I was very interested in the recent (now locked) thread on the 777 incident that re-opened the debate on the jet plane on conveyor belt problem. Congratulations to a number of posters who gave very clear explanations and illustrations of the correct answer - being, of course, that the aircraft will take off. However, I think there is an inherent paradox in the problem (touched on by JohnD) and also believe the assertion that aircrraft wheel rotational speed will be 2 x aircraft forward speed is wrong. I acknowledge the very helpful analogy, given by BAS, of the super-engineered toy car on a supermarket checkout conveyor belt, which (i found) makes it much easier to visualise what's happening.
So - imagine the toy car with frictionless wheel bearings on the belt. I am standing alongside the belt and supporting the rear of the car so it cannot move backwards. I start to walk forwards at 2 mph. Necessarily the car, supported by the 'thrust' of my hand moves along the belt at the same speed, so its wheels have a rotational speed of 2 mph. In the conditions given by the OP, the belt must match the wheel speed so the belt starts to move at 2 mph in the opposite direction. However, the thrust I impart by supporting the car while walking forward is still maintained so the wheel rotational speed must increase to belt speed + 2 mph and so the belt speed must also increase and so on. It seems that if I maintain a forward thrust on the car, the belt speed and the wheel rotational speed will continue increasing towards infinity. That's the paradox.
But, if the belt is continuously accelerating towards infinite speed, the rotational speed of the wheels at any moment must be equal to belt speed PLUS the forward velocity imparted by the 'thrust' of my forward movement (i.e not 2 x the forward velocity).
Not absolutely sure I've got this right but still finding this interesting.
So - imagine the toy car with frictionless wheel bearings on the belt. I am standing alongside the belt and supporting the rear of the car so it cannot move backwards. I start to walk forwards at 2 mph. Necessarily the car, supported by the 'thrust' of my hand moves along the belt at the same speed, so its wheels have a rotational speed of 2 mph. In the conditions given by the OP, the belt must match the wheel speed so the belt starts to move at 2 mph in the opposite direction. However, the thrust I impart by supporting the car while walking forward is still maintained so the wheel rotational speed must increase to belt speed + 2 mph and so the belt speed must also increase and so on. It seems that if I maintain a forward thrust on the car, the belt speed and the wheel rotational speed will continue increasing towards infinity. That's the paradox.
But, if the belt is continuously accelerating towards infinite speed, the rotational speed of the wheels at any moment must be equal to belt speed PLUS the forward velocity imparted by the 'thrust' of my forward movement (i.e not 2 x the forward velocity).
Not absolutely sure I've got this right but still finding this interesting.