Tuesday, July 2, 2024

BDA Diagrams (Before, During, After) on Desmos

My inspiration for this project was a Twitter tweet from Brian Frank. In that tweet he showed a photo of a graph sketched out on a whiteboard. It was a graph of a collision between two objects, and it showed the velocity and momentum of the objects before, during, and after the collision, as well as the force the objects applied on each other during the collision. This photo is from his blog:

Here's a link to the full blog post:
When considering collisions, one usually compares the momentum before the collision to the momentum after, demonstrating the conservation of momentum. Rarely does one consider what is happening during the collision (because it's complicated). What happens during the collision is usually saved for a discussion of impulse, where it is revealed that during the collision the objects exert equal and opposite forces on each other (and this is why momentum is conserved). What Brian had done that was exciting to me was to present it all in one beautiful set of graphs: momentum, velocity, and force, for both objects, before, during, and after the collision.

I decided to model this in Desmos.

I used as my default view a 6-axis view rather than Brian's 3-axis view. I thought that the 6-axis view would be less confusing for my students. The 3-axis view is more elegant, though, so I built a "switch" into my simulation so you could switch back and forth between views (scroll down the left panel to lines 18 & 19).

I have also added other kinds of interaction: the masses and initial velocities can be changed (lines 8 - 11); the collision can be switched from elastic to inelastic and back (lines 12 & 13); the time duration of the collision (impulse!) can be changed (lines 14 & 15); and there's an adjustable scale which is helpful when the lines all start to overlap (lines 16 & 17).

I added a collision simulation of two balls at the bottom that is timed with the graphs. When the balls collide, they simply overlap, I didn't try to build a realistic collision model.

Here's the link to the project: https://www.desmos.com/calculator/xtr8shdagl

Line 3 in the left panel has the play button -- just click on it to play the simulation. You can minimize the panel on the left (click the "<< " symbol). You can also manipulate the right panel to change the viewpoint.

And here's a link to a newer version with vectors in the bottom simulation!
BDA Diagrams (Before, During, After) on Desmos, with vectors!

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