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 
Heck, if we’re making groups up now:
I’m group 87.
The madman who unleashed the velociraptors.
The fools at the university laughed at me. Well, who’s laughing now, fools?
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@Mwang: Hoorah for the subjunctive!
I suppose I shall follow the great Mr. Munroe’s advice, and tell a story.
Today I drank four cups of tea and worried about a situation that I’m in, in which I am in love with my best friend and he with me but neither of us know what to do about it. I baked two loaves of bread, both of which are a little saturated with tears. I haven’t slept in a month and that isn’t going to change for at least another week. But guess what? I’m still in love, even if it makes my ribs crack open when I breathe. And that is what is wonderful about life!
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How lovely, Jekkelo!
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THANK YOU!
I will now be able to link to this post in the future which should help clear things up. When I first heard the problem I thought the conveyor belt was supposed to keep the plane in place. Of course it wouldn’t take off!!!
PS. Wanker Watson made this page very wide.
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My kitten died last night.
I buried her down in the hollow by the creek.
The good thing about burying kittens is that you don’t have to dig a very deep hole.
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Hey I wrote this at yer livejournal, but I’ll drop it here too:
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the problem with group 3 is they ignore friction… sometimes 😉
if the friction slows down the plane then it’s accounted for
if it burns up the wheels (because friction converts movement energy to heat) then it’s ignored
ps – There must be a better term than movement energy
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@Louis
A treadmill moving infinitely fast pushes the air infinitely fast as well. Since the airspeed you need across the wings in order to take off is FAR less than infinity, therefore it takes off?
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I think part of the reason the airplane/treadmill question has so many different camps is because the question has been worded a number of different ways.
For example, the way the question was posed to me in my dynamics course removed thrust from the equation. A plane sitting on a treadmill (which then begins moving backwards relative to the plane) will move backwards and fall off the back of the treadmill if the coefficient of friction between the wheels and treadmill is greater than zero. It’s like putting a hotwheels car on a human-sized treadmill.
I’ve heard it the same way except with a tether to keep the airplane from moving backwards.
I’ve heard it with the airplane producing thrust, but with the treadmill not being motorized (in neutral).
I’ve heard it with the treadmill producing an updraft under the wings that may or may not be enough to lift the airplane.
I’ve also heard it with a tailwind equal in speed but opposite in direction to the treadmill, and in a vacuum.
Clearly, not all of these have the same answer. What might be the correct answer for one of these (or even more than one), is certainly not correct for all circumstances (which I believe is the purpose of asking the question in these different ways).
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The Straight Dope has a better explanation of the airplane/treadmill problem: http://www.straightdope.com/columns/read/2642/a-plane-is-standing-on-a-runway-no-its-not-heres-why
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One thing has always confused me about this problem. If you take the ability of the treadmill to keep the plane stationery for granted, you are basically keeping the wheel struts fixed in place with the full thrust going. If you buried the wheel struts in concrete; either the connection point of the wheels or the engines would be torn off. Never mind the treadmill where the hypothetical perfect nature allows some unrealistic applications of materials technology; you’ve got a normal plane that would be torn to bits by the experiment.
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If you ignore the friction of the wheel bearings it just becomes a nice example of telling x machine to achieve infinite speed without actually effecting the environment around it (i.e. since it has no effect on the rest of the airplane it can be ignored) put simply infinity times 0 is still 0.
As an aside, is anyone else bothered that one of the recurring arguments on websites claiming the lhc will kill us all is that scientists are being arrogant?
I mean, since when does saying that reasonable scientific investigation, theory and analysis by experts in the field is more valid than armchair scientists saying “But you admit you don’t know absolutely!” been arrogance?
Ooh, that’s a bit angry, will temper it by telling everyone it’s my birthday and I got an email this morning from my girlfriends aunt completely in rhyming couplets
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I think the “arrogance” accusations come from the fact that the CERN kids have mostly tried to convince other particle physicists the contraption is safe, and expected everyone else to accept the expert consensus, which, given the fallibility of mere humans, some people aren’t willing to do. I wish they’d put more effort into their empirical cosmic ray calculations (which can be fairly evaluated by non-particle physicists): I’m sure they could’ve obtained a tighter bound than the 1 in 50 million everyone keeps quoting. As it stands, when you’re talking about ZOMG GAME OVER, I consider 1 in 50 million to be “uh, guys, let’s think about this” territory rather than “it’s negligible” territory.
But rhyming couplets! Awesome!
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I do not understand how the setup of vC=vW+vB in Interpretation #3 is valid. This equation expresses mutual dependence between the speed of the aircraft relative to the ground (or air*) and the speed of the wheels relative to the treadmill. This mutual dependence is relied upon to solve for the speed of the aircraft at the conclusion of the interpretation.
This mutual dependence is invalid because:
1) Kinetic friction does not depend on relative speed.
2) The friction of the wheel bearings is the only force transmitted between the wheels and the aircraft.
2) This friction is less than thrust. If wheel bearing friction was greater than thrust, the aircraft could never accelerate to take off, even on a normal runway.
4) The aircraft will be accelerated by thrust minus wheel bearing kinetic friction. This sum is the net force acting on the aircraft itself, and its magnitude is positive. Thus, the aircraft will accelerate forward.
5) The aircraft’s speed as a function of time will depend soley on this acceleration.
5) Steps 1) through 5) do not depend on the speed of the wheels. By induction, the speed of the aircraft does not depend on the speed of its wheels.
These are the only forces considered by the problem. Accordingly, the aircraft will eventually reach sufficient speed to take off.
*The distinction between airspeed and groundspeed is also irrelevant as it is not considered in the problem. If it was considered, the effect would only be a shift by a constant, again not producing linear or any other kind of proportionality between wheel speed and aircraft speed.
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My mind is thoroughly blown. Over a hundred comments on the airplane/treadmill thing?!?
Were the wheels acting to drive the airplane forward the plane would pop a wheelie. They don’t propel the plane- that’s the job of the engines, which are pushing against the surrounding atmosphere rather than the ground. The wheels merely coast for the ride.
Or did people really believe that the wheels are what makes the airplane move forward?
I need to go defecate.
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I’m with you in the #5 crowd. If you don’t like the problem, change the rules until you do. (:
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At first glance, I did not think that second picture down was a brachiosaurus. Too many biology textbooks at school, see.
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Turn the plane around and take off in the other direction.
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“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.'”
This reminds me of a story a friend of mine tells about attending an event in honor of Philip Glass. The punchline of the story is, “Just because you’re the most famous minimalist composer in the world, that doesn’t mean you get to cut in front of me at the mini-taco bar.”
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My brain melted after reading half of that…
I actually thought I was pretty smart too…
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@Guy (trapped in Italy):
Technically infinity times zero is undefined. 🙂
But I get what you’re saying. Intuitively it should be zero.
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Hey DJ3500, the link you posted:
http://hasthelargehadroncolliderdestroyedtheworldyet.com/
Check out the source code. Brilliant.
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There are 10 kinds of people in the world: those who understand binary, and those who don’t, and those that start counting at zero.
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It took me a while but I think I finally see most of the problems with the question. In the case of group 2, is the question confined to the plane’s activity while over the treadmill? What’s at the end of the treadmill? Is it a high brick wall?
Jonas, I hate to be a downer, but the reasons for burying bodies six feet deep have little to do with their size. Mostly they have to do with the population of large scavenging mammals in the area.
In happy story news, I bought a very special pizza made with no gluten or milk, and I shared it with my sister, which made her very happy. It tasted surprisingly like cheese.
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@JC
If you are looking for romance novels, check out Joanna Watkins Bourne. She has two books out and a third coming soon. The first two are ‘The Spymaster’s Lady’ and ‘Her Lord and Spymaster’. Well written, well researched, good characters. Great stuff!
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Two Envelopes problem sounds a lot like what Deal or No Deal plays off of. “You can Open up the case up there or the one here, but you have to trade your case away first.”
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WHERE’S THE ALT-TEXT?!?
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This is just the kind of thing that happens when there’s God around who likes to damn things.
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Randall, I really hate you some times. Why must you melt my brain so often with these problems? Even when I fully grasp what you’re speaking of, it still wastes so much of my time looking at the problem from every angle trying to determine if my thinking could possibly be flawed in some way, or trying to find a more efficient answer than what I came up with. So, in effect, you’re nerd sniping me every time.
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i would just like to say that probably a collective 15 hours has been spent waiting for alt-text
until we all went ooh ooh page source!
and then went
awwwwwwwwwwwwwwwww
a tradition was broken dammit!
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oh there it it
did i just miss that completely?
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I think the most important part of the explanation for me is that the thrust isn’t applied via the wheels, but via the engines. Therefore it doesn’t matter what the conveyor belt is doing, the plane will move forward. This is why you don’t use a dynamometer to measure the horsepower of a jet powered car. It would just fly off the dynamo and into the wall.
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This is the first time ever that i wrote anything, anywhere on xkcd, so Mr. Munroe, thanks for xkcd 😀
It’s funny, because this was the most popular conversation in my group of friends (except when they go “LHC will destroy us all, OMGWTFBBQ”) in the last weeks…
The thing is, first time i saw this problem, was a .jpg of a plane in a jogging belt with the question “Will it fly?” (Anon will now quite well what I’m talking about).
So i thought what the problem, so stated, meant, was to trick you into thinking that the plane is rolling, full throttle, on the belt but without moving from its place. That is, no windspeed over the wings, no lift, no take-off.
(This is quite obvious, of course, but one guy among all the people i discussed with, held that the plane would indeed take off, no windspeed whatsoever…)
So for me it was always a more basic physics question (how does lift work), in which the trick was done by presenting you a setup in which you imagine the plane stationary, as you would imagine a stationary jogger. Of course, you can run faster than the belt and the plane could go faster, but it’s somehow implied on the fact that it is a running belt that the plane doesn’t move, and not a treadmill.
So, it’s more a question of visual semantics, so to say. At the end, it’s more interesting even how the people tend to interpret and discuss the thing, than the thing itself…
Of course, the way you presented it (the way it’s usually discussed) it’s entirely another problem. And one in which the people tend to shout “yes!” or “no!” depending on the assumptions each made.
Appart from what you said, if this were not a liner, but some RC plane in which, for example, the thrust of the engine were not enough to overcome the friction of the wheel’s axle, then the plane would not take off but, possibly, be pulled backwards by the treadmill.
So, the thing depends quite on each one’s assumptions.
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Owcrap, that was long, sorry!
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You are on a Cessna 150 is on a 1500 ft by 50 ft treadmill. The treadmill is moving backwards at 20 ft/sec, while the Roll-Royce engine produces 150 HP. Further complicating the problem is the improvised explosive being hung over the Cessna, which has a roughly 5 kiloton explosive payload, however the exact specs are unavalible because the bomb was made by evil alternate universe Macgyver. Due to Macgyver’s inter-universe travel, a large spacetime anomaly visually resembling a large sandwich is advancing from behind you a rate of 42 ft/sec. Your copilot, normally adept at these strange situations, is currently unable to help because he is currently attempting to decipher the bizarre rules of french grammar, particularly frustrated on a problem involving irregular verbs. Will the Cessna:
A.) Explode
B.) Take off
C.) Rise straight up off the runway
D.) Cease to exist
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I love XKCD, first time in the blag so i think i must say it, to all programmers reading this Happy Programmer Day , if you don’t know why http://www.programmerday.info/
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I have no one to celebrate my twenty-first birthday with!
Oh, wait, that isn’t a nice story. That’s a sad story. The internet has me confused.
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I’d just like to say thank you very much for explaining this. As a group #2 member, I could never find a group #3 member to explain to me why they believed that. It’s a forbidden topic in 90% of forums, IRC channels, and newsgroups, and the other 10% has no idea what is going on anyway.
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A fun thing happened to me the other day in /b/: The switching doors dilemma was cleared up to me. It turns out changing your door doesn’t change your chances of getting the big prize. This is because the door you choose before changing your choice doesn’t matter in any way, contrary to popular belief.
What do you think you get when you pick the first door? You get nothing, literally nothing, other than the choice to stick with that door or pick the other one. This is the significant choice, the other is an illusory choice from which nothing can be gained or lost.
This should be even more obvious with the envelopes, where the distracting third door is preemptively removed from the equation. No matter how many times you change your mind, your final answer is the only one that counts.
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Randall you silly billy! Because of your failure to include an alt-text in today’s comic, people kept coming into irc and asking US why there wasn’t one, despite the fact that it said so in the topic >_>
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It’s different.
Given:
Wrong door 1
Wrong door 2
Right door
* First choice is Wrong door 1
Then Wrong door 2 is removed
Switching ALWAYS returns Right door
* First choice is Wrong door 2
Then Wrong door 1 is removed
Switching ALWAYS returns Right door
* First choice is Right door
Then either Wrong door is removed
Switching ALWAYs returns *a* Wrong door
If your initial choice is truly random, that is, if choosing whatever door is equiprobable, then, given that the second choice is always switching (100% probability), the probability of getting the correct door by switching after the initial choice is 2/3 (66.666%).
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Of course, given that there are three doors and that one wrong door is removed after the initial choice. Those are the rules of the game. If no wrong door is removed after the initial choice, this doesn’t apply.
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Well, it depends. If we forget the math for a moment, here’s how I see it:
The plane using a jet engine flies because of the difference in wind velocity under and over the wing. This difference pushes air up, which causes the lift. The most important pre-requisite for this concept to work is a non-zero relative velocity of the PLANE and AIR (not ground).
Now, if we interpret the treadmill to be set up such that it exactly retards the motion of the wheels, produced by the jet engine, then the plane doesn’t really move, relative to treadmill or ground or air. Hence, there will be no upward lift and consequently, the plane will not take off.
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@RSB
That’s why this is so confusing; it all depends on how you interpret the question.
How about this: Imagine a craft capable of self-powered flight 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 part(s) of the craft that is/are touching the conveyor belt, moving in the opposite direction. Can the craft take off?
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Am I the only one who sees the two envelope paradox as an equation error not a paradox?
Specifically, if you label your envelope “A”, then the other envelope is 1/2((2/3)A) + 1/2((4/3)A), and not 1/2((1/2)A) + 1/2(2A), because you don’t know if A is the larger or smaller value, so you treat it as 3/2 of the smaller value. Done this way, the probabilistic value of the other envelope is (1/3)A + (2/3)A = 1A, or probabilistically the same as what you currently have.
Now, I’m no math major, so am I totally wrong about this?
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>Hylas says:
“Ok, but what if you put a helicopter on a giant turntable….”
Brilliant sir. Brilliant. One thousand internets to you.
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See this for feynman sprinkler experiment results, showing that it does rotate: http://www.physics.umd.edu/lecdem/outreach/QOTW/arch4/q061.htm
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RANDALL IS A CUNT.
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I’m not sure you read these, but I just want you to know I saw someone with the Raptor-Free Zone t-shirt today. I never thought I’d see an XKCD shirt.
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