I have to take a moment to acknowledge how hard simply solving a system of equations is. I was puzzled by how the velocities of two colliding objects are calculated, so I looked it up. Turns out both the conservation of momentum and the conservation of energy hold true in an elastic collision (in which no energy leaves the system of two colliders).
That's two equations involving masses and velocities. The Wikipedia article, which is really good, waves it through and says with these two equations, you get the handy one that tells you the velocities after the collision.
I tried a couple times on my own to derive the handy one on my own from those two. I kept ending up with huge, crazy equations. I realized there are quite a few paths to choose at each step of solving a system of equations.
In the end, I had to look it up and follow along with this
answer on Socratic. The trick I had forgotten about was that terms in the form x^2 + y^2 can be factored. Getting rid of squares and avoid square roots is the key here.
I think I'm in for a bumpy ride in my quest to understand acoustics.
I have to take a moment to acknowledge how hard simply solving a system of equations is. I was puzzled by how the velocities of two colliding objects are calculated, so I looked it up. Turns out both the conservation of momentum and the conservation of energy hold true in an elastic collision (in which no energy leaves the system of two colliders).