## more math (coalesce notes)

m_{1}v_{1} + m_{2}v_{2} = m_{1}v_{1f} + m_{2}v_{2f}

m_{1}v_{1f} = m_{1}v_{1} + m_{2}v_{2} - m_{2}v_{2f}

m_{1}v_{1f} = m_{1}v_{1} + m_{2}(v_{2} - v_{2f})
* v_{1f} = (m_{1}v_{1} + m_{2}(v_{2} - v_{2f})) / m_{1}
Then solve for v_{2f} in the other equation & substitute? where exactly does the cosθ come from?
I'd have to figure that out, anyways, and I'm not 100% sure how. (Hell, I don't have a clue.)
m_{1}v_{1}^{2} + m_{2}v_{2}^{2} / 2 = ... wait ... why the x/2? Both sides of the equation are divided by 2. Couldn't I just get rid of it? The answer to that, of course, is yes.
but basically I'm solving for *4* unknowns. 2 vectors = 2 sets of 2 values each. Either force + angle or dx + dy. And I *can* convert between those.
m_{1}v_{1}^{2} + m_{2}v_{2}^{2} / 2 = m_{1}v_{1f}^{2} / 2 + m_{2}v_{2f}^{2}
m_{1}(v_{1}^{2} - v_{1f}^{2}) = m_{2}(v_{2}^{2} - v_{2f}^{2})
m_{1}(v_{1}^{2} - v_{1f}^{2}) / m_{2} = v_{2}^{2} - v_{2f}^{2}
v_{2f}^{2} = v_{2}^{2} - m_{1}(v_{1}^{2} - v_{1f}^{2}) / m_{2}

So I should treat all collisions (that don't destroy both objects) as collisions between spherical objects, at least as far as determining the post-collision movement vectors.
Then I need to determine the line connecting their centers, and calculate the objects' movement vectors (x,y components) relative to *that* line. The y component will remain unchanged.
*[some notes pertaining to a graph that won't make sense w/o the image]*
v_{1f} =

- v
_{1}+ m_{2}(v_{2}- (v_{2}^{2}- m_{1}(v_{1}^{2}- v_{1f}^{2}) / m_{2}))) / m_{1}*[sic]* - v
_{1}+ m_{2}(v_{2}- (v_{2}^{2}- (m_{1}v_{1}^{2}- m_{1}v_{1f}^{2}) / m_{2})) / m_{1} - v
_{1}+ m_{2}v_{2}- m_{2}(v_{2}^{2}- (m_{1}v_{1}^{2}- m_{1}v_{1f}^{2}) / m_{2}) / m_{1} - v
_{1}+ m_{2}v_{2}- (m_{2}v_{2}^{2}+ m_{1}v_{1}^{2}- m_{1}v_{1f}^{2}) / m_{1} - v
_{1}+ m_{2}v_{2}- m_{2}v_{2}^{2}/ m_{1}+ v_{1}^{2}- v_{1f}^{2} - v
_{1f}+ v_{1f}^{2}= v_{1}+ m_{2}v_{2}- m_{2}v_{2}^{2}/ m_{1}+ v_{1}^{2} *[this next equation looks wrong]*

this doesn't help me!
a^{2} + b^{2} = c^{2} ? (Do I know either a or b? I don't think I do.)
This *can't* be that hard. Hell, if it helps I can even tweak the equation so one body is at "rest".
I need a good geometry / physics book. But I can't. I'm already behind on my bills. There's the internet, granted, but that's not the same.
*[many more trig. notes that would be useless w/o the diagrams accompanying]*