It doesn’t matter how fast the conveyor belt can go. It doesn’t matter how much drag the bearings are adding to the equation, because that force is insignificant compared to the thrust provided by the prop. Unless you’re on the conveyor long enough to cause enough friction to literally make the bearings explode and the plane to crash to the ground, which won’t happen, because the plane will have reached sufficient velocity w.r.t. the ground long before that point. The plane will take off. Period. The whole “continually accelerating” thing is moot. It only exists at this point so people can still rationalize their wrongness.
OTOH, most race series have strict rules about aerodynamic elements, particularly underbody elements. While they can generate massive downforce, they can be unstable at speed. If a car is relying on it to stay stuck to the ground, then upsetting the suspension and getting a lot of air under the car can launch it.
I believe this, but I want to see this too. Adam Savage: back to you…
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I have debated this a few times with smart people. The confusing comes from the wheels and the belt. Most vehicles most people will ever drive are powered by the interaction of the wheel on a surface. Put a car on that conveyor belt and it will never ever go.
Planes of course only have wheels to have something between the surface and the fuselage. There are after all planes on floats or skis…
A plane taking of and flying only depends on airspeed, air density, wing shape and surface etc… Whatever stuff hangs from the bottom of the fuselage is only relevant if it creates too much drag.
Another point for pedants: Many descriptions of this problem are confused with the word speed. Do they mean rotational speed or linear speed and if linear what is the reference? For example the speed of the wheels will always be exactly the speed of the fuselage or they will fly off… 
He’d have to build some sort of giant Hot Wheels loop.
And yes, I’d love to see that, too.
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Remember the scene in ‘The Italian Job’ where the minis drive down a storm drain? If you had a racing car whose body fitted a tube rather than a flat surface, and a tube long and straight enough, it ought to be possible.
A radio-controlled model car might generate enough lift, though these things do not scale.
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Obligatory:
“Original or remake?”
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I did not know there was a remake. How could they? shudders
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If you want to bring practicality into it, we already don’t have treadmills for 747s. I always viewed it as a theoretical excercise, where the belt can’t fall apart. It would have to get up to enormous speeds to continue arresting the motion of the plane through just spinning the wheels backwards (eventually hittting light speed where it really falls apart). Under that framework the plane stays stopped as the belt begins to approach the limit. As it gets infintesimally close, I don’t know what happens (I guess that friction boundary layer thing mentioned above).
I feel like you’re making some unconcious assumptions here about valid ways to think of the problem. This is really the entire problem with this problem. It requires you to make a bunch of assumptions to even think about it, but they might not be the same assumptions as the guy next to you (for whatever reason, I imagined the question required an indestructible, infinitely powerful treadmill, but basically normal physics otherwise). It really doesn’t have a single answer unless you define way more specific rules (more specific than Adam Savage did, really - which is why I mentioned the conception I had).
Much like the Wicker Man, one pretends there was not. it’s better that way.
OK, then suppose your indestructible, infinitely powered treadmill. My 747 with unlimited thrust and indestructible tires and frictionless wheel bearings will still take off. You seem to assume that the plane will just stay in place, and then either rise up off the treadmill, or not. That isn’t what would happen. Regardless of the speed of the treadmill, the plane will move forward, get to takeoff speed, and lift off as normal. Even if you add in a layer of air moving with the treadmill thick enough to envelop the plane, that would only add to the lift of the plane as it moves forward wrt the ground. And again, we’re in a fantasy world with indestructible objects and no friction.
Sorry, but I don’t live in the realm of the theoretical. There is no real-world, practical scenario in which the plane does not take off. And I find this kind of discussion, with zero to no practical application, frivolous and boring.
The rotating Earth is like a giant treadmill. If your plane is on an East-West runway at the Equator, the plane and the ground will be moving at 1000 mph with respect to the Earth’s centre. The atmosphere is largely moving with the ‘treadmill’ apart from some large scale Coriolis effects. Or, if 1000 mph is not enough, you could use the speed of the Earth around the Sun, which would be 67,000 mph. Or the speed of the sun around the Galactic Centre, at 512,000 mph. These aces of reference were used for Michaelson’s experiments to show the uniformity of the speed of light.
I am not sure any of us can convince you that the aircraft would still take off. Perhaps you might wonder why you are so certain it could be stopped when nobody has a treadmill large enough to try.
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I feel like you’ve really managed to miss my main point, which is that there is no single, canonical, correct answer to this question. And anybody who says “this is definitely what would happen” with this impossible, poorly defined hypothetical is overreaching. The “correct” answer to this question is a tree of possible answers, where you spell out your assumptions for each one. The correct answer is that the question is flawed.
I was just telling you my first conception of this problem. Under those assumptions, the plane doesn’t move because the magic treadmill pulls it back. Under other assumptions, that doesn’t happen. Neither is more correct than the other, because the question doesn’t give you the information you need to decide that.
Physics gives you all the information you need to know. Either you’re in the real world, or you’re not. And if you’re not, you can assume all sorts of things that render the question moot, and thus uninteresting.
In the real world, where we’re trying to answer the question, the treadmill simply cannot impart enough force to stop the plane moving forward. And no amount of solipsism about “assumptions” is going to change that.
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But it doesn’t because the wheels under a plane during take-off roll freely. They are not powered, a plane is not a car.
Worst case, the really, really worst case the wheel bearings catch fire or even break but that should not happen even at twice V1.
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If you’re going to constrain the problem so much that the treadmill has to adjust and top out at a practical speed, then you might as well set it on a runway. Yes, planes can take off from runways. Did that really make it more interesting?
In my head, I’ve had the version of this question where the treadmill is being adjusted intentionally to frustrate the airplane. I’m just saying if your initial impulse is to answer “no”, I can think of one physically-possible conception of this that allows you to do so (actually without bringing the speed of light in to it).
You can model this so that the treadmill accelerates to a finite speed at which drag (not frictional - my mistake) forces on the wheel and the acceleration from the engine balance out (before the air cushion thing happens). It’s a high speed with most airplanes, probably unreachable in many realistic cases, but not theoretically impossible.
The answer isn’t “the plane never takes off”, but it isn’t “the plane always takes off” either. Now, I release you from any obligation to tell me how uninteresting this all is again, BTW.
P.S. Can’t ever be sure, but I do believe you probably continue to exist when I’m not paying attention.
I’m just far from convinced that any of that could happen.
If you’ll forgive me stating things you know and agree with for a moment: The point of the answer that the plane will take off is that the plane’s forward motion is independent of what the ground it is on is doing. It’s like pulling a wagon along a conveyor belt by a rope. If you pull the rope the wagon moves forward with you, what’s happening under the wheels doesn’t matter.
So is it possible that we put a conveyor belt under the wagon that spins so fast that I’m left there straining to pull the rope but I can’t get the wagon to budge?
Maybe if we properly model the forces of the conveyor and the wheels of the wagon some way we could actually impart enough force to the wagon from the rolling wheels to counteract my strength. But when I try to imagine that in real life it always goes from I can walk along pulling the wagon no problem to the conveyor belt hurling the wagon into the air and never gets to me standing there putting my weight into pulling the rope and standing still as the wheels spin. My intuition is that the belt would have to be moving very fast (the calculations are beyond me, but I imagine it happens at speeds so extreme that in reality they’d rip the ground apart, let alone the wagon). And at what point does the conveyor become so hot it just melts the plane/wagon to scrap?
The conveyor belt had better be perfectly smooth or the wagon will hit some imperfection and go flying. The wheels on the wagon will have to be perfect too. I’m not an engineer, but at this point I think I’m assuming frictionless wheel bearings without assuming zero friction between the wheel and the conveyor. We have to assume the conveyor belt does not pull any air along with it (otherwise the plane/wagon takes off for that reason) but also assume that the plane’s wing still interact normally with air (since otherwise the answer is automatically that the plane won’t take off). It feels like a super weird set of assumptions.
If the point is that the conveyor wouldn’t have literally zero effect but would have some negligible effect then I guess that’s true. But it may be more likely that the conveyor stops the plane from taking off by being insufficiently smooth a surface for the plane to take off on (since, like the wagon, any lack of smoothness is magnified by the speed the conveyor is moving).
I think Adam Savage’s explanation of where the confusion comes from is very good. The point is to to realize that the force moving a place forward is independent of the ground. But we know many planes can’t take off from a bumpy field (not to mention lava, a minefield, an extremely powerful electromagnet), so the conditions of the ground can’t be totally ignored either.
See, that’s an interesting point. In the real world, things like smoothness come into play when velocity reaches extreme levels. But you’ve hit the crux, for me at least, in that whatever other forces are acting on the plane are not enough to counteract the huge force represented by the engine thrust, even when we’re in the realm of the nigh-theoretical.
The question is, “Can a treadmill stop an airplane from taking off?”
The answer is, “No.”
It’s only interesting because so many people get the wrong answer.
