Originally Posted by gnofliwr

I like the VTOL solution. It is the only one that gets the plane into the air within the stated conditions. I'm still not sure it can move horizontally with respect to the belt without violating the conditions, but if you consider hanging on the prop as taking off, then it can.

Barring the VTOL solution, the theoretical plane will never take off from the theoretical belt.

- Roger

Originally Posted by theKM

Say the diameter of the wheels are the same as the diameter of the belt axis at one end. If you put a tachometer on them they will read the same as long as the plane doesn't move. Now if the wheels move forward on the belt relative to something that is not on the belt (as what is needed for the plane to fly), the tachometer on the wheels will need to measure rotation higher than that of the belt. Therefore... the wheels need to rotate faster than the belt in order for the plane to take off. But because condition of the question states that the belt matches the speed of the wheels, the plane cannot move while this condition is true.

Originally Posted by Montag DP

Let's do a little dynamics problem.

Say you have a wheel with frictionless bearings on a conveyor belt. The wheel has diameter d and moment of inertia I. You apply a force F on the axle of the wheel to get the wheel to accelerate at a rate A. The belt is accelerating backwards at a rate A1. What is the value of the force F in terms of all the other parameters?

I worked through it and got a value F = 4I(A+A1)/d^2. On the other hand, if the belt is not accelerating, the force is just 4IA/d^2 (obviously). If A is 0, then the force (the "thrust") is 4IA1/d^2 (in this case, the wheel is stationary or moving at constant speed). So clearly, the accelerating belt produces a force on the wheel, pulling it backwards.

The same could go for a plane, so in theory a belt could hold a plane in place by accelerating at a high enough rate.
This is just one example that shows that the accelerating belt imparts a linear force to the axle of the wheel, even in the absence of bearing friction.

Originally Posted by scaredsilly

If the speeds of the belt and the wheels do not match, than the preconditions of the question are not met. The question is if the plane can take off if the speeds do match. The answer is no.

Originally Posted by willhaney

Here’s the question the “yes” voters are answering:

Imagine a plane is sat on the beginning of a massive conveyor belt/travelator type arrangement, as wide and as long as a runway, and intends to take off. There is no wind.

Can the plane take off?

Originally Posted by gnofliwr

Will,

You are correct. The "Yes" answers have answered a different question that the one posed in the problem. It is difficult for most people to think abstractly when the problem they are solving is outside of - or more importantly - contrary to common experience. The problem as stated is defilitely outside the box of common experience. On the other hand, the "Yes" solutions are outside the box of the problem.

I don't know which side is thinking outside of the right box.

- Roger