And how will the plane remain stationary wrt still air? This problem is defined purely by the assumptions you make; randomly making assumptions without stating them and drawing consequently random conclusions is exactly what fuels this discussion.

@Sanchises I did state the assumptions, which come straight from the condition that the conveyor belt and the wheels have the same speed.

But it's not always possible to match the speed of the wheels. Are you assuming ideal frictionless wheels? Then your system is noncausal and the problem has no well-defined solution. Are your wheels not idealised? Then you will break your landing gear, do a belly flop and be launched backwards by the treadmill. So, your solution only works if you assume unbreakable wheels with friction that increases without bound as the wheel speed increases. Which is an entirely arbitrary set of assumptions.

@Sanchises Your first sentence is correct. The question is a thought experiment, which is often done in physics and engineering. Regarding the frictionless wheels, what could possibly make them spin as the question asserts (neglect motors)?

Can you remove the downvote if you have doubts about your own reasoning?

I upvoted. Don't know why there was a downvote

downvote because he assumes planes take off by pushing against ground, like a running man (or a car)

@Ben You're saying that the propeller thrust doesn't create forward motion of the wheels?

@Radu094 Says who?

I'm not saying it's the wrong set of assumptions, so your answer may be valid given that set. However, in my experience the underlying assumptions are what causes the debate, and any successful answer to the question must list the assumptions (which is also often done in physics and engineering). Anyway, this question is a duplicate of an off-topic question, so why bother xD

And, I meant frictionless bearings, no wheel inertia (although that's not that important) and zero rolling resistance, but of course the wheel should not slip, sorry if I wasn't clear.

I did list the assumptions.

This answer was voted down because it is incorrect. The treadmill analogy is totally wrong. Get on a treadmill, strap a jet engine to your back. You WILL move.

@MikeSowsun Mike, your example doesn't even come close to the question. You should really read the question first (just like reading the fucking manuals in some professions), and then try to understand it before saying that the answer is incorrect. I don't doubt the enormous thrust of jet engines, but if the treadmill moves me back as strong as the jet engine pushes me forward, then I won't move relative to the air around the treadmill. But this example is already totally off because it doesn't involve aerodynamic lift and spinning wheels.

@jjack Your answer is flawed. In order for the treadmill to keep the aircraft from moving forward, it would have to exert at least an equal force against the aircraft in a direction opposite the thrust of the engines. Since the treadmill merely matches the wheel spin rather than influencing it, no force is transferred from the treadmill to the aircraft... the aircraft will accelerate forward.

@SteveH The treadmill would accelerate the aircraft backward if there was no thrust force on it.

@jjack No it would not. Your concept of a treadmill implies a solid connection between the treadmill and the aircraft. Perhaps if the parking brake were set giving the belt and aircraft a solid connection. But that concept is not involved in the question. The only connection (far from solid) between the conveyor belt and the aircraft is the bearing friction and tire friction and inertia of the spinning wheel mass. these are acting on the aircraft very minimally compared to the thrust of the engines. The conveyor can move any speed or direction and not impart enough force against the thrust

@SteveH When you run on one of those running treadmills and stop, what happens with your body?

@jjack That's the point, because we have a solid connection with the treadmill, we are thrown back. The aircraft through it's freely spinning wheels does not have that solid connection so it behaves differently.

@SteveH The aircraft's wheels experience friction too.

@jjack yes they do. How much friction do you suppose they experience and how would it influence the forces acting on the aircraft.... compared to the thrust of the engines?

@SteveH If the plane moves on the conveyor belt and doesn't move with respect to the ground around, then there is no aerodynamic lift. The plane won't take off.

@jjack the comment immediately after mine is a better-worded way of saying what I meant, but now that I’ve had some sleep I’ll give it another go. Imagine you have a toy car on a treadmill attached via a string to your hand. If you pull the string the car will come towards you, regardless of the treadmill speed. That’s because the car is getting that propulsion from something not on the treadmill - you. It’s the same in an aircraft - the propulsion is coming from the air being sucked into the engines and pushed behind. The tyres will spin very fast but the engines and wing don’t care.

@Ben The question mandates that the wheels spin and the speed be equal to the conveyor belt speed and opposite.

Downvoted for defending the indefensible, and for the ridiculous but now edited remark about down-voters.

@Airsick You're obviously an ardent downvoter.

Oh my god I accidentally spread this and now the same is happening here.....

@EricBlueWilson Good questions are never a disease. In your case however I think you might have had such a result in mind young man...

@jjack, You are ignoring the fact that the entire premise is impossible. As soon as the aircraft is moving at all, even at one tenth of one mph, it is impossible for the premise of the question to be satisfied. No matter how much the conveyor belt accelerates backwards, the wheels are attached to the aircraft and to the conveyor, and so will ALWAYS be one tenth of one mph faster than the conveyer. They can NEVER, EVER, be equal again. They can only be equal if the aircraft ground speed is zero. So the entire premise of this silly puzzle is logically impossible.

@CharlesBretana You seem to be confusing the thought experiment part and the logic with what is physically experimentally implementable.

@jjack, How so? How are you going to physically implement this? as soon as the pilot pushes the throttles forward, the aircraft is going to move, and the premise becomes logically, demonstrably, false. The wheels will be faster than the conveyer, and for each mph that the conveyer accelerates, the wheels will spin one mph faster, and will ALWAYS remain faster than the conveyer, by exactly as fast as the aircraft ground speed. Unless you have attached a big chain to the aircraft that you are not telling us about?

The premise (that the conveyer will always be at the same speed as the wheels), can only be satisfied if the aircraft cannot move with respect to the ground. It is impossible. So the only way you could "physically implement" this is with a very large chain.

@CharlesBretana The chain is yours. I have nothing to do with it.

And there we have it. A totally lucid, analytical, substantive comment, that speaks to the issues at hand in a clear, step by step, understandable argument. Come on, forget the chain, How are you going to implement this without the aircraft moving? You do understand that if the aircraft moves across the ground, your premise is false, right? Or are you saying that making the wheels spin furiously will somehow stop the aircraft from moving?

@jjack, You also realize that if the aircraft engines are not running (or if it is tied down to the ground, (not to the conveyer) that no matter how fast the conveyer (and the wheels) are moving , your premise is true and remains true (by definition!).

@CharlesBretana There is no trick involved or anything. No tie-down, an engine with thrust on the plane producing an acceleration that is matched by the acceleration of the conveyor so that the wheels spin according to the statement in the question.

I'm not saying there's a trick @jjack, What I'm saying is that the premise is logically not possible. Please, take a moment and calculate the speed of the wheels (or the rpm that this equivalent speed represents) when the conveyer is moving "according to the statement in question" when the aircraft is moving at one foot/sec. For the sake of argument, say the conveyer is moving at 10 ft/sec, and the wheels are 10 ft in circumference. What rpm exactly, are the wheels spinning at?

But I think I see where you might be losing the connection. I suspect you may be misunderstanding the word acceleration. Acceleration is (or should ALWAYS be) measured in a single consistent frame of references. You have to decide, before you start, which frame of reference you are going to use. You cannot switch back and forth between the frame of reference of the earth (which is an inertial frame), and the frame of references of the conveyer, which is an accelerated frame of reference. And if you wish to compare measurements made in both, you must apply the appropriate transformation.

So your statement that " ... thrust on the plane producing an acceleration that is matched by the acceleration of the conveyor... " can only be interpreted to mean that both accelerations are measured in the Earth frame of reference, which means that the plane is accelerating in the earth frame, and therefore is indeed accelerating in the airmass, generating airspeed, and will take off. Since it is moving forward over the earth, it's acceleration in the frame of reference of the conveyer is the SUM of it's acceleration in the earth frame and the acceleration of the conveyer in the earth frame.

The wheels roll on top of the conveyor belt with a circumferential speed relative to the top of the conveyor equal to the speed of the top of the conveyor relative to the ground. It's just "playing with" relative velocities. I think you're trying to use specific numbers to grasp the concept, you should try to picture it instead.