. Find a treadmill, put roller-blades on, and have a friend push you as soon as you turn the treadmill on. (Your friend is the ‘propeller’) It really doesn’t matter what speed the treadmill goes, because the wheels will spin instead of you falling back. (Whatever you do, don’t step on the brakes!)
In the second scenario, we will assume that the treadmill is somehow able to affect the plane’s movement. This means that there is some resistance in the wheels that gets transferred to the plane (a much more realistic approach, as zero-friction wheels are in short supply).
Now the question becomes which is greater, the thrust of the engine or the pull of the treadmill? Keeping the same assumptions as before, minus the friction of the wheels, we need to figure out if the engine can get the plane up to 150kts with the pull of the treadmill. The equation for flight is Vp – (COFw*Vt) >= 150kts where Vp is the speed of the plane, COFw is the friction in the wheels, and Vt is the speed of the treadmill. (At this point I’d like to point out that I’m pretty sure I’ve made a mistake, but I don’t really care to verify my work at the moment. Feel free to do so in your subsequent comments). We can see that the COF is vital to whether the plane can take off. We need to know what the COF is, and more importantly, how much thrust the engine can create to overcome the friction.
At this point it is really pointless to determine how much friction there is or how much thrust the plane can generate. We would either come to logical answers (low friction, high thrust, the plane takes off), or ridiculous hypotheticals (well, what if the wheels were locked up, but the plane had 5 million pounds of thrust? In that case my friends, the engines would take off on their own, leaving the plane behind, and we would be forced to argue whether or not an engine on its own constitutes a plane or a highly illegal firework).
In all but the most bizarre cases, the plane should be able to take off on its own, without any hindrance from any sort of over-sized piece of gym equipment. But what if it was on an aircraft carrier going 150kts North and you tried to take off from it heading south…?
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Danny-O: In both of your scenarios, the plane would not be able to take off. The treadmill exerts force on the plane through the axle as long as the wheels have mass. This doesn’t require friction in the wheel assembly; think about the case where you pull a piece of paper from under a marble, where there isn’t even a wheel assembly.
Since you’re assuming indestructible components and zero friction except between wheel and treadmill, the treadmill will always exert exactly enough force to counteract engine thrust, while the wheels spin faster an faster.
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Watch this! http://www.youtube.com/watch?v=YORCk1BN7QY
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Think about it, what is the relationship between the wheels and the plane? Why do the wheels moving at all effect the planes speed? If the wheels exist in a frictionless enviroment (except the friction between the wheel and the treadmill) then the treadmill moving will just spin the wheels regardless of wither the plane’s engines are on or not. Taking relativity out of the maths, you can increase the treadmills speed to infinity and the plane will just sit there as there is no mechanism connecting the energy from the treadmill to the rest of the plane. The wheels will be spinning dam fast but this has nothing to do with the rest of the plane. The plane can set off as it normally does.
The only reason this is hard to think about is because they’re is normally friction between the wheels and the plane meaning all of the energy is not transferred into rotational energy but into kenetic energy of the plane moving backwards
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@sriracha,
I don’t think I understand the point you’re making with the marble. You’re saying that if I pull a piece of paper from under a marble, the marble will move with it, correct? This is because there is friction between where the paper contacts the marble and the marble itself. The reason I assumed there was no friction was because the wheels effectively absorb all force from the treadmill. Of course realistically there will be resistance in the wheel assembly, which is what I explained in the second example. Then it becomes a matter of which is greater, and will the engines be powerful enough to overcome any friction AND reach takeoff speed.
Of course if the treadmill spins fast enough to exert a net negative force (say, if the plane is generating 1000 lbs of thrust, but the treadmill is exerting 1050 lbs of force against the plane, after figuring out what the resistance is) then of course the plane will not take off. Reasonable assumption of wheel resistance would mean that the treadmill would have to spin many times the IAS of the plane to get enough force to the plane. My scenario stated that the treadmill would only go at 1x the IAS of the plane.
Example:
Say the engines generate 1,000 lbs. of thrust forward at t-0. The plane begins to move and so the treadmill kicks in. At t-1 the plane is at 10kts IAS. The treadmill is spinning at 10kts backwards. After calculating the resistance of the wheels, we’ll say it’s exerting only 50lbs. of force against the plane (since the treadmill could not be 100% efficient in transferring energy through the wheels). The net force is 1040lbs. in favor of the plane, and it will move forward. This will continue until the plane reaches 150kts.
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Can’t… post… nice… story… …must… argue…
If we assume some control system setting the speed of the conveyor, then the set point is the speed of the wheels. Increasing the wheel speed increases the conveyor speed so the system is in positive feedback. If the system components are ideal (i.e. unbreakable) then the system quickly reaches relativistic velocities and the increasing weight of the conveyor components (assuming that they’re much more massive than the aeroplane wheels) requires a cosmic amount of power to drive it. Which either isn’t supplied and the plane takes off (assuming it can still lift the weight of its wheels) or the plane gets sucked into a black hole 🙂 (Skilfully skipping around smiley parenthesis problem 🙂 damn…
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Danny-O:
no – I am assuming that there is no friction in the wheel assembly. all that matters is that there is friction between the wheel and the treadmill, which you explicitly assume in your original post, and that the wheels have mass.
As for your assumption that treadmill goes at 1x IAS, this is unwarranted. the question explicitly states that the treadmill speeds equals the _wheel_ speed, which contradicts your assumption.
let’s make it clearer. assume that we have a tiny almost weightless plane with enormous heavy wheels. assume there is NO friction in the wheel assembly, but infinite friction between wheel and treadmill. now it should be clear that the treadmill can exert non-zero force on the plane by moving backwards, just like in the marble example. In fact, by accelerating quickly enough, it can counteract the force exerted by the engines. (this acceleration would of course have to be very large, but we are assuming ideal conditions here.)
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A very tiny airplane with very large, very massive wheels is not an airplane, it’s a monstrosity.
Consider also: if the treadmill speed responds to the wheel speed–even instantaneously–the engines are still exerting a net force on the hull. Three possibilities: ground friction is quickly broken and the plane takes off, the wheels are ripped from the hull and the plane belly-slides off into oblivion, or the engines are ripped from the hull (and do whatever they damn well please after that).
I love this discussion.
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eximus:
the net force on the plane would be zero. the engines exert a force on the hull, but the wheel axles exert an opposite force on the hull as well, “transmitting” the force exerted on the wheel surface by the treadmill.
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This problem is simple. The wheels provide frictionless contact with whatever is below them. Thus it is equivalent to a plane on skis sitting on an a sheet of ice.
It’s obvious what happens. Even if you devised a treadmill made of ice it won’t matter.
No math needed.
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I’m tired of these motherfucking airplanes on these motherfucking treadmills.
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“Moreover, [Feynman] only brought it up to start an argument to act as a diversion while he seduced your mother in the other room.” Well, he seduced my mother on a beach. At least that’s the way he told it when I asked him how they met.
–Carl Feynman
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http://www.airplaneonatreadmill.com/
Went to the pub last night, with a few friends. Great laughs!
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tom: wrong. your claim is only true if the wheels are massless. if the wheels have mass, then the treadmill has to exert a net force on the wheels in order to induce rotational acceleration (which obviously occurs), and thus the treadmill exerts a net force on the entire plane.
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Pretty simple, in my mind: We’re not talking about a car here, which produces forward motion by physically turning the wheels against the ground to produce forward movement. The fact that the wheels turn when the jet moves forward is circumstantial at best. The jet engines are not directly driving the wheels; indeed, they’re forcing the vehicle through the air in front of it by exerting force on the air behind it, and they are going to push that jet forward through the *air* regardless of the existence of the treadmill or how quickly it is moving. The wheels of the jet won’t produce enough friction to come anywhere near canceling out the massive thrust that the engines produce. Danny-O is right – the plane would take off and the wheels would simply be spinning faster.
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t-rav: wrong.
As long as the wheels have (angular) mass, the motion of the treadmill will exert a force on the plane. this force doesn’t come about from friction in the wheel axle or kinetic friction generated by the wheels “skidding”; although it relies on static friction between the wheels and the treadmill.
if the treadmill accelerates sufficiently quickly, it will generate enough force on the plane to cancel out engine thrust. of course, as per this sort of question, we assume that we can construct a sufficiently powerful treadmill or sufficiently indestructible wheels.
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Sriracha, I think we’re finally getting somewhere. There is some minuscule amount of force applied to the plane from the treadmill, through the wheel assembly. This is given. If an unpowered plane were sitting on a treadmill and the treadmill started up, the plane would move the direction of the treamill. This is realistic. Hypothetically a wheel assembly that turned ALL of the linear energy of the treadmill into rotational energy (what I described as a perfect, frictionless wheel) would keep the plane perfectly still, regardless of the treadmill’s actions. Obviously real life tells us that no such wheels exist, so some of the force from the treadmill will transfer to the plane. You stated that if the treadmill went fast enough* it could counteract the force of the engines.
Your quote:
f the treadmill accelerates sufficiently quickly, it will generate enough force on the plane to cancel out engine thrust.
My quote:
Of course if the treadmill spins fast enough to exert a net negative force (say, if the plane is generating 1000 lbs of thrust, but the treadmill is exerting 1050 lbs of force against the plane, after figuring out what the resistance is) then of course the plane will not take off. Reasonable assumption of wheel resistance would mean that the treadmill would have to spin many times the IAS of the plane to get enough force to the plane.
Now it’s just a matter of how fast will our treadmill spin. You said the treadmill should match the wheel speed. This is a little ambiguous. Are we measuring from the axle of the wheel, or from a point on the wheel’s edge? I guess it really doesn’t matter. I used IAS because it illustrates the speed of the entire plane through the air – wheels included. I suspect a treadmill going even 5x the wheel speed, IAS, cheetah speed, or whatever other measure we want to use relative to the plane would NOT be enough to hold the plane back. We would need a very fast, powerful treadmill to approach the possibility of stopping the plane. It’s really just a math problem from this point. Figure out exactly how much force the treadmill can apply, and how many times the multiple of the plane’s speed would the treadmill have to go in order to hold it back.
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I’m pretty sure that anyone that argues that the plane doesn’t move is just trying to egg folks on. It just *has* to all be a big joke, because no one can be that ignorant after reading all the evidence.
Although there is the argument of…
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danny-o:
– A “perfect, frictionless wheel” that “that turned ALL of the linear energy of the treadmill into rotational energy” would in fact allow the treadmill to exert a force on the plane (you seem to disagree with this). We don’t need friction in the wheel assembly to generate this force. To see this, draw the force diagram.
– the IAS is zero in this scenario – so equalizing IAS with treadmill speed makes no sense. In fact, there is no steady-state treadmill speed that keeps the airplane in place – what is required is constant force exerted, which entails non-stop _acceleration_ of the treadmill. We can, of course, solve an ODE to obtain the required acceleration profile.
– this is, of course, a hypothetical where we assume ideal indestructible wheels and treadmills that can perform at arbitrary velocities. In particular, many (wrong) counterarguments against the plane staying in place assume as much.
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The plane would not take off. The max Take off weight of a 747-400 is around 850,000 lbs. The thrust generated by a PW2040 engine is 41,700 lbs. It would take 21 of these engines to take off when your speed relative to air movement is effectively 0. The way a plane flies is by creating lift. In this scenario, there is NO LIFT CREATED. There is a reason why only a few military aircraft have the power to weight ratio required to do vertical Take offs.
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As for the airplane, I am a Mechanical Engineer and will try to give some insight. First, about jet engines: they create thrust and push the plane with air. Basically air enters through a large inlet, and is forced out of a smaller outlet. This means it leaves at a higher velocity so it accelerated… by Newtons 3rd law there is acceleration on the jet engine and therefore on the aircraft wing at the point where it is mounted. If the wheels have no mass, no speed or acceleration of the treadmill can keep the plane from taking off. If you assume the wheels have mass, then by changing the velocity of the treadmill, you can make the wheels generate a force on the airplane (changing the velocity creates a force on the wheel, some of that will go to accelerating the wheel and some will get put through the axle). This makes friction irrelevant to the absolute solution of this problem as it can basically be lumped in with this force from the wheel on the axle. What matters is that if you accelerate the treadmill to infinity you can hold the plane in the same spot. There is one problem is that this accelerating treadmill creates a huge boundary layer which will create an airspeed over the wings. Therefore even in the event of a treadmill accelerating off to infinity an airplane will take off.
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ryan – your point on the wheels having mass is exactly correct.
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How about… you’re all wrong.
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next time, take the train
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Rofl this post talks about how you keep arguing with each other then you start here
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Well, it all depends. If Alan Tudyk is the pilot, the plane will take off, no matter what else happens. If my friend Andrew is on the plane, it will catch fire and the conveyor belt will melt and the plane will take off and explode. If Leonard Nimoy is involved, the plane will not take off, because he and his hippy chicks sing about Hobbits and distract everyone involved. On the other hand, his mere presence might cause the plane to move forward with incredible speed and thus achieve lift. By the way, everyone who took this argument seriously on this thread should rickroll themselved eight times (Here you are: http://www.youtube.com/watch?v=DBCJeQ7FLxE) And read all the comments on a youtube video about the “public option.”
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The plane will take off. And this is why. A huge fraction of the power generated by the jets is transferred through the plane to the wheels to the treadmill. Imagine a treadmill powered by a jet engine. The plane is remaining at a standstill. No air movement over the wings means no lift means no flight. Except: Ryan is right. Via friction, the treadmill takes air with it as it moves. At some point the air moves so fast and at such volume (relative to the height of the plane and length of the wings) that lift is “created”. The plane takes off. And then immediately crashes, because the plane’s true airspeed drops to approximately zero once the boundary layer of air induced by the treadmill is left behind.
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Claude, the jet engine DOES NOT transfer power to the wheels. Jet engines work by accelerating air backwards, not by spinning wheels. Imagine the Free Body Diagram for this case. The Jet engine forces act on the wings in a forward direction… for the treadmill to cause a reverse force on the airplane it must constantly ACCELERATE in a backwards direction. The amount of acceleration depends on the mass distribution of the wheels and the mass of the airplane and the friction of the wheels on the axle. Because the wheel mass is almost negligible compared to the airplane mass, the treadmill would have to accelerate faster than the airplane would on a stationary runway just to hold the airplane back. Assuming that you could somehow accelerate a treadmill like this (VERY UNLIKELY) you would quickly get a boundary layer. Once the plane takes off, its airspeed does not immediately go to 0, because the velocity field is not discontinuous without shock waves (only above mach 1). Instead, the plane would sit at whatever altitude holds it in the air as it accelerates forward out of the boundary layer. No matter how you work the problem, I cannot see the plane not taking off… sorry.
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This is the situation as I understand it. Let’s assume the problem starts at a full stop and we’ll not confuse ourselves with wind. The engines give thrust which pushes the jet forward, but without lift the jet moves forward along its contact with a stationary object. Since the stationary object in this situation is a mobile treadmill that matches the wheel’s speed the plane will not move forward. Without wind to provide lift a plane must create it’s own wind by traveling forward through the atmosphere, and since the treadmill prevents this, it cannot “lift” off.
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Trickneal what does that even mean “the jet moves forward along its contact with a stationary object”? You just defined the airplane as stationary and then said because it is stationary it cant take off. But until you provide a reason as to why the plane is stationary in the first place, your argument falls apart. Make a free body diagram… the engines cause a forward force ON THE AIRPLANE. How can the treadmill cause a FORCE on the airplane? The only two ways that I can think of are: friction between the wheel and axle, and acceleration of the treadmill. Accelerating the treadmill forces the wheels to spin faster as well as cause a reverse force (this is from dynamics). Friction would most likely be negligible even at high treadmill speeds in comparison to the force of the jet engines. Therefore the treadmill would have to be accelerated backwards and you get the situation I explained above.
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This problem is all about friction. The fact of the airplane not being driven by the wheels is important, not only because it means that the plane does not exert any force or thrust on the runway/treadmill except for friction, it also means that the runway/treadmill has a maximum and relatively small effect on the speed of the aircraft limited by the coefficient of friction. in reality i bet most of that energy is converted to heat within the wheel bearings and such. If the plane were driven by the wheels initially, if the wheels had some sort of transmission/power train assembly this would cause the direct application of the force of the treadmill to the plane through the wheels, causing a stationary plane as the treadmill would match the aviliable driving force (thrust, except through the wheels).
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airplane. just picture it as a closed system with basic forces acting on the plane. the air is still. no matter how fast the plane is moving on the spedometer it has to be moving that fast relative to air to achieve lift. now given jet engines force vectors cannot be matched oppositely practically with a conveyor belt (to my knowledge). not only that, but they pull air so quickly off the wings that lift is achieved. so to idealize make it some sort of plane with no propeller or engine. if a glider is pushed fast enough it will achieve lift, like a kite. but to get a kite to fly it must move relative to the AIR not the GROUND. so a glider on a treadmill will never ever take off if it is stationary relative to the ground, no matter what is pushing on it. THAT is basic physics. however, the problem does say a plane and given that yes it will take off because the propellers and jets make the air move relative to the plane, producing lift. if mythbuster’s force calculations were correct then the plane would not have moved relative to the ground, can’t believe they didn’t realize that this neglects the conditions of the problem. thought experiment that helped me: if a giant grabbed a plane with the wheels spinning just as fast as his giant treadmill, i.e. no forces involved as both objects don’t need to accelerate, the plane will not move relative to the air which is ideally still relative to the ground. my students are die hard plane will take off fanatics initially but they all seem to come around to logic once the numerous ambiguities are removed from the problem.
Prof. M.T.Y.
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To simplify – the engines move the plane forward causing the air to move over the wings. It is the movement of air over the wings that creates lift. If the treadmill couteracts any forward movement created by the engines then there is no air going over the wings, therefore no lift.
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Here, I have proved the answer in finality because this problem was /actually/ resolved by the military in the late 60’s
This is absolutely, 100% proof.
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Ryan, it is possible you misread my comment. I said that the plane *would* take off due to the effect of the treadmill inducing air movement over the wing via friction between the treadmill and the air.
Once the plane left the treadmill it might remain within the quickly flowing boundary layer for a short time, until the plane’s jets moved the plane out of the layer.
We must all remember: this problem has nothing to do with reality. There are no engineering constraints.
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Sriracha:
Wrong: You are a troll.
Anyone who thinks that the rotational mass of the wheels is intended to be a part of this problem is grossly mistaken. Or trolling.
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Sriracha is correct. If we consider the mass of the wheels, it is possible for the treadmill to put a backwards force on the plane, and therefore to keep the plane stationary.
This is an interesting physical point, not trolling, Tom, unless you think it’s against the point of the discussion for some people to learn something.
TLDR explanation: This effect doesn’t kick in until the wheels of the plane have mass and the treadmill is constantly accelerating at some very fast rate (not just running at the same constant speed). Therefore, you probably don’t have to worry about it.
And finally before I get into a real explanation, I’ll repeat Randall’s sentiment: *practically* the plane is going to take off (or possibly burn up). We’re operating without the constraints of practicality for the purpose of figuring out some basic principles of mechanics.
A longer explanation:
The plane’s engine provides a force pushing the plane forward. This forward force exists regardless of what the treadmill is doing. Unless the treadmill can exert an equal backwards force on the plane, the plane will move forward and take off.
If the wheels are massless and their axles are frictionless, the treadmill cannot put any force on the plane, and cannot stop the plane from taking off.
However, consider just a (initially stationary) bowling ball on a treadmill (or test it at home if you don’t trust your mental picture). If the treadmill starts running backwards, the ball will also travel backwards, starting to roll as it does so. The direction of the roll will be forwards, but not fast enough to keep the ball stationary.
Now put an frictionless axle through the ball, stand next to the treadmill, and hold the ball stationary, preventing it from rolling backwards. How do you do this? By putting a force on the ball through the axle.
Now let go. The ball remains stationary. Once the ball is spinning forward fast enough to become stationary, it can keep stationary without any additional force. If you increase the speed of the treadmill, though, the ball will start moving backwards again.
Now pretend that axle is connected to an airplane. The plane can keep ball from moving backwards just like the person standing next to the treadmill: by putting a force on it. The plane needs to run its engines to do this–only barely, though, since it doesn’t take nearly a full jet-engine worth of force to keep a bowling ball stationary on a treadmill.
Even if the treadmill is really, really fast–fast enough to really send that bowling ball shooting back–the plane just needs to apply enough force for long enough to accelerate the ball from a backwards velocity to a forwards velocity, and then to a forward velocity large enough to take off.
But if the treadmill keeps speeding up, it will accelerate the bowling ball backwards, more than the plane can keep up with using its engines. Then the plane will never take off.
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Well, of course, there are ambiguities in the question that make the problem confusing. But in the the xkcd examination of the problem there are further issues. In his equations, he’s using V, which is usually shorthand for a vector, when he’s usually referring to the speed, which is not a vector, so I’m not sure what his math is saying.
But back to the problem. “The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction.” The “conveyor belt” speed is in the opposite direction as the wheels. What is the speed of the wheels? The speed of the axel relative to the ground, or the to conveyor? The speed of the point on the wheel where it touches the conveyor, relative to the ground, conveyor, or plane?
The problem needs to be rewritten.
Suppose we just said, “the conveyor moves horizontally such that the plane stays still relative to the ground (until it flies.)” What then? I assume the plane won’t leave the ground (or move at all) unless the upward thrust from the engines is sufficient to overcome the weight of the plane, which is possible if they are pointed correctly.
Also the question assumes a perfect conveyor control system which can instantly counter the forward motion of the plane. This is impossible. Any control system has errors and delays. Thus the plane will achieve horizontal motion of some sort, and once this is accepted, the problem breaks.
Back to xkcd Case 1: Vb=Vc: Suppose the pilot locks up the breaks, but supplies full power to the engines. The plane leaves a lot of rubber behind, but eventually achieves flight by skidding down the unmoving conveyor until liftoff. Case closed.
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Jason:
Although you are correct that wheels with non-zero angular mass would allow the treadmill to apply horizontal force to the plane…
1) The problem never mentions “a plane with wheels of angular mass ‘I’.” Which is what any physics book would do if the author wanted you to take the angular mass into account.
2) The problem is obviously set up to test beginners who are just getting their first experiences with free body diagrams. And at this level “wheel” means frictionless contact unless otherwise stated.
Thus I believe Sriracha is either trying to stir up trouble, or desperately wants attention and thinks he’s super smart by bringing up angular mass.
OR you and Sriracha are one and the same person, in which case Sriracha is indeed a successful troller.
I hate the internet so much sometimes.
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NO —> plane will CRASH and most likely kill people inside too stoopid to believe it will be able to takeoff.
Ok the speed is as in #2, the 747 tires spin at about twice the speed.
And there is no wind to affect takeoff. Air being blown along the conveyor not counted.
The plane will go forward to takeoff speed, which is 175-185 mph
Being accurate about the takeoff speed won’t matter too much, well because:
THE 747 TIRES HAVE A SPEED RATING OF 235 MPH!
At umm, 160mph the tires are going at about 320 mph.
Lets see if they’ll fail, YES, remember what happened to concord when its tire(s) shredded?
A couple tires will shred, hits from the high speed chunks of tire whacking into other tires, about to pop too, causing a domino effect and a CATASTROPHIC FAILURE.
That’s the right answer. The solution to make it fly is to get landing gear that can take over 360 mph for a normal takeoff, (skis?). The stall speed of a 747 is about 160 mph, but the absolute minimum speed some nut could take off at…
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Forget about the airplane and the treadmill. Instead, consider whether a truck with a model helicopter in the enclosed cargo area will weigh the same whether the copter is hovering or powered off and sitting on the floor. In my aeronautics classes, I found this scenario much more interesting than the plane/treadmill.
Extra Credit: what if the truck has an open top?
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Sriracha, Ryan & Jason are not trolling; they give the only recent replies that answer THIS wheel-speed question. Most people are answering the question as if the conveyor matches the plane’s speed, in which the plane takes off. The plane speed question is very prevalent on the Internet and is much simpler. Having the conveyor match the wheel speed adds a more complex and difficult to comprehend concept that significantly alters the answer.
Here is a glimpse into how a treadmill pushes a wheel back as it accelerates. Note the set-up in the .jpg photo: The fire extinguisher is an anchor (overkill, I know) for the rubber band that is tied to a wire that is looped through the axel of the wheel. To keep everything aligned, the wire goes through tubes that are taped to the green stool.
The wheel is resting on the belt sander. When the sander is turned on, the sander and the wheel gain RPM for less than ½ a second. During this time, the wheel shoots to the right, stretching the rubber band. When the sander and wheel stop accelerating and the RPM become constant, the wheel is no longer gaining significant energy from the belt and the rubber band pulls the wheel back to the left where it spins merrily in a steady state of energy. http://hallbuzz.com/images/unlinked/wheel_on_sander.JPG Watch the movie (1.2 MB ~ 3 seconds) http://hallbuzz.com/movies/wheel_on_sander.AVI The acceleration of the wheel stretched the rubber band in the direction of the treadmill (belt sander). This is an example of how a treadmill of unlimited speed could load energy into a wheel of unlimited strength (and through a perfect bearing) through rotational acceleration. Since the force is only applied to the bottom of the wheel where it contacts the treadmill, it is not balanced. A vector of the force is applied to the axel in the same direction of the belt. Note that it will not prevent the plane from moving if it only accelerates for ½ a second. The acceleration (increase in RPM) must be constant, and must be massive.
Watch the movie and imagine things on a much greater scale.
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Here’s a story that relates to the problem above: (Note that the term wheels in this story refers to wheels and tires)
Identical triplets Al, Bob and Chuck buy three identical bush planes. Since they live in Alaska, all three brothers buy and install large balloon “tundra tires” and wheels. The wheels, planes and brothers are identical. All three planes will take off from a normal runway in exactly 100 feet and at exactly 50 mph. The brothers fly their planes to an air show in Wisconsin. At the air show Bob finds and buys a set of fantastic wheels. These wheels are exactly like the wheels he has on his plane in every way except they have half the mass. Their mass is distributed in the same proportion as the wheels that he plans on replacing. Al thinks Bob is silly and is content with his old wheels. Bob thinks that Al will eventually want a set, so he buys a second set to give to Al on their birthday.
Bob finds a buyer for his old heavy wheels and installs a set of his new lightweight ones. He loads the second set into his plane so that it is balanced just as it was before. Bob’s plane now weighs exactly the same as Al’s and Chuck’s, but its wheels have half the mass.
Meanwhile, Chuck runs into a magician who sells him a set of magic wheels. These wheels are exactly like the wheels he has on his plane in every way except they have no mass. Chuck installs his magic wheels. He loads his old set into his plane so that it is balanced just as it was before. Chuck’s plane now weighs exactly the same as Al’s and Bob’s, but its wheels have no mass.
When the brothers leave the air show they request a formation take off. They line up wing tip to wing tip and apply power at exactly the same time. All three planes weigh exactly the same and must hit 50 mph to lift off. When Chuck’s plane lifts off his wheels have no mass, they also have no rotational inertia. When Al’s plane lifts off his heavy wheels are spinning at 50 mph and have considerable rotational inertia. When Bob’s plane lifts off his half-weight wheels are spinning at 50 mph and have exactly half the rotational inertia as Al’s wheels.
Where did the rotational inertia and energy in Bob’s and Al’s wheels come from?
How did the rotational inertia and energy now stored in Bob’s and Al’s wheels affect the take off distance of their planes?
We know that Al’s plane will still take off in exactly 100 feet; where will Bob’s and Chuck’s planes take off?
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Some people are getting hung up on the semantics or exactly how the conveyor adjusts its timing to match the wheel speed. They are missing the point. Consider the same plane-speed/conveyor question where the conveyor exactly matches the PLANE speed. How does it match it exactly? It also does not matter. If a light plane normally takes off at 50 MPH, it does not matter if the conveyor is moving at 49, 50, 51 or even 100 MPH. The plane will take off. The point is to understand that the conveyor speed in this range of speeds will have almost no noticeable effect; the plane will take off at any of these speeds. Reconsider the same wheel-speed/conveyor question where the conveyor exactly matches the WHEEL speed. How does it match it exactly? It does not matter if the conveyor’s regulator allows 1 millimeter or 1 meter of error to adjust and match the wheel speed or it somehow anticipates the wheel speed and gets it perfect. The point is to understand the forces at work and to comprehend how a huge amount of energy via rotational acceleration can be loaded into the wheels and keep the plane in place, even with the engine at full power. Such a conveyor could even move the plane backwards with the engine at full power. Note that the energy in the form of rotational inertia/momentum being absorbed by the accelerating wheels will equal the power output of the engine/propeller. The force will be coming from the conveyor, however. If the airplane sat at full power on this contraption until it ran out of gas, the energy loaded into the wheels/tires would be equal to the energy of a tank of gas minus the inefficiency of the engine and propeller.
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Now I’m not that good at aerodynamics but seeing as you may have to have a treadmill going to infinity or very fast to say the least, and it keeps the plain still. Have you ever thought of the air movement the treadmill will make (I am sure its not much but once you get it rolling fast enough) the airplane just take off at a stand still.
Though maybe through my disbelief that a plane would not take on a treadmill a guy that ask me the question with my idea that if the plane was tethered off (could not move) what speed would the treadmill have to move for the plane to take off? Although a treadmill belt is flat it will have friction to the air. Like I said it would not be much but if you get it spinning fast enough I think the plane would take off at a stand still (especially since we are dealing with infinity).
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