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Path to obtain the shortest traveling time
I've been told that if one would want to make a ramp to get a ball from point A to a lower point B (at a certain horizontal distance from A), the best shape to make this ramp in - by which I mean the ramp that makes the ball reach point B the fastest - would be a cycloid path. Why is this?
To simply explain the curve consider what it does. It allows the ball to drop first to pick up speed and then transitions to more horizontal motion to span the distance from A to B.
If the ball were to transverse across first and then drop it would do so slowly. It is simply a matter of optimization to get the correct curve.
As @dmckee noted in the comments, this area of math is called "calculus of variations" because the variation of the ball path is considered.
