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The curve of fastest descent is not a straight or polygonal line (blue) but a cycloid (red).
In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'),[1] or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696 and solved by Isaac Newton in 1697.
The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. However, the portion of the cycloid used for each of the two varies. More specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp. In contrast, the tautochrone problem can use only up to the first half rotation, and always ends at the horizontal.[2] The problem can be solved using tools from the calculus of variations[3] and optimal control.[4]
Balls rolling under uniform gravity without friction on a cycloid (black) and straight lines with various gradients. It demonstrates that the ball on the curve always beats the balls travelling in a straight line path to the intersection point of the curve and each straight line.
The curve is independent of both the mass of the test body and the local strength of gravity. Only a parameter is chosen so that the curve fits the starting point A and the ending point B.[5] If the body is given an initial velocity at A, or if friction is taken into account, then the curve that minimizes time differs from the tautochrone curve.
https://en.wikipedia.org/wiki/Brachistochrone_curve
Vsauce takes a while to get to the point here but *bonus* Adam Savage appears!
https://youtu.be/skvnj67YGmw?si=4OR7HQQ_ht4vzBa6
While vaulters are not necessarily looking to get to the end faster the brachistochrone offers some intriguing possibilities when assessing the vault trajectory. It's vertical ending pairs nicely with achieving a high "skinny" parabolic flight. It's not a given but the best way down in terms of velocity vs trajectory length seem an interesting start for looking at the best way up.
Honestly a proper physics analysis of this is on the level of masters thesis work or PhD. Personally my intuition is that the best path probably lies inside of this, but not by much. The vaulter swinging through is a "steering up" of the system so there's a path change there, a bit of a kink up in the potential energy curve.
I do think about this one once and a while.