Problem Solved

I suppose what makes this "airplane-treadmill" question (via Anthony) so compelling is that each side's so sure they're right. Here's the basic scenario:

Imagine a plane is sitting on a massive conveyor belt, as wide and as long as a runway. The conveyer belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?

There are now 852 comments at the NYT site. I haven't read them all, but, really, what an exercise in talking scientific-sounding bollocks. Of course the plane won't fucking take off. Jesus Christ!

OK, the controversy's now over everybody. Time to return to your normal lives.

Comments

[jeffrey Mushens]

Of course it can take off when it gets to the appropriate speed for the airflow over the wings to provide sufficient lift.

[Dom]

Actually, I love these paradoxes, and I collect alot of them. Here's a new one.

Say you are in a boat, with a slow leak. If I bail the water out at least as fast as it gets in, the boat never sinks. There is energy in the water as it falls from my pail back to the ocean, but it is equal to the energy I exert when I bail. That's how it should be -- no energy lost or gained.

Now, let's say I place a hose over the hole in the boat, and curve it over the side. Same situation -- the water is leaving as fast as it enters, so I never sink. There is still energy in the water leaving the hose, like a tiny water-fall. Only now I'm not working. Have I gotten free energy? Do I have a perpetual motion machine?

[Mick H]

Jeffrey - well it never gets to the appropriate speed, does it? Cos it's on top of a conveyor belt which, by definition in the setting up of the problem, makes sure it stays stationary wrt the ground and the air. And no, the movement of the air caused by the engine - jet or propellor - wouldn't be enough on its own. If the plane was attached to the ground so it couldn't go forward but could go up and down, then however hard you revved up the engine, you'd never see the plane start to rise up.

Dom - how could the water go through the hose, from lower (the leak) to higher (above the surface of the surrounding water)?

[P. Froward]

Picture your car on a conveyor belt. Take it out of gear and take your foot off the brake, so the car can roll freely. Start the belt. The car will move in the direction the belt is moving. If the belt starts with a jerk, the car may roll forward a bit relative to the belt; if the belt starts smoothly, the car's liable to roll very little. In either case, the car's wheels are not frictionless, and if you keep it up, the car will move in the direction the belt is trying to move it.

Since it's a question about physics, you might as well assume that the laws of physics all apply, and that includes friction.

Here's the rub[1]: Picture a 747[2] blasting away like the dickens with all four engines. How fast must the conveyor belt move to counteract that much force? Quite possibly faster than is physically possible, or at least faster than the current state of conveyor-belt design and construction can support.

This is either a) a physics question in which the laws of physics are optional -- which is idiotic; or b) a question about the design and construction of very large high-performance conveyor belts -- which can be addressed meaningfully only by a specialist, and whose answer is not final because the state of the art will change.

Pointless, either way.

But yeah, if the plane is limited by the laws of physics and the belt is *not*, the plane won't take off. If the wheels have no more problem with friction than the belt has, the plane can take off just as it pleases. The question either way is what, if anything, is counteracting the thrust of the engines?

[1] Gedit? Gedit? Friction, rub?
[2] Airbus products, designed as they are by government employees (more or less) in a socially just social democratic socialist society, are presumably less affected by the laws of nature.

On the hose issue, anybody ever heard of a siphon?

[P. Froward]

I think people are taking the speed part to mean that you can take it as a given that one of the ground rules is that surface of the belt, and the outer surface of the wheels, are moving at the same speed. "Same speed" is "same speed" whether it's zero or greater than zero. In other words, they think the "designed to" clause is a handwave which, in plain English, translates to "for reasons unknown, the plane remains motionless".

So the question they're answering is this one:

"An airplane is motionless relative to the ground. Is it going to take off?"

Well, duh! Neither is a bowling ball, or Spain.

Er... Unless there's one hell of a headwind. Anybody consider that? And a Harrier could just float straight up, smirking. Somebody at the link mentioned that, naturally.

[Mick H]

Well yes, exactly. That's the way I read it, and that's why I was amazed that anyone could think the plane would take off. But what you say in your first comment is interesting - the key question is, what's counteracting the thrust of the engines? People are saying, well, the friction in the wheels can't be enough, so the plane has to move forward, ie fly. But I'm reading the way the problem is set up as ruling that out. So, whatever, it's got that ambiguity built in...

As for Dom's boat - siphons work from higher to lower, but as I understood it, this is lower (below the waterline) to higher (above the surface, "like a tiny waterfall").

[Dom]

"how could the water go through the hose, from lower (the leak) to higher (above the surface of the surrounding water)?"

The water does that anyway. If there is a hole in the boat, the water spouts up -- and then lands in the boat to sink it. But what if the spouting went off to the side (via the hose) so that it lands not in the boat but back into the ocean, leaving the boat dry. There is nothing to sink the boat, and nothing to stop the leak.

[Dom]

Say I have an infinite supply of balls labeled 1, 2, 3... Now at 1 minute before noon, I place balls labeled 1 through 10 in an urn, and throw out ball #1. At half-minute before noon, I place balls labelled 11 through 20 in the urn, and throw out ball #11. Then 21 through 30, throw out #21. And so on. How many balls are in the urn at noon. Infinite. Name any ball and I can tell you if it is in the urn or not.

Now say at 1 minute before noon, I place balls labelled 1 through 10 in the urn and throw out ball #1. At half-minute before noon, I place balls labelled 11 through 20 in the urn, and throw out ball #2. Then 21 through 30, throw out #3. And so on. This is the same operation -- get 10 balls, throw out 1. All that has changed is the label of the ball thrown out each time. But how many balls are in the urn at noon this time? None. Name any ball and I can tell you the exact time it was thrown out.

[Mick H]

"If there is a hole in the boat, the water spouts up -- and then lands in the boat to sink it." It wouldn't spout up higher than the water level of the sea - which is what you're supposing.

As for the infinite balls - well, it's one of those Achilles-and-the-tortoise type deals, isn't it? Playing around with the meaning of infinity.

[DaninVan]

Uh, folks; planes don't need wheels to take off, skis or pontoons work just fine, thanks very much. The point is LACK of friction with the ground, until the thrust generated by the engines generates enough forward speed to allow liftoff.
We're looking at apples and oranges when you compare planes with ground vehicles which DEPEND on friction with the ground, but ONLY IF THE ENERGY IS DELIVERED THROUGH THE WHEELS. A perfect example of both cases is a hovercraft which of course can't 'fly' but also doesn't contact the ground (or conveyor belt).
The only issue with the problem presented is whether the wheels could sustain the higher rpms likely caused by the conveyor driving them; they have no mechanical connection to the momemtum of the plane, they're free wheeling during takeoff.