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The Head

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However, when I mathematically solve it, I technically get two solutions for what happens after both collisions. When solving for the velocity of the second ball, it factors and I get a zero and a non-zero solution. For the first collision I know we choose the non-zero solution because it moves. For the second, we choose the zero solution because of symmetry. But is there a better reason for this?

If it's useful, in the simple case where the balls are equal masses, after substituting my momentum equation into the energy one, I worked out that v_2(v_2 - sqrt(2gh))=0. I'm not concerned about the actual values of the velocity, but it appears to say that either v_2 = 0 or sqrt(2gh), where 'h' is the initial height that ball one was raised to. The same equation occurs after the second collision. How do we know to choose v_2 = sqrt(2gh) for the first collision and v_2=0 for the second?