The **brachistochrone curve** is a classic physics problem, that derives the fastest path between two points A and B which are at different elevations. Although this problem might seem simple it offers a counter-intuitive result and thus is fascinating to watch. In this instructables one will learn about the theoretical problem, develop the solution and finally build a model that demonstrates the properties of this amazing principle of physics.

This project is designed for **high school students** to make as they are covering related concepts in theory classes. This hands-on project not only strengthens their grasp on the topic but also offers a synthesis of several other fields to develop. For example while building the model, students are going to learn about optics through Snell's law, computer programming, 3d modelling, digital frabrication and basic woodworking skills. This allows an entire class to contribute dividing the work among themselves, making it a team effort. The time required to make this project is around a week and can then be demonstrated to the class or to younger students.

There is no better way to learn than through STEM, so follow on to make your very own working brachistochrone model. If you like the project do **vote** for it in the classroom contest.

## Step 1: Theoretical Problem

The brachistochrone problem is one that revolves around finding a curve that joins two points A and B that are at different elevations, such that B is not directly below A, so that dropping a marble under the influence of a uniform gravitational field along this path will reach B in the quickest time possible. The problem was posed by **Johann Bernoulli** in 1696.

When Johann Bernoulli asked the problem of the brachistochrone, on June 1696, to the readers of Acta Eruditorum, which was one of the first scientific journals of the German-speaking lands of Europe, he received answers from 5 mathematicians: Isaac Newton, Jakob Bernoulli, Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l'Hôpital each having unique approaches!

**Alert: **the following steps contain the answer and reveal the beauty behind this fastest path. Take a moment to try and think about this problem, maybe you might crack it just like one of these five geniuses.

it's an amazing way to learn math!

Also, it's funny how our projects are kind of "cousins". Yours involves CAD/CAM and Arduino, mine uses trash and Makey Makey :-)

https://www.instructables.com/id/Grand-Prix-a-STEA...

Very interesting piece of applied mathematics

But, no explanation of the logic of finding the solution and derivation of the curve equation.

Would you kindly publish (or email me) that.

Regards.

In Pic 18 above, look at the gap between the bar that holds the balls, and the 'floor' of the track where the balls will be contacting the track. With the balls loaded, it takes less travel of the retaining bar to release the steeper ball than the shallower ones.

Below is a screencap from the vid. The two steeper balls have already left the gate while the shallowest one is still there.

I'm not saying this will change the outcome of the race, only that it's less accurate than it should be.

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One way to think about why the steepest track should be the fastest is to think of the exact opposite: Imagine a very shallow 3-4 deg grade, for about 3/4 length to the target, then a very steep dropoff to the finish. Yes, that finish will be very fast, but the way it dawdled along the first 3/4 causes it to lose.

(Just a small thing: the 'heat sinks' that you are using are more conventionally called 'threaded inserts'.)

I'm looking forward to making a (modified) version of this!