Written by Yuju P. (GAFL'18)
Edited by Muchang B. (KIS'19)
━━ August 1st, 2018 ━━
Finding the Curve of Quickest Descent: The Brachistochrone Problem
A problem that often appears as itself as a question or as a part of solving other math questions is to find the shortest distance between two given points on a Euclidean plane. It is evident that the shortest distance is the length of the straight line connecting the two points, which can be found with the use of Pythagorean theorem. However, what if we were to find a path such that it takes the shortest time for an object to move from a point to another? A similar but more rigorous question was posed by Johann Bernoulli in 1696, named the brachistochrone problem:
Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time?
The path of the light may be thought of as a straight line. Although it can be correct, it is only limited to situations where the density of the medium between the points is uniform. However, as no such assumptions on the density has been made, cases where the density changes also have to be taken into consideration. By assigning arbitrary values for each density, both cases where the medium is uniform and non-uniform can be considered. In physics, it is known that the speed and the direction of light changes when the density of medium changes, as in Figure 1.
Although we did not find the exact solution to the brachistochrone problem as the constant M remains unknown, it is possible to find graphically. We simply have to vary M until the curve intersects with the final point (excluding M where the final point intersects with a curve segment different from the one that includes the initial point) and taking such value of M will give the exact equation of the brachistochrone curve in search.
There are a number of ways to solve the problem and the method we just used is only one of many. Although the property of light was used to derive the solution, it should be made clear that such additional knowledge from physics is unnecessary. The problem can be solved with an assumption that there exists an object such that its path is of the shortest time, while being knowledgeable of the Fermat’s Theorem allows us to formulate such an idea more easily. The brachistochrone curve can be found in real world, such as in the flight path of a descending bird, in fish scales, and in construction of slides and roller coasters. The brachistochrone problem is significant in that it led to the development of multivariable calculus which is found to be very useful in natural sciences and engineering and is worth looking into.
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Bibliography:
- Kunkel, Paul. “The Brachistochrone.” The Brachistochrone, whistleralley.com/brachistochrone/brachistochrone.htm.
- “Desmos | Graphing Calculator.” Desmos Graphing Calculator, www.desmos.com/calculator.
- Giancoli. “Giancoli 7th ed Physics Chapters.” Giancoli Physics Chapters, fcis.aisdhaka.org/personal/chendricks/IB/Giancoli/Giancoli%20Chapters.html.
- MacDonald, Paige. “The Brachistochrone Curve.” 16 May 2014, https://mse.redwoods.edu/darnold/math55/DEProj/sp14/PaigeMcDonald/FinalDraft.pdf
- Babb, Jeff. “The Brachistochrone Problem: Mathematics for a Broad Audience via a Large Context Problem.” Academia.edu, www.academia.edu/19539363/The_Brachistochrone_Problem_Mathematics_for_a_Broad_Audience_via_a_Large_Context_Problem.
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