Let us take two points on the wall, A and B, and a marble which accelerates from A to B, beginning from the state of inaction, only under the influence of gravity. What shape must the path that connects the two points, along which the marble moves, have for the marble to arrive in the shortest time?
This is a famous problem due to its interesting and unexpected solution. Many important world scientists investigated it and found quite original solutions. The problem was posed by Johann Bernoulli in 1696. He not only set to solve the problem, which he named the brachistochrone problem, himself, but he also posed it to the world’s leading mathematicians and readers of the Acta Eruditorum magazine.
No one sent any solutions for a year and a half, so Bernoulli sent the problem to Isaac Newton. Newton solved it in one evening and sent the solution back by post, anonymously. After receiving the anonymous solution, Bernoulli immediately recognized the author and famously stated „You can recognize the lion by his claws.“
Johann Bernoulli himself guessed the correct solution, but the proof he gave was incorrect. He challenged his brother Jakob to find the required path. When Jakob did it correctly, Johann tried to replace his incorrect proof with his brother’s.
It is interesting that the best path is actually a curve called cycloid, along which the marble must in some cases go uphill.
Cycloid (https://hr.wikipedia.org/wiki/Cikloida)
EXPERIMENT:
Release marbles at the same time from the starting point and observe which path is the quickest.
There are three possible paths in front of you: one of the is straight, while the other two are curved. Write down which of the paths is the quickest.
The curve that most quickly connects the path between two points A and B in different heights is called the brachistochrone curve.
Brachistochrone curve (Greek: βράχıστος: shortest and χρόνος: time) is a curve lying on the plane which connects two points of different heights in a gravitational field on which a material point, moving without friction and only under the effect of gravitation, would arrive from the higher point to the lower one in the shortest amount of time.
In this case, the curve is a cycloid; if points A and B are one above the other, the curve would be a line.
