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I'm looking for the real solutions to the system

\begin{array}{rcl} 1 & = & v_1+v_2+v_3+v_4+v_5 \\ 1 & = & v_1^2+v_2^2+v_3^2+v_4^2+v_5^2 \end{array}

Background: I'm looking at a Newton's cradle with 5 balls, each of mass ##m=1##. The first ball is pulled away and let go such that it hits the others with velocity ##v=1##. Above equations are conservation of momentum and energy. I know that Newton's cradle is more complicated and restricts the solutions much more, I just want to find out which dynamics are allowed from momentum and energy conservation only.

Some solutions I've found:

And of course all permutations of these solutions.

- ##v_1=v_2=v_3=v_4=0,v_5=1##
- ##v_1=-\frac{1}{3},v_2=v_3=0,v_4=v_5=\frac{2}{3}##
- ##v_1=-\frac{1}{2},v_2=0,v_3=v_4=v_5=\frac{1}{2}##
- ##v_1=-\frac{3}{5},v_2=v_3=v_4=v_5=\frac{2}{5}##

Are there more? If so, how can I find them?

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# Solutions for a quadratic system of equations

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