Can anyone explain this "mathematically"? i.e. how to calculate the time in each cases?
Yes, but not at B level. The math required is called variational calculus. You will need multivariable calculus and differential equations as prerequisites.
At B level, I would say that it is a combination of maximising the speed and minimising the path length. The upper path is shorter, but the ball accelerates slower in the beginning. The lower path gives a larger mean speed, but is longer.
Sorry, I did the mistake. It should not be in "B" level. But I've corrected that.
This is related, but neglects the rotational inertia of the balls:
Here is the full clip BTW:
If one makes the simplifying assumption that the balls roll without slipping then one can model the motion by using an "effective inertial mass" which bundles the effect of the ball's inertial mass together with its moment of inertia. The effect is the same as if the acceleration of gravity was reduced. Unless I am missing something, the Brachistochrone should still be optimal in such a case.
I thought about this a bit and I think it is correct as long as the ball radius is small in comparison to the curvature of the track such that the angular velocity of the ball is directly given by the ball's radius and the velocity of the ball.
Good point about the radius. I'd missed that.
Thanks to all of you (specially @A.T.). Actually I skipped some portions in Classical Mechanics due to (my personal) exam. strategy in completing my Graduation as I am not a good student. That’s why I could not solve this problem. But now, I have found this (The Brachistochrone) in my text book! Wow! It’s very interesting topic.
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