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u signifies vector velocity before the collision

v signifies vector velocity after the collision.

Usually, we either only know the velocities before or after a collision and would like to also find out the opposite. Correctly solving this problem méans you have to know what kind of collision took place. There are two basic kinds of collisions, both of which conserve momentum:

Elastic collisions conserve kinetic energy as well as total momentum before and after collision.

Inelastic collisions don't conserve kinetic energy, but total momentum before and after collision is conserved.