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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 ith sphere before the collision and v_{i} the velocity vector after the collision. (Red symbols stand for vectors.)
Conservation of linear momentum requires that
For elastic spheres kinetic energy is also conserved and hence
This reduces to
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.,
Then the change in momentum is given by
This means that the conservation of kinetic energy can be expressed as
Since m_{1}(u_{1} − v_{1}) = ak it holds that v_{1} = u_{1} − (a/m_{1})k. Likewise v_{2} = u_{2} + (a/m_{2})k.
This means that the conservation of kinetic energy requires that
since k·k=1 the above equation reduces to
Everything on the righthandside (RHS) of the above equation is know; therefore a is determined. With a known v_{1} and v_{2} can be computed.
Note that the distance between the two spheres is given by
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