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Elastic collision

Newtonian mechanics

There's nothing within the basic idea of Newtonian mechanics that would allow us to quite directly derive the notion of elastic collision. All we can do at the start is to simply use general arguments to see what would happen if they indeed happened.

First consider a set of free particles indexed by $i \in I$, each of them with their own positions and masses, $\{ \vec{x}_i, m_i \}_{i \in I}$. At the event of collision (for simplicity for now we will assume that at $t_c$, $\vec{x}_i(t_c) = \vec{x}_j(t_c)$), their momenta are $m_i \dot{\vec{x}}_i(t_c)$. By conversation of momentum, up until the next collision,

Elastic collisions for point charges

A more satisfying model of elastic collision involves the collision of point charges.

Consider two point charges