Monday, September 8, 2008

Welcome, XKCD readers!

To all of you that got here from the forums at fabulous welcome. And yet, many of you make me despair, as I did on that fateful January 31 day. Even as you read the main argument below, the debate is raging anew at the forum of your favorite webcomic. Read the latest comments for the explanation and the debate even continues there! Some of you still don't get it. I die a little inside every time I hear someone say that the plane will not take off, and cite their experience as a "commercial airline pilot", or boast their thousands of hours of flight time, as if some desperate appeal to authority will make up for a complete lack of understanding of basic physics. So I implore you, please - if you read the entire article below, and still think the plane doesn't take off, read it again. And if you still think it doesn't take off, read it again, only make sure you read it this time. We all know you didn't pay attention in physics class, but hopefully you at least stayed awake for a tiny portion of reading comprehension. Read it again, and again, and read the comments, and the thousands of other websites that explain why the plane takes off, and if you still think it doesn't, then please stop clogging the tubes of the internets with mindless drivel. Try one last video that might convince you otherwise instead. Happy flying.


Anonymous said...

It is utterly disgusting that all the truth on this marvellous subject is not yet written on Wikipedia...

Anonymous said...

I found the video explained it best.

But seriously, I had never heard of this debate before, and after reading this post don't understand how it can be debated (assuming your physics are correct, as the skateboard example seems to demonstrate).

CJ said...

Although I agree that with any practical treadmill, the plane takes off. There are so many subtleties in this question.

You say that there are two correct answers:
No, the plane can't take off
Yes, the plane can take off

I disagree. The question says This conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction)
so if the plane is moving forward then the control system has failed.

Therefore the two possible answers are not fly / no-fly they are:
No, the plane won't take off
The question is invalid since you can't build such a treadmill.

But this is a thought experiment, we can build whatever treadmill we like. Therefore as soon as the plane starts to accelerate relative to the ground our treadmill spins up to crazy speeds and all sorts of effects will provide some extra friction and stop the plane.
At the practical end of the scale we'll see effects like the grease in the wheel bearings contributing to the friction (in moving through fluids faster is definitely harder); obviously at some stage the wheels will melt which will probably stop the take-off. At the extreme end of the scale eventually relativity will mean that the kinetic energy of the wheels will give them enough mass to either exceed the plane's take-off weight or just to increase the friction within the bearing.

Anonymous said...

Christopher if the treadmill goes in the same (but oposit speed) of the aircraft the when the plane moves forward slowly the the treadmill moves backwards slowly. not very fast in order to keep the plane stationary since the plane is not moving very fast forward. This means that the wheels only need to remain intact at twice the take off speed.
(take off speed forward + take off speed backwards)

CJ said...
This comment has been removed by the author.
CJ said...

Well that's only true if you assume that "the plane speed" means its speed relative to the ground not the treadmill.

That's an odd assumption though. If you assume that then you don't need a plane, you can just walk off the runway. Walk forward at 2mph, treadmill moves back at 1mph and you're going forward relative to the ground at 1mph.

I suppose it's still a possible interpretation of the question but it means that you don't make use of any of the special properties of planes. Presumably the original question was supposed to demonstrate some sort of aeronautical point so I don't think this interpretation can have been the original intention.

Also, it's not consistent with the normal use of a treadmill. If I said that I "went to a gym, got on a treadmill and adjusted the speed so that it was the same as my running (but in the opposite direction)", would you assume I ran in to those TVs that they have in gyms playing MTV at half my normal running speed or would you assume I stayed still relative to the gym floor?

Anonymous said...

The point you're ignoring christopher is that the plane's speed has no bearing at all to it's friction on the ground. It's friction remains constant (only increasing by a slight degree due to the friction of the grease and bearings) while it's speed increases. The treadmill cannot transfer enough force to stop the plane from moving relative to the ground, since a plane moves by pushing air, not the ground. If it was a car with wings, then yes, you'd be correct. But as it uses a jet engine (or a propeller, or what have you) it doesn't increase it's surface friction to increase speed.

Southwest said...

I hope you don't mind that one of the first few posts in the respective xkcd forum thread was me posting a link here. It helped me understand the thought experiment, and I figured that the benefits of having more people well-versed in this problem would outweigh the possibly overwhelming traffic.

That said, thanks for the site! I found it incredibly helpful in wrapping my mind around this.

Anonymous said...

All you 'yes fly'er's, get of your little skateboards and learn about relativity. If you put your skateboard on your treadmills, yes they will stay relatively stationary without any external force. Now give your skateboards a push. Yes they will over come the treadmill moving backwards and move forward, and I guarantee you that the speed of the skateboard moving forward will be equal to the speed of it's wheels minus the speed of the treadmill. This is called relativity. Also note that whilst we exist in a world with gravity, the forward movement of the skateboard will always be dependent on the speed of the wheels.

Do you see what has happened here, the external force (not applied through the wheels, like an airplane's thrusters) increased the speed of the wheels for the skateboard to move forwards relative to the treadmill.

So, if there is a command in the question stating that the treadmill will travel at the same speed as the wheels of the plane, by some complex control system, the force imparted to increase the speed of the plane, and hence it's wheels, will cause the treadmill to increase speed also, but in the opposite direction.

So, assuming no external lift, every time the pilot increases his thrust, he will increase the speed of his wheels, which will increase the speed of the treadmill, which will increase the speed of the wheels, which will increase the speed of the treadmill, which will increase the speed of the wheels ... do you see where this is going?

Anonymous said...

No, the wheels of a plane are free wheels. A free wheel isn't going to speed up the treadmill.

Well, minor correction: if the axle is completely frictionless it won't have an effect on the speed of the treadmill. Since there is a small amount of friction in the bearings the plane's engines will speed up the treadmill a little bit, but the additional speed will not increase when the pilot increases to full throttle.

Anonymous said...

Now give your skateboards a push. Yes they will over come the treadmill moving backwards and move forward, and I guarantee you that the speed of the skateboard moving forward will be equal to the speed of it's wheels minus the speed of the treadmill.

I think you're getting your causality wrong. You're assuming that:

Speed of Skateboard = Speed of Wheels - Speed of Treadmill

In fact:

Speed of Wheels = Speed of Treadmill + Speed of Skateboard

When you push the skateboard, it starts to go forwards, which means the wheels spin faster. If you then increase the speed of the treadmill, you will not succeed in reducing the speed of the skateboard, only in making the wheels spin *even faster*.

Thinking about it, this is basically a restating of Xeno's paradox, in which Achilles and a Tortoise are replaced with an Aeroplane and a Treadmill. At low speeds, the 'plane could theoretically match velocity with the treadmill such that there was no movement (scenario 2 in the original post) but there comes a point where the treadmill simply ceases to be a factor.

Ken Sikora said...

Wow. Rickrolled.


Seriously, I've never actually been successfully Rickrolled. I normally expect it. But this was classic.

Also. I fully expected that link to be to the Mythbusters video on the topic.

Anonymous said...

Thanks for turning me from 'no-flyer' to 'flyer', I never truly understood the problem. Even the mythbusters did a pretty crappy job explaining the problem. Thanks for laying it all our for me.

CJ said...


"only increasing by a slight degree"

There's the crucial point. Over certain ranges friction is _almost_ independent of speed. Almost but not quite and that fraction is enough. There's no limit to the speed that the treadmill can run so if there is even the slightest real life deviation from the ideal constant friction wheels then eventually the treadmill will be able to either balance the thrust of the engines or (probably more likely) destroy the plane.

Josh said...

Christopher, draw a Free Body Diagram of the wheel. What causes the wheels to spin? It is not the thrust of the airplane, which acts on the center of gravity of the wheel, it is the static friction from the treadmill, f=ks*N, which gives it a torque, causing rotation. The friction force doesn't increase with speed, because it is a based on two constants, the coefficient of static friction and the weight of the airplane. So, as long as the thrust from the engines is greater than the static friction of the wheel on the surface, it will always take off.

Anonymous said...

the whole point of this blog was to get you idiots to stop debating it. you could at least have enough class to debate it elsewhere.

CJ said...

You're right, for an ideal wheel.
However, what speed will our wheels and treadmill be rotating at once the plane starts to move? It should be rotating at it's maximum speed to try to keep the plane still but if it's an ideal treadmill it doesn't have a maximum speed.

So the question really is whether you consider that the treadmill is ideal or the plane.
I took the treadmill to be ideal because we're told in the question that the treadmill CAN track the plane's speed. The three rewordings make it even clearer by referring to the treadmill as "very powerful".

I think that most of the problem here comes from the fact this site claims to be the definitive analysis but then goes on to give a single yes answer. The definitive analysis would surely but a matrix of assumptions with an answer for each. Unfortunately the original question is so hazy that it will be quite a large matrix.

Anonymous said...

Hey, just wanted to say Randall just blatantly ripped you off:

You might want to send him a nice email about this.

Anonymous said...

I just got rickrolled.


Greg Ley said...

If I didn't laugh my ass off every time I was Rickrolled, I'd come to your house and cut you.

Anonymous said...

If the turbines are spinning, won't they push the air across the wings regardless of the planes movement?

sbunny8 said...

It seems to me that the original problem was this: "If you have a plane that needs a 2000ft runway to get up to 70mph before it can take off, and you didn't have enough room to build a 2000ft runway, could you build a 40ft treadmill, crank it up to 70mph, and take off straight up?". I think we can all agree unanimously the answer to that question was NO IT WOULDN'T WORK.

Then some unknown smart-aleck said "oh yeah? What if the treadmill was 2000ft long? You could rev up the plane's engine and move forward along the treadmill and yes you'd be able to take off at the end of it, even if the treadmill operator cranked up the speed of the treadmill way faster than 70mph." And guess what, the smart-aleck is right. To which I say SO FREAKING WHAT?? Why on earth would anyone want to build a 2000ft treadmill when they could just build a 2000ft runway? You've defeated the whole point of the original puzzle.

Anonymous said...

the definition on the XKCD page DIFFERS from the one in this blog.

This blog discusses "The speed of the belt is matched to the airplane's speed"

XKCD's version says "conveyor belt is designed to exactly match the speed of the wheels".

It's important, because the answers are actually different

BugoTheCat said...

I wasn't much aware of the problem and when I tried in the past to read it and read some answers I never quite got it. Initially I would think no-fly but that's because I didn't have an idea of airplanes and the wheel not being mechanical. And of course remembering it's the air thrust that moves it and the wheel just for rolling nicely.

But the bicycle/skateboard explanation made it for me. It's everyday things that we are more familiar, like when you have a grinding wheel or a motor, and on the other hand you put a sellotape ring on your pen, keep it stationary so that the grinding wheel friction rotate it backwards, no matter how faster you didn't need to excert more counter force. It reminds me of little games we played, I remember how simple the force it has to be to just counteract the friction.

Anyway, was a great read, fortunately I never had an opinion on this problem when I came here, so maybe I wasn't biased and that's why I read and tried to understand it. I had an easier go with Monty Hall problem, but this one I always avoided it or never knew what the fuzz was about.

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