Natural Sciences >> OtherThe Brachistochroneby Patryk Gasior
Submitted : Spring 2020
In this project using optimization, physics and trigonometry, I will explain how to find the path of least time. I first took an optimization problem and show how one would find a solution to a path of least time when traveling through two different mediums. By using some simple calculus, the answer is found, transitioning to the second part of using Snell’s law and the laws of the world to help find this perfect optimized path. After proving how Snell’s law works with an algebraic solution in addition to a physics one, the paper moves on to the main topic of the brachistochrone. The application of Snell’s law by Johann Bernoulli is demonstrated in the paper, and then a modern-day solution done by the mathematician Mark Levi is demonstrated as well. Lastly a demonstration of the Tautochrone curve, showing that no matter where an object starts on this curve, it would end up at the bottom at the same time, wraps up the paper. From this paper the results show how the path of least time can be found and it is applied and demonstrated in many ways. This is useful for engineers as optimization is a very important to be able to make sure that something is efficient to get a maximum output.
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