Bouncing balls
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The states (on the y-axis) are y and v_y

y'   =  v_y
v_y' =  -g

Note that this follows the standard form.


Elastic collision
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           m1      m2
           O->     O->
Before:    u1      u2
After:     v1      v2  


Conservation of momentum


(1)  m1 v1 + m2 v2  =  m1 u1 + m2 u2
                      _______________
                      =I (known value)


Conservation of kinetic energy

(2)  m1 v1^2  +  m2 v2^2   =   m1 u1^2  +  m2 u2^2
     -------     -------       -------     -------
        2           2             2           2

Elimination of one variable (v1 or v2) gives a second order
equation. If u1<>u2 this equation has two solutions, one for the
solution before the collision, the other for the solution after.

For the solution after collisions we obtain from (1) and (2)


v2 -v1  =   -( u2 -u1 )
            ___________
            =R (known value)

that is, the relative velocity changes signs at the collision. v1 and
v2 are then obtained from the system of equations:


m1 v1 + m2 v2  = I

v2 - v1 = R


which is easier to solve.