Asymmetric Newton's Cradle

Is there a way to calculate the result? I know that both kinetic energy and momentum must be conserved, but what if there is more than one solution? How would you go about picking the "best" solution?

There shouldn't be a situation where multiple solutions exist. If it's asymmetric, then the moving ball won't stop, it'll bounce back (if it's lighter) or keep moving (if it's heavier) compared to the one on the opposite end.

To be specific, I was reading this. It provides several scenarios.

Is there a way to calculate the result?

Lagrangian Mechanics is the exact subject you seek and is typically taught in an intermediate mechanics college junior course. You learn how to deal with some interesting things like double pendulums, or beads on strings... on trains, or systems of springs. The weird stuff that you think of in introductory physics 1 but would have no clue how to even approach, that type of stuff.

And this problem type doesn't have multiple solutions as long as it's well posed and you're careful about your q's and such.

I've actually never heard of Lagrangian mechanics, but after reading over it, how does Langrangian mechanics solve the problem? If it does solve the problem, is it worth my time learning in two weeks if my teacher will just blow me off again? (I'm taking the most basic of basic mechanics qq.)

I'm a physics major. I just looked over it some more. If I had 4 week's time I'd definitely try. I really do like the idea of a generalized coordinate system. Shame I don't. I'll attempt to learn it over Spring break, I guess.
V

When you 'input' how your particles are constrained to move and what their initial potential and kinetic energies are, Lagrangian mechanics will basically spit out a time variable acceleration vector. That's how it's solved, by telling you what kind of fucked up acceleration each particle undergoes. Honestly, if you aren't a physics major (or don't wanna be), it's probably not worth the time. If you are, this and introductory GR were by far my 2 favorite physics subjects. But like I said, junior in college level.

Your other thread looks like something you wouldn't need Lagrangian mechanics for, because you don't constrain the particle's occupiable space to a subspace of whatever set of dimensions you're working in. An asymmetric newton's cradle can only swing back and forth, and it'll probably be chaotic (which is why introductory mechanics courses don't do it), but the balls aren't going to detach from the frame. So... More like Lagrangian mechanics is a branch of the other thread, where you would say your particle is a bead on a fixed string / restricted to a hula hoop / something like that. You could apply the concepts in the other thread to this and probably wouldn't be able to do some harder Lagrangian stuff without them.