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The Elastic Collision of Spheres

Consider two spheres. For i∈{1,2} let m_{i} , x_{i} and u_{i} be the mass, position vector
and velocity vector of the i-th sphere before the collision and v_{i} the velocity vector after
the collision. (Red symbols stand for vectors.)

m_{1}(u_{1}·u_{1}−v_{1}·v_{1}) = −m_{2}(u_{2}·u_{2}−v_{2}·v_{2})
and hence
m_{1}(u_{1}−v_{1})·(u_{1}+v_{1}) = −m_{2}(u_{2}−v_{2})·(u_{2}+v_{2})

In a collision the change in momentum for each sphere is in the direction of the vector
between their centers at the instant of contact. Let k be the unit vector in that direction;
i.e.,