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I am preparing a seminar on mathematical discovery for beginner's students, and one of the topics I plan to cover is the brachistocronic/tautochronic curve. For this purpose, I would like to let the students experiment with various curves connecting two given points P and Q for a while, trying to minimize the time it takes a ball starting at P to arrive at Q.

Since I cannot think of a feasible way to do it physically, I am looking for a suitable piece of simulation software. Ideally, one should be able to draw a curve and then have it simulate a ball rolling along that curve, showing the time it took at the end.

So far, I have been able to find this:

https://www.physicsclassroom.com/Physics-Interactives/Circular-and-Satellite-Motion/Roller-Coaster-Model/Roller-Coaster-Model-Interactive

This is about what I want, but for my purposes, it is too "coarse" with just four adjustable points. Does anyone know of a piece of software that does this? Since it should be used by the students during class, it would be good if it was freely available online.

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    $\begingroup$ Good question! One thing that came to mind is desmos and their marbleslides. (Example link: teacher.desmos.com/activitybuilder/custom/… ) I have used these in activities where students have to adjust the function so the marbles roll down the curve and "collect the stars." I am not sure if it's possible to modify this to exactly your needs, but I can at least guarantee this kind of thing allows students to modify the function formula and then click a button to set the marble in motion. $\endgroup$ Commented Feb 23, 2022 at 19:54
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    $\begingroup$ I was also thinking of desmos. Can you write the functions as functions of time, to include the physics? $\endgroup$
    – Sue VanHattum
    Commented Feb 23, 2022 at 22:35

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The Energy Skate Park by PhET seems to come quite close: enter image description here

In the "Playground" section you can draw arbitrary curves and add a stopwatch. The simulation has to be manually stopped at the end of the curve and you have to deal with a person (or a dog) skating down the curve, but otherwise it should do everything that you need.

You could also use the "Bumps" setting in the simulation that you suggested.

Finally, there is this product (german supplier, but maybe "magnetic curve track" yields results for your country as well) to create real curved tracks on any magnetic surface:enter image description here

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    $\begingroup$ Yes, that is quite exactly what I was looking for, thank you very much! Also, the "hardware" version looks great and should not be hard for me to obtain, as "my country" happens to be Germany. :-) $\endgroup$
    – M Carl
    Commented Feb 23, 2022 at 18:34

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