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I remember how confused people were by one of the Fast and Furious films where cars were driven into a cargo plane.
A simple way of looking at it is that if a jet car was put on a rolling road and was pushed, or pulled, by an external force then would it move forward.
The answer is yes.
No it's not. If it were on ice it would take off because the jets would propel it forward. The conveyor actively moves with the wheels, it doesn't just slip.
What is the source of the force (or forces) in the updwards direction that overcomes the downward force of mass times gravity?
If the only source of force is the jet engine, and that force starts in the horizontal, then there is no component in the vertical (orthogonal) direction because obviously cos(90deg) = 0. Hence, there would be no lift.
If the jet engine is tilted with respect to the horizontal, such that it has a vertical component then there will be some lift, but it needs to be enough to overcome mass times gravity.
If there is an airflow across the wings that is enough to create a upwards force that overcomes mass times gravity then the aircraft will lift. The 'airflow over the wings force' needs to be enough to lift the nose then the upward force from the tilted jet engine will also combine with the lift from the airflow over the wings. But that implies forward motion or a bloody strong wind.
However, due to the relatively low friction in the wheel bearings and a well inflated tyre against the belt, the conveyor would have to accelerate at a practically impossible rate to keep up with the accelerating plane, so in practice I would have thought the conveyor mechanism would fail before the undercarriage.
However, your point is a very much valid one and should such a conveyor system be feasible (e.g. we could have 3 separate smaller units, one under each wheel, thus reducing the amount of stress forces that would be experienced in a single, large system), it could well result in the failure of the undercarriage/tyres before the belt itself.
I think that the spirit off the question has indestructible, infinite-speed-capable, infinitely powered aeroplanes and conveyors. Which is why the plane would just stand still while the universe is annihilated by the jets.
Supportact said: [my style is] probably more an accumulation of limitations and bad habits than a 'style'.
Reverse that process and then let the plane take off normally from the runway. Problem solved.
Sheesh, it takes an acountant to work out a practical answer.
If we accept the infinite, theoretical stuff then the answer is undefined since 1) infinity - infinity (the horizontal force balance) is undefined. 2) its theoretical anyway.
If we don't accept this conveyor belt can exist, but look to actual real-world components, then the plane can take off for the fairly simple reason that the conveyor belt can't match the wheel speed because it can't know the wheel speed because aeroplanes don't have wires dangling from them, and don't have a radio link which is constantly providing the data about the wheel speed. (And even if they did, ie a remote engineering/monitoring/data acquisition system, the data is encrypted).
OK so the jet engines would cause some AIR 'suck' over the wings and the wheels would have friction blah blah, but without the air splitting above and below the wing airfoil section there is no lift. 'THE CONVEYOR IS DESIGNED TO EXACTLY MATCH THE SPEED OF THE WHEELS'
No it's not, because the car is driven via the wheels. it's more like a bike on a treadmill with somebody on the bike but not pedalling. You hold them upright as the treadmill starts by pushing them on their back. The only thing stopping you then pushing them forward (because even at a relatively high speed of treadmill it would be easy) is the limitation of the question. If you pushed them very hard, the treadmill would have to go incredibly fast to stop forward movement, which in essence the paradoxical limitation of the question.
My issue with the question is it is not at all real world, thrust would easily overcome drag, in practise. The treadmill would have to increase exponentially with respect to the thrust, putting the speed of the treadmill at full thrust beyond the bounds of possibility.
I'll put it another way, it's a paradox, not a physics question.
the conveyor would in this magical situation be capable of sustained rolling of 700+ miles per hour... without the tyres or belt melting...
but air is being pushed at 700 miles per hour in that situation - that air has to go past the wings and would create lift - the plane would at first lift vertically and then go forwards... the mechanical lift generated occurs not because of the movement of the plane, but movement of the air over the stationary plane. Yes the greatest movement of air would be through the engine, but it would force the air around those 4 small horizontal columns of air to move too
In a real situation of as described the belt/tyres would not be capable of sustaining the speed of movement long enough to work... and so would allow the thrust of the engines to overcome the drag of the wheels... but I think in these hypothetical situations it's fair to assume we're ignoring lots of factors. which probably means we're supposed to ignore the drag the air will have on the surrounding air...
This is obviously not a real world situation, but if it were to be I think that the question would be thus:
If a plane applied thrust smoothly as it normally would to get up to 200mph over 60 seconds and the conveyor belt that it was on accelerated smoothly up to 200mph in the opposite direction over 60 seconds would the plane take off?
I think that the answer is "no" for the reasons Maynehead has explained.