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c0nrad's blog

Calculating Collisions Pt. 2

Programming the equation we derived in the last post.

Background

So in the last post we were able to generate the following equation. It can be used to determine the resulting velocities of a collision of two spheres:

$$ v_{1f}^2 + v_{1f} \frac{(- 2 p_{i} m_{1} m_{2})}{m_{2} m_{1}^2 + m^2_2 m_{1}} + \frac{p_{i}^2 m_{2} - 2 m^2_2 K_{i}}{m_{2} m_{1}^2 + m^2_2 m_{1}} = 0 $$

Results

This morning I codified the equations into simp, and they turned out pretty nicely!

They line up to the math we worked out in the last blog post.

Vector Fun

In the last post I assumed that the vector math would just work out. Which ended up mostly being true.

But, it took me some time to figure out what I was supposed to do with the vectors when multiplying them together.

For Kinetic Energy, it almost seemed to make sense to take the dot product of the velocity vectors. Do we really want the “kinetic energy” for all the individual components? It feels weird to directly multiply two velocity vectors together.

$$ K_{i} = \frac{1}{2} m v^2 $$

$$ \begin{bmatrix} v_x \\ v_y \\ v_z \\ \end{bmatrix}^2 = v_x^2 + v_y^2 + v_z^2 $$

But this didn’t work. When I treat kinetic energy like a scalar instead of a vector, I had to add/subtract scalars from a vector (during the quadratic equation step) which was also weird (and the results I got were less than ideal…):

So instead I just multiply the vectors across:

$$ \begin{bmatrix} v_x \\ v_y \\ v_z \\ \end{bmatrix}^2 = \begin{bmatrix} v_x^2 \\ v_y^2 \\ v_z^2 \\ \end{bmatrix} $$

And it seems to work. /shrug

Next up

Turns out there’s another big problem!

The equations don’t take into account how the spheres are collide!

Back to the drawing board. It has something to do with the normal forces, but I’m not 100% sure how the math will look yet. Stay tuned!