We are all familiar with head-on elastic collisions. They conserve energy and momentum according to the formulas:
Many texts expect the student to solve these two formulas simultaneously to find the final velocity of the two objects after the collision. Other books give the students the formulas to find the velocity of each car after the collision. These formulas are derived from the laws of conservation of energy and momentum with the additional but very limiting parameter that v2 = 0
(i.e., you must hit a stationary object). The formulas are:
The problem is that not every object that is struck is stationary. Stationary objects are just easier targets.