• Author / Uploaded
• phurhiufukhfurh urhfuh

Citation preview

Mathematics Mock IA ~Pulkesh Venkat Prayaga

Introduction Johann Bernoulli posed the problem of the brachistochrone to the readers of Acta Eruditorum in June, 1696. Initially created over 300 years, the brachistochrone is the solution to an intriguingly simple question: “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.” Interestingly, the solution to the problem was solved by five of the greatest philosophers: Bernoulli (Jakob – the brother of Johann), Leibniz, de L’hopital, von Tschirnhaus and Newton. The first thought to anyone when asked the question would assume that a straight path is the answer, as it is the shortest distance. However, the brachistochrone curve looks for a path which will cover that distance always in the shortest time. Consider the adjacent picture and the ball rolling down from A Figure 1 to B. If the slope was shallow, then the ball would roll really slowly. However, the ball could be made to drop vertically from A to O to gain speed, it would then travel at constant velocity from A to O and then from O up to B. In the extreme when the slope from A to B is horizontal, the ball would not move. Therefore, this showed that a straight line is not always the shortest time. Theoretically a cycloid will give the fastest half-pipe if the friction is neglected. The solution of the fastest curve would be the brachistochrone curve in this case. Such a curve in its pure form has infinitely short verts and is π times as wide as it is high. Figure 2

Aim At a superficial level, the aim is to explore mathematical tools such as the brachistochrone curve, the tautochrone curve and cycloids. At a more profound level, the aim of this investigation is to see one of the theoretical applications of the brachistochrone curve, the tautochrone curve and cycloids in designing half-pipes. A half-pipe is a structure used in gravity extreme sports such as snowboarding, skateboarding, skiing, freestyle BMX, skating and scooter riding. Figure 3

Rationale While scrolling through YouTube, I was watching some videos on a channel named Vsauce. I found a really interesting problem called the brachistochrone curve, the video talked about what the curve was and then the history behind how it came into existence by mathematicians and physicists asking themselves simple questions. As I watched more of it started learning more complex terms like the tautochrone curve and cycloids, but what really was going on in my mind was where could this curve by applied in the real world?

1|Page

Background theory for the concepts in this investigation This report talks about the brachistochrone curve and the tautochrone curve. In mathematics and physics, a brachistochrone curve, means '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 under the influence of a uniform gravitational field to a given end point in the shortest time in a system without consideration of friction. The problem was posed by Johann Bernoulli in 1696. A tautochrone, also known as the isochrone curve (from Greek prefixes tauto- meaning same or iso- equal, and chrono time) is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point. The curve is a cycloid, and the time is equal to π times the square root of the radius (of the circle which generates the cycloid) over the acceleration of gravity. The tautochrone curve is considered a mathematical property of the brachistochrone curve for any given starting point to end point. A cycloid (see Fig. 2) is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping, a curve generated by a curve rolling on another curve. Figure 4

In an article I read about ramp designing, it said that the designers of skate ramps knew that the ramp they designed was the fastest ramp. But apparently, there was no realization of this same ramp being a cycloid curve/involving the brachistochrone curve. The point of the report is to prove that the brachistochrone curve is part of a bigger curve that is known as a cycloid. And the skateboard ramps designed in today’s world had to somehow use the concept of the brachistochrone problem. Figure 6 shows, the cycloid formed out by a point P on the rim of a circular wheel rolling on the ceiling. The point where the wheel touches the ceiling is C. Point C acts as this instantaneous centre of rotation for the trajectory of P. It’s as if, for that moment, P is on the end of a pendulum whose base is at C. Let the point where y is Figure 5 touching the ceiling be F. Therefore, 𝑃𝐹 = 𝑦 Since the tangent line of any circle is always perpendicular to the radius, the tangent line (𝑃𝐶′) of the cycloid path of P is perpendicular to the line PC. This gives a right-angle inside of the circle (∠𝐶𝑃𝐶′). Any right triangle inscribed in a circle must have the diameter as its hypotenuse, D, (𝐶𝐶′). In conclusion, the tangent line always intersects the bottom of the circle. A pair of similar triangles is found (∆𝐶𝑃𝐶′ ≅ ∆𝑃𝐹𝐶). Let θ be the angle (∠𝐶𝐶′𝑃) between the tangent and the vertical line. The length of 𝑃𝐶 = 𝐷𝑠𝑖𝑛(𝜃). Using the second similar triangle, the length of 𝑃𝐹 = 𝐷 sin(𝜃)2 . Hence,

Figure 6

𝑦 = 𝐷 sin(𝜃)2 √𝑦 = √𝐷 sin(𝜃) 1 √𝐷

=

sin(𝜃) √𝑦

2|Page

Since, the diameter of the circle stays constant throughout the rotation, the

sin(𝜃) √𝑦

is constant on a cycloid.

This is the proof for the Snell’s law property that shows that the brachistochrone curve is actually a part of a larger cycloid curve.

Conclusion and evaluation This investigation can be further applied to how the Brachistochrone curves are useful for engineers and designers of roller coasters. To accelerate the car to the highest speed possible in the shortest possible vertical drop is a designing challenge that can be overcome with using the Brachistochrone curve. As proved, the Brachistochrone path will give the quickest way to get between two points. However, these engineers would have to keep in mind the resistance force of friction acting on all bodies would lead to the results being off. Compared to the theoretical calculations, the findings would not be reliable and accurate. All the values and theories are not considering friction as a resisting force. But however, in the real world these calculations done show us the real picture.

References https://en.wikipedia.org/wiki/Half-pipe#Skateboarding,_freestyle_BMX,_and_aggressive_inline_skating https://sinews.siam.org/About-the-Author/quick-find-a-solution-to-the-brachistochrone-problem-1 http://www.mathcurve.com/courbes2d/brachistochrone/brachistochrone.shtml http://www.mathcurve.com/courbes2d/tautochrone/tautochrone.shtml http://datagenetics.com/blog/march32014/index.html https://engineeringsport.co.uk/2010/10/29/surfing-the-brachistochrone/ Weisstein, Eric W. "Tautochrone Problem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TautochroneProblem.html Weisstein, Eric W. "Brachistochrone Problem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BrachistochroneProblem.html https://en.wikipedia.org/wiki/Tautochrone_curve https://en.wikipedia.org/wiki/Brachistochrone_curve https://en.wikipedia.org/wiki/Cycloid https://www.youtube.com/watch?v=Cld0p3a43fU

3|Page