The plane on the treadmillwill too take off.
Ahem. I and many others had it right all along.
Not really, at least as far as that other thread is concerned, unless you qualified your answer to modify the question. As Cecil says, the question is often worded in a way that, taken literally, means the wheels and conveyor would spin at infinite speed if the plane started to move forward. And that is precisely how the question was worded in the OP of that other thread. Since infinite speed would require infinite energy to overcome infinite angular momentum in the plane’s wheels, the plane couldn’t move.
Your answer was the right answer to a sensibly worded version of the question, sure.
Malarky. Cecil didn’t say any such thing. Here’s what was written:
And the damned plane takes off.
It’s only a paradox if the airplane takes off. 
Seriously. Framing the problem like this creates a paradox as soon as the airplane attains the slightest bit of forward velocity. But not if the airplane remains fixed in space! What was aruged for nine pages in that trainwreck wasn’t “A = A + 5”, but rather “A = A + x, thus x = 0 to avoid a paradox, so it follows that the airplane can’t move.” At that point, the “magic conveyor” was contrived as a thought experiment to define a way – any way, no matter how implausible – to apply a reaction force to the wheels and keep the airplane stationary. Hence the rapidly accelerating belt and all that jazz.
Oh boy, my confidence in Cecil has taken a gigantic hit today:
Its unfortuate that Cecil has fallen into the common trap of believing that the velocity of the treadmill in this case is what is important. Its not, what is important is the acceleration of the treadmill. I have no more desire to argue this but I swear, on pain of retaking physics before I graduate as an ME, that if I accelerate the treadmill at a rate of 2*(Force from engines)/(Mass of tires) that the plane goes absolutely nowhere.
You did, at least, convince me. Thanks, and chalk one up!
Wrong Forum, though.
I beleive that infinity could also be a mathmatical answer, as could be the speed of light due to spacetime effects (mass being everywhere at one moment of time) a physical answer, but these assume a non-slip condition.
So basically as the plane moved forward the treadmill would have engage at a faster speed then the wheels are going in the opposite direction, the only way this could work is if we allow the wheel to slip. And the wheels would have to slip during the entire takeoff. If the plane has enough power and stability, and could withstand the high winds created by the treadmill it would take off, if not it wouldn’t. The tires would be toast however, and perhaps part of the landing gear, if not the belly itself.
Another solution is wait till the tires blow off, the treadmill then would stop (as no tires exists anymore), then take off on the hubs.
You are absolutely right. And I’m beginning to regret having posted it at all.
Since this is now comments on one of Cecil’s columns, I’ll move this to CCC for you.
DrMatrix - GQ Moderator
Infinity doesn’t work, but slipping does. If you rewrite “A = A + x” as “V_belt = V_plane + V_wheel,” (relating the velocity of the belt, the forward velocity of the plane, and the tangential velocity of the tire, assuming rolling motion), there are only two conditions that set both sides equal: Either the plane can’t move forward, or the wheel can’t turn.
So, we have the following possible scenarios. Just to simplify things, no outlandish assumptions about the plane will be made. It’s just a basic C170 with field approval for aftermarket skis.
Group I: "Conveyer belt is exactly matches the tangential speed of the wheels."
[list=A]
[li]Wheels don’t skid. Plane, by definition, has zero air velocity. Requires invention of “magic” conveyor to apply ~200 pounds of restraining force to tires, which undergo rapid angular acceleration and seize their bearings. Plane remains on ground, requires repairs.[/li][li]Wheels skid – the normal scenario. Lock the brakes and apply takeoff power. A 160 hp O-320 engine might be able to overcome friction between the wheels and a relatively slick belt, but just barely. Plane does not achieve takeoff velocity. Remains on ground.[/li]
[li]Wheels skid – the skiplane scenario. Strap skis to the wheels, put the airplane on a few inches of snow, and apply takeoff power. As long as the conveyor is covered in snow, the airplane can move forward. Plane takes off. [/li][/list]
Group II: "Conveyor moves backward at the same rate that the plane moves forward"
I don’t think anyone’s disputing this scenario. The wheels would spin twice as fast as normal (130 knots vs 65 knots tangential velocity) and result in a little more drag, but the plane takes off.
Sigh.
As I recall, you bowed out of the earlier thread because you didn’t understand the physics.
Unfortunately, Cecil seems to have the same problem.
The answer changes depending on what your initial assumptions are. The plane will take off with certain initial assumptions. No argument there at all.
But under other assumptions, the conveyor will apply enough force to the plane to offset the engine thrust, and the plane stays in place. And there’s no reason to invoke “infinity” or “paradox,” either.
KeithT has done a pretty good job of rounding out the most reasonable possibilities for initial assumptions, and what they lead to.
You mean such as the assumption that the treadmill is made of unobtainium and the wheels of inflated unobtainium foil, both polished to Johanson-block standards?
In pretty much any normal airplane, the wheels, even if locked, will not be able to sustain a sufficient force against the engine.
Yeah. So? Lookit, that’s a perfectly valid answer. For a real-world situation, if that’s your assumption, it’s unlikely you can force the plane to stay stationary. I doubt any such treadmill can actually be built (assuming, parenthetically, we’re talking about an actual passenger plane, and not a tiny RC model).
But this is a thought experiment. No one’s proposing to actually build such a thing. The inability to construct an experimental model doesn’t invalidate the question. How about the question, “what happens to a spaceship as it approaches the speed of light?” Or, “what happens if you dig a tunnel through the center of the Earth and jump in?” Is it sufficient to note that we can’t build a spaceship that will approach the speed of light, and that structural materials won’t withstand the heat and presesure at the center of the Earth?
So when a 747 spools up its engines at the end of the runway with the brakes on I get to see it skid out of control down the runway with tires smoking? Cool. 
Actually even military aircraft have sufficient braking power to hold themselves stationary with military power on. Burners might be a different story…
But Cecil, unlike you, read the question.
It doesn’t say, “…has a control system that tracks the engine thrust and tunes the acceleration of the conveyer belt to exert exactly the same force (but in the opposite direction)…” Nor does it ask if there is some method of operating a treadmill that would prevent the plane from taking off. I think under any interpretation of the question, as asked, the plane flies because insufficient force is generated by the treadmill to prevent the plane from generating sufficient wind speed to take off.
No, the interpetation of the question I speak to is the one in which the speed at the edge of the tire is the speed of the wheels and is equal to the speed of the treadmill.
To be specific, the interpretation treis speaks of was presented in the original GQ thread by bouv.
This does not imply that the treadmill must use force control or speed control; either technique could be used so long as it satisfies the boundary condition of the belt and wheel speeds matching.
I suppose that raises another question. If you put a dead horse on a treadmill, will it still take a beating?
Using cecil’s assupmtion if you had a plane on the belt, just in ‘neutral’ with out brakes applied at 0 velocity, then started up the belt the plane would just stay in place as the belt just rotated the wheels.
Just because a object (plane) is acted upon by a force (engines) does not mean it can’t be acted upon by another force (friction in the wheels).
Absolutely.
Look, it is relatively simple physics. To paraphrase Tries, what does the velocity of the treadmill have to do with it.
I think we all agree that if the plane accelerates relative to the air, it will eventually fly. Right? What are the forces on the plane? There is the force due to the engines spitting out hot, fast air. The existence of a treadmill is immaterial. What other force acts on the plane? The force from the treadmill. How does the treadmill produce a force? Through the force of (rolling) friction between the tires and treadmill surface. What difference does it make if the treadmill is moving relative to the air? (Even if the treadmill is off, it moves relative to the wheels and the plane.) The force exerted by the treadmill on the plane is the product of the coefficient of friction times the weight of the airplane. It is independent of the treadmill’s speed, acceleration, and motor size. The net acceleration of the plane is exactly the same, whatever you do with the treadmill. Only if you increase the force of friction, by say, using the brakes, does the plane not take off.