Adam Sobel Week 6 - The Brachistochrone Curve

    Have you ever wondered what the fastest way is between two points where the starting point is higher than the ending point, an equal force of friction, and if there is a constant downward force of 9.81 m/s^2? If you have, you aren't alone.

    Most people would assume that the fastest way between the points would be a straight line, because most people are aware that the fastest way between two points would be to go in a straight line. You wouldn't want to go left, then straight, then right if you only have to go straight. It just takes less time.

    What do you think would happen if you started by simply dropping immediately to gain speed the quickest, and then you would start moving towards the other point when you are on the same level or y value as the second point. At first, it wouldn't make sense because, if you looked at a right triangle, the hypotenuse would always be less than the sum of the two legs of the triangle. However, that would be a measurement of distance. Don't forget that we want to find the quickest way to a point, not the shortest way. By dropping down to the same level as the lower point first, we would be gaining speed the quickest, which means that it would take less time to travel the extra distance that we would be using than if we just traveled along the hypotenuse.

    However, there is still one more possible solution to try to get to the second point: the brachistochrone curve. This curve is a concave up curve that gets you between the two points the quickest. The brachistochrone is a cycloid, or the path of a point on a wheel as it rotates. This curve maximizes a short distance with picking up speed early to get you between the two points the quickest.

    Even more fun is that a brachistochrone is unique in that, if you start falling from any point on the curve, you will reach the end point at the same exact time as any other point. For example, if you start falling at the original point, and have something else start falling halfway along the curve with an equal mass, you will reach the endpoint at the same time. This isn't true for the other two options or any other option.

    Below I linked a video to explain the brachistochrone curve but more in depth with Michael Stevens and Adam Savage. They built this scenario in real life and demonstrated the properties of the brachistochrone curve. Additionally, there is a diagram if you are bad at visualizing or would just like an example.

    Do you find the brachistochrone curve entertaining? Do you think you would use this concept ever, or did you just find it interesting to learn about?

V-Sauce Video