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.
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Hehe. I brought this up in Theology class at my school, right in first period, by supposing it to a student. Some overheard and started arguing too. Eventually the class split in two and the teacher joined our team, and we spent the first ten minutes of class debating it, with no clear resolution….
Of course, it appeared in second period, too, and then third and fourth and so on, and everyone was talking about it in lunch, too, and no one even came to a consensus until the NEXT day.
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How about the viscous friction in the axis of the wheel? It is proportional to rotation speed and able to create a force which increases with increasing speed. This friction is able to fully compensate the thrust of aircraft engines.
And of course we must take into account the friction in this problem – otherwise the wheels will not spin, but will slide.
There are some really good points you made in your post…very insightful
The wheels of a plane only apply torque on the plane’s landing gear (and by extension the plane) via sliding friction between the wheel and its axel (assuming the parking brake is off). Sliding friction is not dependent on speed so the magnitude of the force counteracting the planes take-off would be the same as if it were on the ground. The pilot wouldn’t even notice that conveyor belt was even there unless he looked out the window.
Also the wheel wouldn’t move infinitely fast, it would have an angular velocity of (2Vw)/r. Twice the wheel/plane velocity (because the difference in speed between the conveyor and the plane is 2 times the velocity) divided by the radius of the wheel (because thats how you get rotational velocity).
sorry for double posting but i read the comment 2 up from me and i must defeat this troll
I think force of friction of a wheel on its axel still follows the Fnμk formula (force of kinetic friction times force normal) but i could be wrong. I don’t see speed there though.
*RAWR triple post thats supposed to be coefficient of kinetic friction times force normal
We were able to stop arguments over this by requiring anyone who tried to discuss the “takeoff” problem to first answer the question of “what happens when the same airplane lands on the treadmill?” This brings to the forefront the question of “what is the treadmill actually doing?”
Very clear explanation. And I kept mind on the contrary
Its funny how simple this really is. The plane engines push on the air, they do not power the wheels. The plane will move forward and take off regardless of what the wheels or the tredmill does.
Myth Busters solved this one already.
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Forget the treadmill. A plane preparing to take off in a westerly direction on an east/west runway is already on a treadmill going 1,000 miles an hour against the intended take off direction. Does anyone think an airplane of any type cannot take off to the west?
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What this analysis fails to take into account is that a constantly accelerating wheel with non-zero moment of inertia will produce a constant torque acting as a brake on the plane even with frictionless wheels. So since we are working in an ideal world where plane sized runways exist, if the wheels have mass and are frictionless and operate in a vacuum and there are no relativistic effects, the treadmill feedback loop described by interpretation #3 could continuously accelerate the wheels and hold back the airplane.
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Plane will not launch because it needs to be pushed by air, which will not happen beause plane is not moving against the air. It doesn’t matter how fast are the wheels rotating.