Game engines use state to handle collisions, constantly updating the positions of particles until a collision is detected. This approach is logical for games because graphics are displayed chronologically, but this isn’t necessarily the best system for education-oriented physics simulations, as to get the state for a time far in the future we’d need to simulate all ticks up until that point. Instead, we can use a little bit of math to create a stateless and functional approach to collision simulation.

To demonstrate, I use algebra to calculate the range of times when two particles can collide on the x axis, and use these bounds to calculate the specifics of potential collisions.

The first step is to isolate t in the following equation, which is the solution for all times two point masses will collide on the x axis given uniform acceleration.

$\left(x_1+v_{\mathrm{x1}}*t+\text{0.5}*a_{\mathrm{x1}}*t^2\right)=\left(x_2+v_{\mathrm{x2}}*t+\text{0.5}*a_{\mathrm{x2}}*t^2\right)$

After a massive amount of algebra, this gives us:

$t=\frac{\sqrt{{\left(v_{\mathrm{x1}}-v_{\mathrm{x2}}\right)}^2-4\left(x_1-x_2\right)\left(0.5a_{\mathrm{x1}}-0.5a_{\mathrm{x2}}\right)}-v_{\mathrm{x1}}+v_{\mathrm{x2}}}{a_{\mathrm{x1}}-a_{\mathrm{x2}}}$

To account for radii, we modify the first equation to the following:

$\left(x_1+v_{\mathrm{x1}}*t+\text{0.5}*a_{\mathrm{x1}}*t^2\right)-\left(x_2+v_{\mathrm{x2}}*t+\text{0.5}*a_{\mathrm{x2}}*t^2\right)=r_1+r_2$

And now get the following when isolating for t:

$t=\frac{-v_{\mathrm{x1}}+v_{\mathrm{x2}}-\sqrt{{\left(v_{\mathrm{x1}}-v_{\mathrm{x2}}\right)}^2-4\left(0.5a_{\mathrm{x1}}-0.5a_{\mathrm{x2}}\right)\left(-r_1-r_2+x_1-x_2\right)}}{a_{\mathrm{x1}}-a_{\mathrm{x2}}}$

Now we know that if the two particles collide, it will be some time (inclusively) between the two t values. This is done with virtually no computational burden. To account for an N-dimensional systems, we repeat the process N times and brute force the solution in the intersection of time intervals.

What are the consequences of this? That is left as an exercise for the reader.