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 
This is just the kind of thing that happens when there’s God around who likes to damn things.
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The pterosaus. They explain everything.
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So, this was all a build up… to basically tell people “don’t be jerks on the internet”? With some physics thrown in as an example?
XKCD is amazing.
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Why haven’t the mythbusters done the Feynman sprinkler experiment.
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Randall,
THANK YOU!!! There IS a group #4, and it’s people like you and me who can actually tell that there’s vital data missing from the question. Unlike me, however, you have the patience to try explaining it. So from now on I have somewhere to send people when they think they’re so smart by having an opinion about this.
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Surely we’ve all seen the MythBusters episode in which the plane did, in fact, take off?
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i like group 5… i think the answer to what would happen after round 4 ushers in the two envelopes paradox is that rounds 5, 6, and 7 would be delivered in quick succession followed by a clever quip along the lines of “so…. how about that local sports team? they’re really doing well/poorly this season!”
and we’d all laugh about all of those people who are wrong on the internet!!
grins!
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With no intended offense to Master Munroe (and all due exaltation), I would like to point out that the emphatic “was” in the penultimate paragraph should actually be the subjunctive “were.” I point this out only because of its italicization and its proximity to the denial of Internet debates, and because I am among the few people who advocate use of the subjunctive in all applicable situations. I also rally for the inclusion of declensions in Modern English.
Also, I think the story has a subplot with the concussion of “everything is better on treadmills.” I think there’s a hidden agenda here.
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Then there’s the crowd of people who look at the others loudly arguing in front of the buffet line and think it’s all too much effort to get to the miniature brownies .
As for another topic to discuss… has anyone come across any good research on Romance Novels lately? I’m always up for some good recomendations.
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Two astronauts are in a revolving space station, designed to create the sensation of gravity. One astronaut says to the other “Hey man, what’s up?” The other responds: “yeah, I was just thinking the same thing…”
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Is that a Jetpack Brontosaurus?
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Then, we all agree it wouldn’t move because what gives you lift is airspeed (well, what really makes move the plane is the pression before and after the engines…), not groundspeed, aren’t we? Then why aren’t we all on the plane drinking with #5 crowd? Please, Scottish for me!
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I’m sick of these motherfuckin’ airplanes on these motherfuckin’ treadmills.
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Well, 0.999… DOES equal 1. It’s a rather easy, straightforward proof.
I lol’d on #4. I’ve been bickering (like a retard myself) with those goddamned conspiracy retards for over 2 years now. I’m just masochistic like that, I guess.
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Ok, but what if you put a helicopter on a giant turntable…
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There is a list that I subscribe that every time a discussion becomes pointless, and it can be on any subject, someone posts either “the plane does take off” or “the plane doesn’t take off”.
This “paradox” has become a synonym of “stop wasting electrons on this subject”.
XD
By the way, why would someone put a 747 on a conveyor belt?
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> With no intended offense to Master Munroe (and all due exaltation), I would like to point out that the emphatic “was” in the penultimate paragraph should actually be the subjunctive “were.”
Hah! Someone else noticed. I switched that back and forth a few times. The problem is that the next clause puts the whole thing in the past tense, and is referring to something that actually did happen. So the implication is that the plane has already done its thing, and we’re speaking about a consequence of it now. It’s like saying “Even if Neil Armstrong WAS the first person on the moon, he doesn’t get to cut in front of me in line for the tilt-a-whirl.” I’m not sure that’s actually in the subjunctive.
If any grammarians want to jump in and correct me, please do.
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A question with the airplane and a threadmill that moves as fast as the airplane / its wheels detecting the speed of the airplane and without any time delay is of course a nice one, but it is only pretending to be a question about physics. In fact, it’s a logical one. I thought you promised to never squeeze humor out of self-reference again?
Coolbreeze85: Yes, right, but people flying planes mostly assume that the speed of the air around you isn’t high enough to take off without accelerating. Of course, you can also get a huge fan, put it in front of the plane and laugh about the threadmill people. Or use pterosaurs properly and laugh about the threadmill people.
The problem with solutions like “shoot Schroedinger’s cat yourself and therefore be sure it was killed”, is that they don’t actually explain the situation but only take advantage of the inaccuracy of the task. See also the famous Barometer Problem, or the Prime Choice story.
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The pilot will throw up.
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4 basic forces act on a plane as we all know. lift, thrust, drag, gravity.
a plane moves forward when propelled by thrust, (quarter million producing engines would do it.) but in order to TAKE OFF, air must FLOW beneath and past the wings. so, on the treadmill, thrust is happening, but no airflow. the plane would probably shoot forward at takeoff speed, but on one plane (no pun intented) of movement – forward. in order for a plane to effectively take off, it needs air flowing under its wings, a significant amount of air flow is not present on the treadmill.
also, the helicopter idea is interesting, but the directions of the rotors and the turntable would have to opposite or else the rotors would appear to stay in place while the helicoper turns on its central vertical axis. if you have ever seen a helicopter crash because its tail rotor is blown off, you would get the picture. watch blackhawk down, it shows it perfectly twice how the copter simply spins to counter the main rotor spin when the tail rotor is not present
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Hmm… not to contradict you, Randall, but the answer to the sprinkler problem, as given in the Wikipedia article you linked to, is that the sprinkler DOES turn; and moreover, it turns in the reverse direction.
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Shouldn’t it have been pluperfect subjunctive? “Even if the plane had been able to take off…”
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(Actually, from the fact that you don’t have any infomation about the air movement around the plane, but are supposed to consider every physical detail and we think of it as a PHYSICS problem and not a logical one, one has to conclude that the question can not be answered with the given information. Taking the problem as a question of logic would lead to Randall’s interpretation above, “What happens if you take a plane that cannot move and move it?”. That means, the information is either inconsistent (logic) or incomplete (physics).)
Jace: The Wikipedia article states that the result you describe is shown by experiments but it is not yet clear how representative they are, as there are complex effects. So as far as I understood, there’s just no proof of any solution – yet.
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Also, if Monty opens one of the doors with a goat behind it, you should switch your choice.
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Hey, asteroid! Good point– I guess my question was, if Wikipedia’s article (and its references) show that it does move, why did Randall say it doesn’t? Obviously it’s a little more complicated than he made it sound…
As to complex effects… I kinda think that the original statement of the problem (as expressed by Feynman) was, “if you do the experiment, what’s the result?” So I figure that by definition the experimental results the correct answer. 🙂
Now that you mention it, though, I am kinda curious as to what the answer would be in the absence (for example) of all friction.
Wait a second… where’s my mom?
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Depending on what degree of skepticism/exasperation you wanted to express, you could also go with “Even though Neil Armstrong might have been the first person on the moon…”
thought I think bradluen’s response is more accurate.
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I used to be a firm believer that the plane would fly, but I realize the errors of my ways:
The plane has to move forward because of the thrust from the engines. this makes the wheels spin, and the treadmill runs in the opposite direction, which causes the wheels to spin faster, and the treadmill accelerates to infinity. Whatever drag may have been miniscule before is now infinite and pulling air across the surface of the treadmill at an infinite speed, and our good friend Bernoulli holds the plane down instead of up.
Isn’t our magic treadmill cool?
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Wouldn’t the past subjunctive be “even if the plane *had been* able to move”?
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Jace: The question is not only “What happens if you do the experiment?”, it’s more like “Does the same thing happen if you do the experiment several times in different situations?”. Just think of the xkcd comic #242.
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From an alt-usage-english.org FAQ (linked in name):
Possible condition, present: “If I am…”
Possible condition, past: “If I was…”
Counterfactual condition, present: “If I were…” (common usage allows “If I was…”)
Counterfactual condition, past: “If I had been…”
More importantly, does assessing the truth of counterfactual conditions require parallel universes?
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You for got one last option, the plane flies, stationary, in the ground effect produced by the coanda adhesion of the air along the moving belt. It just doesn’t fly very high. 🙂
Hey, if belts with infinite speed are allowed…
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mythbusters did an episode dedicated to airplane on a converbelt…go watch it…would put an end to all the debate
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Any chance of a Pterosaurs For Stability inspired xkcd T-shirt? I love that drawing!
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I second Angus! That is a truly awesome drawing.
And for the record, what actually happens is that the treadmill accelerates to infinite speed, the wheels blow up, the plane hits the treadmill that is moving at infinte speed and is ripped to pieces then gets sucked into the workings of the treadmill and the whole thing goes BOOM. This is could probably be made into a blockbuster movie if you took the unimportant math bits out.
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The plane takes off because Chuck Norris is pushing and Jack Bauer is at the controls.
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Well clarified, sir.
I was a member of the #3 crowd, due to being introduced to the problem by a gentleman who lacked effective language skills when communicating physical problems.
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Now Randall, I’m disappointed. Yes I am.
This is essentially a duplicate of the discussion on the forums about the last comic (where this link was posted, which also has the three explanations: http://www.planeonatreadmill.com)
Sorry, I usually like your blag as well as your webcomic, but that’s because you always brought up something new or something old in a new way.
Btw, I did not take part in the discussion on the forums, so please don’t see this as a personal attack… On other news, the webcomic today was hilarious 🙂
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@Correction: the link should be: http://www.airplaneonatreadmill.com/ , sorry I made a ‘typo’
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ok so for the two envolopes paradox, people forget to integrate time into the equation. someone might say let me think this through for the rest of my life, or they might say “give me the one on the rightleft” because the outcome of the equation is always based on how much time the subject is willing to spend making a decision.
also people like taking the gut choice, say the one that has a bigger bulge….
anyways, i would know nothing about probability if it wasn’t for you so keep up the good work 🙂
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Of course after the 4th round of cocktails the philosophy students make themselves known by bringing up paradoxes involving a cat. Personally I usually end those conversations by declaring that the cat being undead and needing an appointment with my boomstick. Yes, we can take these philosophy majors with science!
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er…that should read “…that the cat is undead…”
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The whole problem goes away when the physics of the problem gets properly defined, when you put a m for the mass of the plane and and Ft to define the trust and Fd to define the drag. Then you can’t have the acceleration a nonzero. Done. Once you do this, there is no way the plane can’t take off. 🙂 I don’t see any paradox, I only see badly defined physics question.
I think I’ll get banned now… What a shame…
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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.
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A nice story: absent the quark-joke pick up lines for what I hope are obvious reasons, I have been able to get the undivided attention of some difficult classes during my last three lessons using exactly the same arguments as today’s comic. It’s slightly annoying that the reason they’re so interested is because they’ve all been convinced we’re about to die, but we can’t have everything.
I assume everyone has seen http://www.hasthelhcdestroyedtheearth.com by now, but in case you haven’t, there it is.
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Oh, and as of 10:23 Geneva time, the LHC works, bitches.
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The logical inconsistency pointed out above results in the plane and treadmill turning into a sperm whale and a small bowl of attractive flowers, respectively. Given the relative weights, the ex-plane does fly, at least briefly.
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I don’t think I really get the envelope problem. I see it as “some-one gives you an envolope with money in it”. This is a win situation. If you switch the envelopes, you also get money. It doesn’t matter! The result is free money, and time spent trying to work out which has the most is time that the giver of free money could use to change their mind.
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@ “post nice things about other peoples? stories”
Let me tell you a beautiful story about a couple of friends arguing whenever a thread stretched between two simultaneously accelerating spaceships would break due to the relativistic contraction…
They were all wrong! Relativistic speed restriction is invalid assumption because all writings of Einstein were done by his wife!
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The number 3 arguement is completely flawed anyway:
If the tredmill is completly stationary, vW = |vB|, which makes vS = 0. As vC is set to match this speed, the tredmill with never move.
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