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 
Helicopter on a Turntable != Airplane on a Treadmill (and neither == Snakes on a Plane)
To make them homologous, you’d either:
Need to put free-rotating ball-bearings along the support-thingies of the helicopter. (Sorry, not a pilot — don’t know the jargon)
–OR–
With a prop-plane, you’d have to have some way for the engine rotating the propeller(s) to be free-rotating, so that you get a Feynman sprinkler effect with the propeller engines. (In both cases, adding snakes will make it more interesting)
In either case — the former doesn’t equal the latter since, as everyone has said, ad nauseam, the wheels aren’t powered by the engines (and thus countering their movement is essentially moot) — whereas the helicopters movement is based on the lift generated by the rotary motion of the blades (and could hypothetically be countered by the turntable — I have no fancy equations or experimental evidence to present for the helicopter scenario, hence “hypothetically”).
Some of the stupid that burns on here reminds me of the type I see over at Answers in Genesis.
The big problem that everyone seems to be ignoring, is that everyone is assuming that Chuck Norris *ISN’T* standing in front of the airplane.
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Im pretty sure this has been proven in mythbusters, the airplane flies.
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How about the freefall-tower you through out of a plane? What about this one here?: http://www.youtube.com/watch?v=XUuwEq98ByM :D^^
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I don’t know about the rest of you but I like a fresh strawberry now and again. So tasty.
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I want a treadmill capable of producing the overpressure required to amplify ground effect.
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I like to think of it like this: there’s a plane traveling at 600 mph or however fast it travels, and it’s in the sky. You pluck it out of the sky, and drag it down to a treadmill. Now why would the plan suddenly magnetize itself to the treadmill and refuse to take off?! Makes no sense. Ergo, plane flies.
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I think that there would not be enough lift generated. All 4 groups got their physics wrong. Airplanes take off due the the lift generated by the pressure difference between the top of the wing and bottom of the wing. The shape of the wing means that the air on top is slower and takes longer to reach the other end, causing lower pressure on top…and the plane takes off. Engines are present to generate motion, that causes air to pass over and under the wings. You could theoretically add a giant fan that generates wind at speed corresponding to Vc…
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Forward motion is required to produce lift under the wings. That is, air passing over and under the structure thereby creating lift due to the difference in surface space on the top and bottom of the wing surfaces.
0 lift under the wings will not allow the plane to fly.
It weill drop like a rock even with the engines running full blast.
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“Forward motion is required to produce lift under the wings. That is, air passing over and under the structure thereby creating lift due to the difference in surface space on the top and bottom of the wing surfaces.
0 lift under the wings will not allow the plane to fly.
It weill drop like a rock even with the engines running full blast.”
EXACTLY! This (more or less) was what i was thinking. It doesn’t matter if the plane can go forward or not, if there isn’t any drag on the wind flaps due to air particles (or w/e) being displaced and whatnot, then how will the plane be able to fly? That’s the whole principle of flight, that’s how birds fly, that’s how paper airplanes fly, that’s how wind tunnels work!
MY solution to the problem would be that the plane cannot fly. It’s physically impossible. Even if the plane were able to outrun (which it can’t, due to the limitations of the problem and such), … well, if the plane were able to escape the conveyor belt, then yes, it could fly, but it can’t, so it cannot fly. The answer is nooooo.
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while it’s not really a solution, it kinda, sorta, does explain it… right? I mean, unless the plane’s in a wind tunnel … or the treadmill’s creating a vacuum type situation, but the bottom/reverse strip of treadmill would cancel that out ..
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oh oh oh, random thought, can there be a “bump” on the treadmill? Or do we assume for the sake of the problem that the treadmill is completely smooth (“massless elephants on frictionless planes?”)? Because, if there were to be a “bump”, the slightest amount of space between the wheels and the conveyor belt would allow the plane to shoot forward, or maybe it’s just that the engines of the plane will tear off of the plane and all motion will stop? too many possibilities … anybody up for investing a few trillion into the research of this problem?
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“It weill drop like a rock even with the engines running full blast.”
if it drops, then the engines can produce thrust, or suck air from the front and out the back, whatever it does, like being in space and tossing rocks out of the back of your spaceship, you will move due to the forces and whatnot, i know i’m not really phrasing it very elegantly, but it’s still the main concept and blah blah. But yeah, if it drops, it’ll fly 😉
p.s. – I seriously think we should invest our tax dollars into research of this problem …
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ahh … shit … i finally figured it out 😦 it doessssssssss take off. wow, that took me 3 days, and neglecting my final exams too, effff me >.<
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The physics problem explicitly mentions a type of plane:
“Imagine a 747 is sitting on a conveyor belt”
The 747 has four wing-mounted engine nacelles which require significant air intake to produce thrust. The thrust is used to produce acceleration which produces lift beneath the wings.
If the airplane doesn’t accelerate, i.e. it stays in place, then no lift is created. If there’s a particularly amazing headwind it might hop forward a bit. That’s the extent of it.
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Isn’t the simple answer that while a car on a treadmill would not move, the airplane will because the forward force isnt a result of the WHEELS? Wheels will spin as fast as they want with outside forces, ie the treadmill + jets.
I don’t understand why its more complicated than that.
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Nice math, too bad you FAIL.
The wheels aren’t doing anything. The treadmill could cause the wheels to spin as fast as you want in a reverse direction, but that only produces a tiny…tiny…tiny fraction of an amount of backwards force due to friction.
Consider this: if the treadmill moves equally as fast as the wheels on an aircraft, THEN IT’S NOT MOVING. The wheels don’t move the plane, morons.
So compare that tiny force to 4 massive jet engines and you need not have me explain the rest.
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The plane engines push against the air, thus accelerating the plane forward. It does not require air to be forced into the engines by the plane moving forward, the engines do that themselves. There’s something in there called a turbine, idiot. It’s purpose is to PULL AIR into the engine.
It does not push off the ground for momentum.
Therefore, you will always have acceleration, thus lift, and thus a flying plane.
If you’re too stupid to realize this, please stop posting and go hang yourself.
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“Fnet = Fengine – uk*Fn – Fair
Fengine is incredibly massive compared to uk*Fn, and Fair would be nothing at rest, so at rest you could approximate as F = Fengine, and the plane will move forward at full speed.”
This makes sense… but since Fnet is positive, the airplane will have positive acceleration. Therefore, the treadmill would have to accelerate similar to the plane. I don’t get how such a treadmill is possible. (This could explain the Mythbusters result–their treadmill wasn’t accelerating!)
Please correct me if I am wrong.
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will the turbines spin without air traveling towards them?
past the initial movement spinning that is?
let me sound scientific but not actually be such at the same time.
doesn’t the instantaneous acceleration (the derivative) rely on the limit of the speed up to that point?
like doesnt air need to be moving faster to attain faster speeds for an airplane?
i know the answer is no but im not allowed to contradict myself, so let me release my ignorance into the world.
(i only realized i was wrong after thinking of a wind tunnel. WAIT A WIND TUNNEL DOES HAVE AIR FLOW! JK!)
OH WELL, im too much of a noob to go read everything else and see if someone has asked this so ill double post just for shits and giggles and then never come back to read this. tootles.
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Here’s an interesting version of the “two envelopes” problem:
There is a game show, where you choose from three doors. One door holds a new car, and the other two hold donkeys. (Don’t ask.) After choosing once, the host would open a door that held a donkey and was not the door you chose. (The host knows the locations)
You can then choose a different one. Now, since I saw this on “Numb3rs” the response they gave is that it behooves you to switch.
Their logic: The first choice has a 2/3 chance of being a donkey.
When one door is opened, the probability of the third door having the car is 50/50 but they neglected to mention that the chances of the first door changed too.
One problem I have with statisticians: They raise themselves above the subject matter. (I’ve been in a statistics unit in school and my teacher was doing this.) She was teaching us things that made mathematical sense but had logical errors. Lots of them. Remember: Statistics emulates the real world, not the other way around.
P.S. I never used “affect” or “effect” in this, so sorry for stealing your fun.
My grammar good is.
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Here is a new topic of discussion, kinda related to reverse sprinkler problem:
Why can you feel your breath when breathing out, but not when breathing in? Go ahead, try sucking out a match instead of blowing it out.
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Hmm well it would take off. You dont even need anything beyond high school physics, never once does the problem say that the plane is moving at a constant speed or constant direction. now i dont know much about planes but it must have some way of creating vertical force. The treadmill cant counteract verticle force, hence the plane could gain altitude easily, at least enough to jump the treadmill and take off. Or i guess if you wanted to you could drop an Infinate Improbability Drive, which wouldnt need a lift off speed…or maybe a Bistromatic Engine would solve the problem…
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That’s really cool!
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Rossiya!!!! There is answer in Wiki. Vsem russkim i ne russkim privet!
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Not having read all the replies, but actually liking to thing about these things, I looked at the paradox from the f=ma POV. Sure enough, the airplane gets away. One interesting point: an unpowered treadmill track would actually slide forward with the accelerating plane. Draw the appropriate free body diagram, and you’ll see this is so.
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Mythbusters did it.
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The wheels don’t push the plane forward, they make thrust easier. It must go forward because the thrust of the engines is pushing the plane forward. Just because the wheels aren’t HELPING the aircraft doesn’t mean it won’t move. All we have to do here is remove the wheels and turn off the giant treadmill. The friction of the treadmill is the same on and off with the wheels (respectively on and off of the plane). If you can safely scrape the bottom of the plane far enough and fast enough down the runway (oh, what beautiful sparks and burnt rubber that would that make), it’ll take off. Wow… and I thought I was just a dumb psychology major…
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I successfully confused two of my (MIT educated) friends with a similar type of problem. Here it is: Two cars traveling in opposite directions, each at say, 60MPH. A tire from one car disintegrates and hits the windshield of the oncoming car. Ignoring aerodynamics and drag, and assuming the rubber leaves the tire instantly, how fast can a piece of tire be going- when it hits the oncoming car- at impact with the windshield? Also, can the rubber from a car going in the same direction as you that’s behind you, still hit your car? The answer is a little surprising. Fun, fun, fun…
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For simplicity one can ignore gravity as well- pesky gravity.
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Everyone is too busy looking at the wheels. In engineering, we are taught to draw free body diagrams of each problems and analyze the forces acting on the body. So here goes the forces:
1)Thrust (acting in the forward direction)
2)Drag (acting in the rearward direction)
3)Weight of the plane (equal to mass times gravitational acceleration) (acts in the downward direction)
4)Lift (actually the addition of two forces, one on the top and the other on the bottom of the wing. The net force (subtracting the bottom from the top force) is lift. This force acts upwards)
5)Normal force on ground created by the wheels holding the aircraft. This force is Normal Force = Weight – Lift. You’ll notice that when the lift created by the wings equals the weight of the plane, the force on the wheels equals zero, as would be expected, as the plane is taking off at that point. This force acts upwards.
6)Friction created by wheels. This force is incredibly small compared to the rest. For example, drive your car up to 20 km/hr or so and put it in neutral. It will coast for quite a while, because the friction is relatively low. The same is true for airplane wheels. This force acts rewards (as does drag, they are both frictional forces).
So after analyzing the forces, here are the conditions we need for takeoff:
1) Thrust must be sufficiently high enough to overcome both Drag and Friction.
2)Enough Lift must be generated to overcome the weight of the plane.
Now based on these conditions, the plane simply needs an engine large enough to overcome both Drag and Friction. With Drag being equal to maybe >98% of the total resistive force seen (and Friction <2%) it is safe to assume that friction created from the wheels is negligible. The plane will fly.
The main assumption that people (wrongly) make is that the wheels are propelling the plane – THIS IS NOT TRUE. The reason we can make treadmills and dynos capable of outrunning people and cars is because both people and cars depend on the GROUND for their thrust. We push off the ground with our feet and impart a force that moves us forward. Cars push off the ground with their tires and impart a force that moves them forward.
Planes push off the AIR with their JETS and impart a force that moves them forward. Whatever the ground is doing is insignificant, as long as it is sufficiently smooth.
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Thank you, Lane…FINALLY, someone who knows what they’re friggin talking about. I propose a simpler version of this problem, so the ‘tards who say it won’t take off will hopefully feel a little more ignorant: You’re sitting in a shopping cart with a couple of airplane wings strapped to it, all of which is on a treadmill. Now duct-tape a jet engine on the back of the shopping cart, and start that sucker up. Are you going to fly forward and take off? Yes, because, just like a 747, shopping carts with jet engines strapped on them are not propelled by the wheels’ interaction with the ground. The power is in the engines’ interaction with the air. Seeing as treadmills don’t have any significant impact on air, the solution should be simple…but as was pointed out earlier, this is the internet…
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@lane. (actually with lane)
YES.
lets break it down like this.
0) what is the difference, if instead of having the engines create the thrust, you shut the off and hook up a steel cable with a winch that pulls forward ? (the winch being located outside of the treadmill)
1) can an airplane move forward 1 inch ? (how about 2, 3, is there a threshold?)
2) can the airplane accelerate ? (1 knot ?, 2? is there a threshold ?)
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Lane, everyone else @Lane, and anyone else who thinks planes are neither driven nor slowed by their wheels,
Planes are not driven by their wheels, because the wheels are not powered. That’s obvious.
However, when the wheels of the plane are in contact with the ground, the friction between the wheels and the ground adds drag to the plane. Just think of a float plane. A float plane isn’t driven by its floats any more than a 747 is driven by its wheels. But a float plane must overcome the friction of water-on-float in order to accelerate.
So instead of an airplane on a treadmill, think of a float plane in a current. It’s the same free-body diagram, except total drag is equal to drag due to airflow PLUS drag due to plane-to-surface friction.
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The question is deceptively formulated and conceals the issue at the heart of this problem: not velocity, but acceleration.
We can assume that the plane starts with a velocity below escape velocity, else it would already be airborne. Moreover, whether it begins at 0mph or somewhere in the middle isn’t important in principle – the goal of the plane, whatever its initial velocity, is to accelerate to the velocity at which its wheels leave the green earth. The goal of the treadmill, therefore, is to prevent any acceleration, as with a non-zero positive acceleration the plane will eventually reach escape velocity. Or more realistically, with a non-zero positive acceleration the question turns into an engineering problem: are the plane’s engines strong enough to accelerate it quickly enough to reach escape velocity, even when opposed by the treadmill?
So, the question becomes: can the treadmill prevent the plane from accelerating, by adjusting to the plane’s velocity?
And the answer is no, it can’t. In order for the treadmill to prevent the plane’s speed from changing, the treadmill must be able to speed up itself. But the treadmill’s speed is tied directly to the plane’s wheels speed and in order to speed up it must detect a change in the plane’s velocity. In other words: in order for the treadmill to accelerate, the plane itself must first accelerate. We know that the plane will accelerate because of the force delivered by its thrusters. The plane has therefore a non-zero positive acceleration, and eventually takes off.
Moreover, the treadmill can’t keep the plane from accelerating even by adjusting for the plane’s acceleration, for it would not be able to prevent a change in acceleration, or, jerk. You could keep rolling down that chain of derivatives forever, becoming perhaps more efficient but never truly keeping that plane on the ground. This is because these systems – adjusting for impulse change in velocity and adjusting for impulse change in acceleration – are identical in concept. An impulse change in velocity is a discontinuous function, which means that not only would you change the function itself, but you change all of its derivatives. Therefore, speed, acceleration, jerk, etc. are all the same thing here. And given a long enough treadmill (or a short enough treadmill) the plane can not be prevented from accelerating. Its acceleration may only be incompletely opposed.
The only way to hold the plane down would be to adjust the treadmill’s speed (and thereby, the acceleration due to the frictional force on the landing gear) to exert a frictional force equal to the thrust produced by the plane’s engines. But then the question – if a plane’s thrusters are perfectly opposed by some other force, can it take off? – is really dumb.
Don’t believe me? Go to http://mouser.org/log/archives/2006/02/001003.html and the MIT grad will explain it to you.
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wikipedia seems to think that a Feynman sprinkler does move, actually.
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Let’s think for a moment about how a treadmill would be able to hold a plane still against the humongous amount of thrust from the jet engines.
If a treadmill were to hold a plane still, it would have to produce a backward force at the landing gear equal to the forward thrust produced by the jet engines. I don’t know the wheel bearings of a 747, but I suspect the wheels would have to be rotating backward at a speed which is multiple times the normal takeoff speed to have this kind of effect. I suspect tire damage is a very real possibility as well.
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Let’s clear up a lot of the crap floating around here
1) jet engines do NOT require an incoming airflow to work. In fact, they work better if the air is stationary (essentially) when it enters the diffser (the part of the engine cowl that sits out around and in front of the first compressor blade)
2) A 747 weighs about 750,000 lbs. The MAXIMUM mu that you’re going to get from AWESOME race tires that are properly heated (lets be clear, this is NOT the kind of tire on a 747) running on perfectly set up asphalt is in the range of .2 This makes the absolute MAXIMUM drag due to the wheels 150,000 lbs.
3) a 747 can produce 266,000 lbs of thrust. More when stationary (The theory here is based on how jet engines work by accelerating a mass of air)
4) This means that, regardless of how fast you spin the wheels, the plane will move forward, because even if you spun the treadmill till the wheels started to slip (because of bearing drag becoming huge) the plane would just drag them along.
266,000-150,000=116,000 In case you don’t understand some of the argument this equation should clear it up. You have 166,000 lbs of ‘extra’ thrust beyond what the tires can resist. The plane moves forward, and will take off.
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Not true about “race tires” having a mu of 0.2.
As a race car engineer, I can tell you that race tires have an “actual” mu much higher (over 1.0). This is because tires are non-Newtonian. If tires had coefficients of less than 0.2, how can a car accelerate at greater than 1g? (the car I work on corners at 1.4g with no aero).
The reason for this is that Newtonian friction doesn’t handle cases where one surface protrudes into another surface – the small rocks/gravel embedded in the road surface actually dig into the tires – providing wayyyy more friciton.
For further reading I recommend “The Racing & High-Performance Tire” by Paul Haney or “Tune to Win” by Carrol Smith. Both are Race Car Engineers who know far more about tires than you ever will.
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Oh, regardless, I believe the plane takes off =)
Just don’t buy into your tire explanation.
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Jason have you suffered a concussion recently? Did the security guard pistol whip after pulling his gun on you?
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Jason doesn’t read so those suggested books will only be good for his mass to energy conversion machine with 100% efficiency.
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The whole airplane/airplane question boils down friction, energy and how its defined for:
1. The treadmill belt/airplane wheel contact
2. The wheel/axle contact
3. Maybe the treadmill belt/rollers contact
4. Air/wing surface contact
5. Air/treadmill belt contact
6. What are the limitations of power for the treadmill rollers for moving the treadmill
7. What is the available fuel for the airplane
8. Tensile strength of the treadmill rollers and belt.
Depending on how you define those (and maybe others) you can get whichever answer you want. I’m assuming the rest of the traditional physical laws apply (like friction/heat conversions, etc…)
You can end up with anything from the plane flying to the plane not to the destruction of the plane and/or treadmill.
The question is flawed in that it posits a theoretical treadmill against a real world plane by saying “The conveyor belt is designed to exactly match the speed of the wheels”. Something that large would take enormous energy and create a huge amount of heat and stress if it is even possible, meaning the treadmill would likely reach its theoretical limits before the plane would (and probably destroy the plane in the process). I’m not even sure materials science is up to creating a belt that could withstand the forces subjected to it at a few hundred miles per hour. Because of this the treadmill to even be a contender must violate certain physical laws.
At this point you may as well say, if a plane that has no engine is on a treadmill that isn’t moving do the mice cry over the loss of bathwater in my aunts house?
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@Ian Barnhart
The funny thing is I made a computer program to simulate this situation. Somehow, it actually does win 2/3ths of the time. It actually boggled me for a while, but I eventually understood it.
So yes, switching does help. By canceling out 1/3 of your options, you get 1/3 added onto the other option. It logically doesn’t make since, but when you though numbers around in a program it works
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Hi.
I like this question and it seems that it has a lot of responses depending on the level of physics familiarity.
If we assume that Vw = 0 by some miracle of a plane being fast without its main engines:
Yes, the conveyor belt is subject to something known as the “no slip condition” which means that the air next to it is trapped into moving at its speed. With the air’s viscosity being non-negligible at high speeds, this means that the air above and below the wings will be moving (with the lower side moving faster even without the wing’s effect due to its velocity profile). This means that even a board on wheels will take off at some speed under these conditions. All of this assumes air or some other fluid is present.
It’s fun to note that the air friction on the wheels will slow them down upon take-off… and outer rim speed is supposed to match the treadmill’s… which slows the treadmill, and then that kills the wind. All of this leads to grounding the plane.
Extra Info on the wheel prob:
Vb = Vc != Vw
Let Ww be the angular speed of the wheel and Rw be its radius.
Ww = (Vw+Vb)/Rw
Vw was not set to 0 in the setup.
Planes either propel themselves around the lot with their wheels or begin takeoff with their main engines. If we start the plane up only with its main engines, the plane will move forward. Since the treadmill is matches the wheel’s speed which means it will not cause the wheels to rotate (Ww = 0 when Vw = -Vb) so long as there is no slip between the wheels and the treadmill. In this scenario, the plane moves swiftly down the treadmill with less friction from the ground than it would see normally.
The answers are both “Yes” and they’re even additive. If you start both modes of propulsion the plane will only lift off sooner.
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i see only one possible solution. empirical testing.
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There’s an option nobody explicitly stated yet: The treadmill is imparting a force to the plane via the tires equal to the sum of all other horizontal components of force being applied to the plane. That would prevent the plane from moving. It is also entirely unreasonable- That same conveyor belt would have to be powerful enough to singlehandedly launch the plane without the engines running.
Note that the aircraft doesn’t care what its ground speed is. Lift is generated by airspeed.
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It should be crystal clear by now to everyone that if the plane is moving forward relative to the ground (and attains adequate speed, again relative to the ground) there will be lift and the plane can take off; and if not, then it cannot.
Determining whether the plane will move forward relative to the ground does not require physics or calculations. It requires us only to interpret the English of the problem, as stated – it appears to DEFINE the plane as motionless with respect to the ground, the treadmill presumably doing whatever it needs to do to accomplish this. If this turns out to be physically impossible, as calculations by previous posters suggest it might be, then Mr. Munroe was correct that it’s a stupid question in the first place. If it IS possible though, then the premise of the problem needs to be granted – the plane stays motionless with respect to the ground no matter what its jets are doing – and so there is no lift, and thus no flight.
A helpful comparison might be to what happens if there is no treadmill, and a different mechanism is used to keep the plane from moving forward with respect to the ground. Let’s say you put a plane on a normal runway, and anchor the back end of it to the ground with a cable of infinite tensile strength. What happens? Well, the plane cannot move forward, so there is no lift generated and it stays put. Assuming of course that the jets aren’t strong enough to rip the plane apart by pulling so hard.
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It doesn’t really matter. The purpose of rolling the plane out is to get the plane to where it generates enough wind force on its wings to create a static drag greater than the weight of the plane. (Because the plane’s wings are curved, the air flowing across the top of the wing has less pressure than the air flowing on the bottom of the wing.) Once the air pressure on top of the wings is less than the air pressure underneath the wings, and the amount of the pressure difference meets or exceeds the weight of the plane, the plane will take off.
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