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Brachistochrone Curves

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What is it's origin?
State the problem with picture.
Explain how the cycloid relates to the solution.

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Solution Summary

Brachistochrone curves are discussed in detail. The solution is detailed and well presented.

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Brachistrochrome Problem

1)What is it's origin?
2)State the problem with picture
3)Explain how the cycloid relates to the solution.

Answers:

1)
A Brachistochrone curve, or curve of fastest descent, is the curve between two points that is covered in the least time by a body that starts at the first point with zero speed and passes down along the curve to the second point, under the action of constant gravity and ignoring friction.

Galileo incorrectly stated in 1638 in his Discourse on two new sciences that this curve was an arc of a circle. Johann Bernoulli solved the problem (by reference to the previously analysed tautochrone curve) before posing it to readers of Acta Eruditorum in June 1696. Five mathematicians responded with solutions: Isaac Newton, Jakob Bernoulli (Johann's brother), Gottfried Leibniz and Guillaume François Antoine de l'Hôpital. Four of the solutions (excluding l'Hôpital's) were published in the May 1697 ...

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