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Special cases of elastic collisions

An elastic collision between two objects with equal mass moving in opposite directions (a frontal collision) has some special properties. The two objects simply exchange velocity: $$$ \begin{align*} & \vecphy{v}_1=\vecphy{u}_2\\ & \vecphy{v}_2=\vecphy{u}_1 \end{align*} $$$

When the mass of the first body is much bigger than the mass of the second, the bigger mass does not change its velocity noticeably. Only the smaller mass noticeably changes its direction and speed. The following is an approximation: $$$\vecphy{v}_1=\vecphy{u}_1 \quad \quad \vecphy{v}_2=2\vecphy{u}_1+\vecphy{u}_2$$$

Two pool balls have the same mass. When they collide head-on, they simply exchange their velocities.
Two pool balls have the same mass. When they collide head-on, they simply exchange their velocities.