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 
The problem of the airplane on the treadmill was resolved on MythBusters.
Because the wheels on an airplane are free spinning — that is, the wheel turning is not what propels the plane, such as on a bicycle — they are unaffected by any motion that might be going on beneath them. Therefore, as long as the treadmill is the actual length of a runway, the plane will get up to speed and take off just like it would on a stationary surface.
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I’m still kinda sad though, Randall made a careless assumption without realizing it! Miguel Ángel Araque pointed it out, and I’m gonna reiterate.
The problem never states whether or not the engines of the plane are engaged. Then there are really two possible assumptions before approaching the problem.
A) Engines aren’t going
B) Engines are going
Case A
The plane is either assumed to be stationary (attached in front with spinning wheels) or has no forward force and arguably moves backwards.
Case B
Randall addressed this sufficiently enough so: See Above.
I think the point of the demonstration, as Miguel Ángel Araque said, is to point out that the plane has to move forward through the air to create lift, but was badly phrased when taken from its original form.
But then again, the “exactly match the speed of the wheels” confuses it, and could arguably create an assumptive forward force. In which case someone tried to create a problem in which the plane could not move forward by restraining the forward movement of the tires.
The way I interpreted the problem was like exactly match the speed of the wheels. You have a model plain tied to a treadmill with some sort of leash, and it sits there, wheels spinning not moving forwards or backwards, and naturally with no lift doesn’t take off.
Then I reread the problem, became confused, realized my confusion was a result of a poorly worded problem, and promptly sought out the guy who wrote it to cut his hand off (I bet he’s smug too until I find him)
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Of course lets not forget the plane wont ever be stationary – assuming the engines are on and generating forward force all the time. As the wheels spin up on the treadmill they are accelerating: in the same concept as a human on a treadmill the acceleration creates a force. Which in this case allows the plane to move forward.
Purely mathematically the plane can move forward and will attain take of speed (not done the calcualtion so I dont know if it can do it in the same length of runway). The wheels just have to spin a *lot* faster (because they are always having to accelerate faster than the treadmill.
Put more simply: the plan will always fight the essentially backwards force of the treadmill causing the wheels to accelerate (as pointed out) and thus the plae to move forward.
🙂 it has been proven mathematicall somewher: I expect a good google will track down the solution.
(personally I like Randall’s 5th group :P)
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So recently, I found this girl who was singing from an upper floor at my school’s library. Her voice shook me to my core, and now we are going out. I think that is a nice thing. We had run into each other once before but the only thing I could stutter in my stupor was “That was nice.”
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Those jet engines generate a lot of kinetic energy and it all has to go somewhere.
Either it is ALL going into making the wheels spin and making the treadmill spin, or the plane is moving forward.
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just wanted to say thanks for such an amazing problem [that only made sense after reading it twice]. i feel like i’ve learned the [little] physics i know here. maybe i’ll bring this up in AP physics next year or the year after…
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There sure is a lot of anal retentive people who write here.
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Well, today I was walking home from my college campus, and I felt the breeze on my face, and it was wonderful. An almost childlike giddiness came over me, from where I know not, but I felt truly aware of my world for the first time in a few weeks.
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Well, technically it’s not just forward momentum that makes the plane take off, but also the wings moving through the air lifting it up? So since it’s not moving through the air at all, no matter how fast the engines go it won’t take off.
Of course, I could be horribly, horribly wrong. It’s happened before, with much more dire consequences…
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Byron, that’s really a very nice story. I really like having a breeze on my face when I walk home from campus. It makes me feel very refreshed.
Yesterday one of my friends decided that he was going to drive several hundred miles to another state without telling anyone. After 18 hours or so of wondering what happened to him, we found a note in the parking spot where his car used to be. We figure if he winds up dead in a field somewhere he has no one to blame but himself. (I get his food if that happens.)
In conclusion, I couldn’t care less about that silly airplane. And I think everyone here is cute.
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w00t fredd loves me
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Another jab at literary critics? You know this means war, right? 😉
I’ll let you have your pick of arenas in which this can be resolved, but I’m thinking either mud wrestling or watermelon eating contest.
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Oh, but it has to be something in which neither of us would have an unfair advantage (no deconstruction or robot lassoing). Not sure if that would eliminate wet t-shirt or not.
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“Thats it! i have had it with these mother****** planes on this mother****** treadmill” should be an XKCD T shirt. i know i’d buy it
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Sapphirous- all the best to you and the girl with the soul-shaking voice.
Byron- 🙂
Not so long ago, a total stranger (an employee at a bookstore that I quite like, but don’t make it too very often) handed me a small note. It read “You’re VERY pretty 🙂 Have a nice day!” And I did. It completely made my day. It also had me spending a lot of time thinking about the impacts we can have on each other, the big differences that small things can make.
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Ummm
Has anyone actually considered the option number 1?
I ran a search on the page and it was mentioned only twice!
Randall, you are wrong about #1. It cannot be rigged with a loose conveyor belt. It ought to come out with the same result as you suggest for #3. All in all, the conveyor and wheels will be doing a load of junk to do with infinity underneath the plane, while the body accelerates and takes off.
There was no point in this post whatsoever as I doubt Randall actually reads these.
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Okay, that’s it. You need to make a shirt of http://imgs.xkcd.com/blag/sauropod.png.
I am serious and will buy it, at the very least for my dad, who used to be an aerospace engineer. Cough it up duder!!
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My friend Jen R and I trade bad jokes. She told me this one today:
Q: Why did the velociraptor cross the road?
A: To eat Randall Munroe.
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My brain hurts…
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What really worries me is that all the people with pilot training get it wrong. Yes, the plane gets lift from relative airspeed not groundspeed, that’s why it was such a good idea that jet engines were designed to push against the AIR, not the GROUND. What the ground, or conveyor belt, or unbroken plain of horizontal naked Republicans does under the jet is utterly irrelevant (although probably rather entertaining).
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If the engines themselves are simply pulling/pushing air, then doesn’t that make the speed of the wheels thus irrelevant? A 747 could be being pulled backwards along teh treadmill from the start, all that would have to occur for it to be able to take of is the windspeed over the wings to increase enough for lift.
Get the engines cranked enough and start pushing enough air around, the whole plane will move. A better question would be if a wind tunnel large enough could be erected could a plane take off with a tailwind that was directly proportional to it’s speed away from the starting location.
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If the engines themselves are simply pulling/pushing air, then doesn’t that make the speed of the wheels thus irrelevant? A 747 could be being pulled backwards along teh treadmill from the start, all that would have to occur for it to be able to take of is the windspeed over the wings to increase enough for lift.
Get the engines cranked enough and start pushing enough air around, the whole plane will move. A better question would be if a wind tunnel large enough could be erected could a plane take off with a tailwind that was directly proportional to it’s speed away from the starting location, in my opinion.
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Thanks DoomBringer, I was going to bring in my hypothetical wind tunnel to defeat my opponents but I found it was much more fun using it to blow their faces into amusing shapes. Now, everyone consider this astonishing hypothetical question (NOTE: NO TREADMILLS REQUIRED!):
Imagine a jet moving down a STATIONARY RUNWAY at 250mph.
Its wheels (complex machines in themselves but we’ll treat them as a simple entity for this argument) are moving relative to the runway at 250mph in one direction.
The runway is moving relative to the wheels at 250mph in the opposite direction.
Will the plane take off?
Answer: NO – the sheer weight of human ignorance and stupidity will keep it anchored to the ground for eternity.
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Spoilers!
The sprinkler spins backwards (whether in air or water) it does not stay still.
The Jumbo can’t take off, flight is provided it’s wings moving through air, which is not happening.
The two envelope problem each envelope has equal chance of having the larger (or smaller) amount.
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By the way Mythbusters should do a show on “the probability of Mythbusters accurately proving or disproving a myth”.
also, 0.999 = 0.999 unless you were brought up believing PI = 3
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(Only assuming the plane in question works like other planes in common use)
It really doesn’t matter what the wheels do during takeoff. Their only function is to provide a nearly frictionless plane/ground contact which could be achieved via other means. So for takeoff, it doesn’t matter how fast the wheels spin, or don’t spin. It all matters about thrust produced by the engines, which push against the air, and not the ground.
Never seen a plane with skis? They tend to take off just fine without wheels. So do sea-planes.
The question I have is would the wheels spin at all, and would they spin faster/slower/same given the setup presented.
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Spoilers!
Many humans are stupid, ignorant and illiterate.
So are kittens, but they’re cute as well, so they win.
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I haven’t been quite as irritating as I could be, so I’ll add this:
If the treadmill moves forward at the same speed as the plane the condition of the wheels and treadmill moving at the same speed in opposite directions is also met, at least relative to each other, i.e. zero.
Also, the pilot is a kitten. In a wind tunnel.
This is fun.
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So, today, I was reading the blag done by the author/artist of one this one hilarious webcomic, and in the comments section there was the MOST interesting and amusing blend of wit, gaiety, indifference, misplaced-arrogance, and random humor one could ever hope to find.
Engrossed in these comments, and all that I imagined they revealed about their authors, I stayed up way past my bedtime. I stayed up even later to compose a little message I hoped would add a dash of je-ne-sais-quoi…or french.
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As for the Mythbusters episode, IIRC, they had a problem working out the spine of the brachiosaurus and still getting that much ballistics gel to set properly. I could be remembering it wrong–they’re talented people.
The two envelope problem is a subset of the paradox of irrelevant information, but I’ll be happy to test it empirically if there’s enough cash in the envelope.
As for nice things: thanks everyone for making me laugh repeatedly, including the line of “I’m sick of these…” and the helicopter on the turntable. Also my phrase to post this was “bigger pool”, which brought back happy memories of the Ball Pit for some reason.
Please have a great day–remember, the summer’s almost over and they don’t roll over into autumn.
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Theres a reason why a plane is a cylinder with two flat things sticking out the side.
A plane moving on a treadmill, NO MATTER HOW QUICKLY, is producing no lift. None at all. It might be producing thrust, but no lift.
Planes operate under the Bernoulli principle. That is to say, they use the concept of fluid speed and pressure in order to create a force that pushes the plane upward.
The plane might be moving very quickly relative to the treadmill, but relative to the air, its speed is still effectively zero.
Thus, i agree with remark. unless the engines of a 747 pointed downward like a harrier, it could not take off.
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Get a job Randall.
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I was one of the people who subscribed to #3, thanks a bunch for explaining it without simply saying I was wrong or just explaining half of the problem and not explaining the part that I was confused on.
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The fact that there’s a squatter website at “planeonatreadmill.com” is really scaring me. Truly.
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This may be a silly question but: why does it matter how fast the treadmill moves? I mean, assuming we’re considering a perfect physics-problem scenario, with no friction between the wheel and the axle and no slip between the wheel and the treadmill — in what way does the movement of the treadmill ever contribute a force on the plane? And if it doesn’t, you could spin the treadmill at any speed from 0 to c and it makes no difference.
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@ Wanker Watson. The Internet is his job.
@ The general population reading this.
The attempt to settle the argument has, thus, sprung the argument anew.
Nice Story: After trying out for a local play, I got the part I wanted and not the part the director was about to force me into. And I didn’t have to confront them! So, I’ll be playing the part I want, and nothing else. Also, a couple old friends stopped by to chat today. We had a good long talk. It was nice.
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Dear god, please will one of the math people (or Randall himself) address the lift issue? I would just love, once, for someone who has “done the math” on all the wheels nonsense to add “of course, the airplane is producing no lift whatsoever since its airspeed is zero, so it wouldn’t fly in that regard.” As a pilot I beg you. This is as frustrating as nails on a chalkboard, or http://www.reformation.org
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so… the treadmill starts spinning, accelerating faster and faster as the speedometer in the plane swings toward infinity… at first, the plane moves forwards, but the treadmill keeps accelerating. its incredibly powerful motors push the rubber surface backwards across the plane’s wheels at a thousand miles per hour, then ten thousand. the friction in the plane’s wheels starts to tell, dragging against forward thrust of the engines. and the treadmill keeps accelerating. the underside of the plane glows a dull red as it continues to slow. the treadmill is now spinning at a hundred thousand miles per hour, providing a backwards drag exactly equal to the million pounds of thrust coming off the plane’s engines – equilibrium. for a moment the plane hangs motionless. then, suddenly, one of its wheels disintegrates. as the plane tilts, the landing strut comes into direct contact with the surface of the treadmill. the plane spins crazily out of control, bouncing end over end down the runway… it reaches the end of the treadmill and hits the asphalt doing something over mach 40, where it explodes with the force of a small atomic bomb. certain pieces of the fuselage achieve orbital velocity. do you call that flying? but that is definitely what would happen.
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GOD, the assertion isn’t 0.999 = 1
It’s 0.999… = 1 – as in 0.999 recurring.
x = 0.999…
10x = 9.999… Multiply by 10
9x = 9 Subtract x from both sides
x = 1 Divide both sides by 9
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Lift. w/o, plane goes nowhere. I have no idea where the airflow would come from if the plane appears stationary on the magic belt which, I’m assuming, would have the same acceleration of the wheels on it.
So unless some wind machine was in front of the thing, I don’t know how the thing would take off.
And I kno the force comes from the engines which come from the wheels but even if the plane is going 737 mph on the treadmill, there’s still no airflow.
When a human runs on a treadmill, they don’t feel a draft as they up the speed or incline. Likewise, the plane’s wings would not be able to produce enough airflow to take off.
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I don’t have a better link to this video, but apparently, you’re not the only one thinking of launching dinosaurs with rockets: http://www.tranism.com/weblog/2008/09/dinosaurs-and-j.html
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@ Dan and Kempn,
The reason the argument exists, and the reason you both seem to be confused, is that there is a group that sees the plane moving forward, and thus providing lift in the normal manner.
The belt is as long and wide as a runway, and the plane treats it just like a runway, more or less ignoring the fact that the belt is moving.
You both state that without the plane moving through the air, it will generate no lift. Which is true. But many people (myself included) believe that (the way the question was originally phrased) the plane WILL move down the belt fast enough to take off.
The wheels spin one direction, the belt spins the other, and this has no effect on the plane’s motion (unless you take into account friction, and make the assumption that the belt is magically capable of accelerating rapidly to infinite speed.)
Imagine it this way: There’s a scale plane with a six foot wingspan on your treadmill at home. You’ve put a bar up behind the plane so it can’t move backwards off the treadmill. Set the treadmill to “fast”, and its wheels start spinning.
Now, how hard would you have to push the plane with your hand to make it move forward on the treadmill? Not very hard, right? Even if the treadmill suddenly started moving that much faster? You could still push it forward relatively easily.
And so can it’s engines.
So, the plane’s engines push it down the runway… I mean belt… just as though the belt were sitting still. Because the forward force on the plane comes from the engines, and easily overcomes the drag produced by the wheels (unless we consider ridiculous speeds), the plane DOES move forward, and WILL produce lift, just as it does at any normal airport.
You both (Dan and Kempn specifically) seem to have a mental image of the plane sitting still because it’s on a treadmill, the same way a runner on a treadmill doesn’t move forward. And I personally think that THAT is the common misconception the question was originally created to address. A runner on a treadmill doesn’t move forward because the forward force on him (or her) is produced by the foot’s interaction with the ground. But the forward force on the plane is produced by the engine’s interaction with the AIR. So the plane can and does move forward under said force.
I understand (more than some others who have posted here, I’m certain) that there is a group who believe that the plane will NOT move, due to the friction force on the wheels, and the fact that the belt can magically accelerate at any rate to any speed. And I grant you, if you take both friction and physically improbable belt design into account, you could produce an extreme situation under which the plane couldn’t move forward. But I think that this is twisting the simple problem to extremes.
Many people picture a plane on a treadmill and don’t see how it could move forward. The original problem (in my opinion) was to illuminate the difference between a runner on a treadmill and a plane on one, and show the falsity of the preconceived mental picture of the plane as stationary. Y’all inventing an infinite case where as soon as the plane moves forward the belt and wheels zip up to infinity and keep the plane stationary isn’t really helping people to understand the BASIC concepts involved.. which was the original point.
[Captcha: considered it]
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@Anglave:
I appreciate your explanation, but I think that the problem with the problem (har har) is that many people are missing the *spirit* in which the question was asked.
Consider:
When someone first asked this question on the internet, long ago, was he/she trying to pose a physics question about the viability of a “super-treadmill” that could keep an accelerating aircraft in a fixed position in space?
Or, someone more logically, was that individual trying to suggest a “wacky idea” in which an aircraft could take off without a runway?
I strongly believe the spirit of the question is “can a treadmill be used as a launch device for an aircraft?”
To which the answer is strongly, screamingly “NO!!!!”
The “short answer” that Randall and all other physicists give is “yes, because the airplane will move.” This is perhaps true, the airplane will bust right off of that treadmill because the treadmill is impossible. Then, the aircraft will begin its lumbering acceleration to 200 miles per hour where the airflow over/under the wings is enough to lift it off the ground.
The problem with the “short answer” is that the people who really were wondering whether an airplane can take off of a treadmill are now convinced that the airplane-treadmill idea works because they read randall’s “yes” and didn’t read the math.
And this makes me want to claw my eyes out.
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Also:
The dinosaur picture is cute, but gives stupid people the impression that a 747 takes off “CUZ ITS ENGINEZ IS JUST SO STRONG!” No. It’s not a friggin’ rocket ship. The 747 takes off because its wings generate lift. The ONLY reason the big badass engines exist is to propel the airplane forward to generate that airflow. The engines don’t “pick up” the airplane.
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Dan,
I think you’re right, we should consider the *spirit* in which the question was asked.
I think it was asked to illustrate the difference in propulsion between a runner (or car) and an airplane.
I’m not proposing any “impossible treadmill”. I’m arguing AGAINST an impossible treadmill as a reasonable interpretation of the question.
And I’m not suggesting a treadmill as a “launch device” for an aircraft… honestly, that had never occurred to me. It’s planely (har har) ridiculous.
I’m suggesting that on a normal, conventional treadmill, a normal, conventional jet aircraft WILL move forward with enough speed to generate lift and take off.
And that this takeoff illustrates the misconception that many people have when they first read the problem, in which the treadmill’s interaction with the wheels somehow keeps it in a fixed position.
I’m certainly NOT saying that a treadmill is a reasonable launch device for an airplane, and I don’t know where you would have gotten that idea. We all agree that the airplane’s wheel’s speed has nothing to do with the lift generated, and that, as you say the planes engines are not designed to lift it straight up. It is, after all, not a friggin’ rocket ship.
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I wasn’t implying that YOU thought an airplane could take off of a treadmill as a launch device, I’m merely afraid that your “short answer: the plane will move forward and take off” will give other people the wrong idea.
Presenting someone (let’s be honest here, someone stupid) with the question and answer as follows gives the wrong idea.
Q: Can an airplane take off of a treadmill?
A: Short answer, people that say “no” are wrong, the answer is “yes” because the airplane will move.
If someone just looks at the above, and has someone like Randall backing it, they go “huh. I didn’t know an airplane could take off a treadmill” because they weren’t considering that a bunch of physics enthusiasts were arguing about the viability of treadmill-to-wheel acceleration.
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Dan,
In more specific response to your:
“Consider:
When someone first asked this question on the internet, long ago, was he/she trying to pose a physics question about the viability of a “super-treadmill” that could keep an accelerating aircraft in a fixed position in space?
Or, someone more logically, was that individual trying to suggest a “wacky idea” in which an aircraft could take off without a runway?”
I think neither of these things. I think it was posed as a physics problem to highlight the fact that planes push against the air while most other things you’d put on a treadmill push against the ground. To sort of generate a mental optical illusion where many people believe (for no reason I yet understand) that the plane will remain stationary.
And, to test our use of vector addition in college physics 1.
You also say:
“I strongly believe the spirit of the question is ‘can a treadmill be used as a launch device for an aircraft?'”
Well, in all the long years of the internet, I’d never considered that interpretation. How could anyone ever think that would work?
“To which the answer is strongly, screamingly ‘NO!!!!'”
I agree.
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Dan,
We’re posting at the same time. Please don’t think I’m not listening.
Actually, I think we more or less agree :>
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