Plane and a conveyor belt

Ooh nice question. But no, even if it left the ground it'd just fall back down again. It'd have no momentum when it got in the air. I think.

Dom

But if the conveyor tracks the wheel speed and matches it, the plane would remain stationary, so you'd never get any airspeed (beyond the wind at the conveyor), as it'd be stationary relevant to the airflow.

You need airflow over the wings to generate lift. No airflow, no lift.

nope, it will take off without issue...

Magic - the aircraft is STATIONARY with respect to the air mass it is within. Unless it's a harrier, it's not going to move upwards.

Think about what happens if you are on a running machine holding a flag. You may look at the speedo and it tells you that you are running at 10MPH and your legs are moving at 10MPH. However, the flag will remain limp as no air is moving past it. An aircraft requires lift which is why it has wings(!) and has to hurtle down a runway at 100MPH in order to create the lift needed to get it off the ground. Just running the engines at 100% isn't enough to get it off the ground.  

But in the question it says that "his 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)" which I take to mean that it will change speed to match whatever speed the wheels are doing. So, on my roller skates I would try to speed up but when I pull myself forward the treadmill would increase speed to match the increase in the speed of my wheels and I wouldn't go anywhere. I would only move forward if the speed on the treadmill stayed the same, which isn't the case here.
If the question was if the treadmill was moving at a constant, fixed rate (say 100MPH) then it could be done as your speedo would say 200MPH when you were actually doing 100MPH yourself (or I could still pull myself along on my skates if I had a rail). But if the speed of the treadmill then increased to 200MPH to match the wheels you still wouldn't be moving?

But the aircraft is stationary, so there'll be no airflow with respect to the surrounding air, so how exactly does that work?

You are saying that you cannot pull yourself forward up the track..

In a roller skates example, you can, as you are creating a differential in speed between yourself and the track, and as your speed exceeds it's speed however breifly, you'll move. That assumes the track doesn't adjust to you pulling.

However, with wings, there is no airflow over them. If you accelerate or travel faster than the track, you WILL get airflow, but if the track constantly matches the speed the aircraft is travelling, the aircraft will remain stationary at the same physical point, hence no airflow.

I'm amazed you guys aren't getting the fact that the aircraft is stationary on a treadmill.

Ask yourself this - if you run really fast on a treadmill, does the wind whistle past your ears???

It's exactly the same principle as "would the aircraft take off with the engines at full thrust but the breaks on keeping the aircraft motionless". Of course it wouldn't!

Great question though!

But it cannot be kept stationary in this example since the wheels are not powered.

But if it's on a treadmill, which matches the planes speed, as detailed in the question, the aircraft will stay stationary at one point.

If it doesn't and moves forwards or backwards, then the control system of the treadmill is defective compared with the description in the question.

Not the same thing. Again, your legs are powering you, same as with the wheels of a car on this conveyor belt. Aircraft are not pushed/pulled along the runway on driven wheels...

But again, that doesn't matter if the control system of the treadmill is matching the aircraft's speed. That implies that the control system is set up in such a way to keep the aircraft at a fixed point on the treadmill.

If the control system is set up so that the aircraft is allowed to move, then the wording of the scenario is incorrect.

I'm amazed a qualified pilot (I think you are aren't you) is thinking the aircraft will remain stationary...

If I built a control system to match the objects speed, then it would remain stationary, otherwise, my control system doesn't work. And in that scenario, it wouldn't take off.