Acceleration downhill, fastest trajectory for a ball

Question

Given 3 ways of going downhill, like in this image:

Would a ball behave like that in real life? Intuitively, it makes no sense. The shortest path here is not the fastest.

Any hints to the math behind this?

Answer 1

You asked for hints, so here they are:

1. Travel time along the ball's path is equal to the ball's speed at each point, integrated along the path.
2. Let the path be defined as $y =f(x)$. The speed $v$ at any point on the path is simply the speed due to gravitational energy change.
   3. You can thus write travel time T as an integral of $ds/v$, where $ds$ is directed always tangent to the path.
   4. Finally, you can solve for the $f(x)$ that minimizes T. This is simple, but amounts to calculus of variations. Look up the Euler-Lagrange equations and you should be able to see how to minimize T.

Answer 2

It does behave like that in real life, sometimes the predictions from Physics are surprising, but they are often right.

In this video Michael from Vsauce in fact builds a Brachistochrone (the name of this kind of curve) and compares it to other trajectories, just like in your animation.

You can find many proofs of the brachistochrone equation, but I think what you need is another perspective:

The straight line is the shortest from A to B, you are right about that, but gravity is an acceleration, the longer you fall, the faster you fall.

In Real Life due to air resistance there is a terminal velocity, a point where gravity doesn't accelerate you anymore, that's why physics is all about approximations, and in this approximation, we are so far from terminal velocity, that we can completely ignore its existence.

The longer you fall the faster you fall, and once you have a lot of speed, it can be redirected in another direction, even upwards.

Seeing all these facts one could consider if it is possible for a path to exist in which we are falling so fast that we would have enough speed to reach point B before something traveling in a straight path, and there is!

Granted, once you consider that it could be possible to "use gravity more efficient" it isn't intuitive at all that the answer is "yes", nor it is intuitive what should be its shape, but that's what mathematics is for.