Fastest route for a ball: Brachistochrone Curve

What are Brachistochrone problems?

Find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the least time. The term derives from the Greek (brachistos) “the shortest” and. (chronos) “time, delay.” The problem was posed by Johann Bernoulli in 1696.

What curve is the fastest?

A Brachistochrone curve is the fastest path for a ball to roll between two points that are at different heights. A ball can roll along the curve faster than a straight line between the points.

The curve of fastest descent is not a straight or polygonal line (blue) but a cycloid (red).

The first one is the shortest, but the ball spends a lot of time moving slowly. The last one speeds up the ball really quickly so it is moving fast through most of the track, but is a lot longer. The middle one is the best tradeoff between high speed and short distance.

The curve will always be the quickest route regardless of how strong gravity is or how heavy the object is. However, it might not be the quickest if there is friction.

The curve can be found using calculus of variations and optimal control

How does a Brachistochrone work?

In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) ‘shortest time’),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.

What is the Brachistochrone curve used for?

Brachistochrone curves are useful for engineers and designers of roller coasters. These people might have a need to accelerate the car to the highest speed possible in the shortest possible vertical drop. As we have just proved, the Brachistochrone path is the quickest way to get between two points.

Here is the video from Vsauce that explains everything. Really interesting video and worth the watch.

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