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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

Picture of 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.

JoakaTech1 month ago
Dudes... am so proud of u guys... well done!!! Love from Germany...
jack tech1 month ago
it's an amazing way to learn math!
M.C. Langer1 month ago
Congratulations for the Grand Prize! Beautifully executed, great design, amazing job!

Also, it's funny how our projects are kind of "cousins". Yours involves CAD/CAM and Arduino, mine uses trash and Makey Makey :-)
ozjon1 month ago
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.


cwb31061 month ago
The brachistochrone is also a tautochrone (or isochrone) curve. That is the travel time from anywhere on the curve to the bottom is the same. If you release two balls, one at the top and the other in the middle, they well reach the bottom together. See for more info.
Technovation (author)  cwb31061 month ago
Yes that can be seen in our second experiment, thanks!
jbarchuk1 month ago
Sorry, I like the demo, but there's a structural problem that causes an error. The balls are not leaving the launcher at the same time.
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.
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.
Technovation (author)  jbarchuk1 month ago
That's a very good observation, we saw the same flaw. We are working on an improved release mechanism, once done we will update the files. Thanks for the feedback!
I thought that too when I first watched the video. But I think it's not really true - what is happening is that the ball on the left is leaving the gate at a very slow speed because of the shallow ramp, and staying visible in the view, whereas the other two are leaving very fast and essentially dropping from sight immediately. Watch the intro video in slow motion from about 1:00. So while it might be that the left ball is leaving late (and probably is a little given the ball holder probably has to rise higher to release it), a lot of it is also related to the initial speed, which in some respects is the whole point of the setup (to get up to a fast speed quickly, since as far as I understand physics, all three should be traveling at the same speed when they finish as they have lost the same amount of potential energy).
JoeS301 month ago
HAHA, I just saw Adam Savage and Michael from vsauce build one of these quite similar to yours, then 2 days later Instructables sends me a link to yours.
Technovation (author)  JoeS301 month ago
Yes they were our inspiration but we have tried to add on other features to make it even better for an experiment. Thanks for the feedback!
subrotos1 month ago
Very impressive indeed. And very, very interesting as well.
(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!
Technovation (author)  subrotos1 month ago
Thanks, will make the changes
Craftcorner1 month ago
Really impressive presentation, everything is so clear and well explained. Keep it up!
Technovation (author)  Craftcorner1 month ago
Thank you!
ShambhaviD11 month ago
Great project, really well described steps with adequate information and detailed images. Voted!
Technovation (author)  ShambhaviD11 month ago
Thanks for the positive feedback!