A billiard ball moving at a speed u1 i strikes, head-on,...

A billiard ball moving at a speed $u_{1 i}$ strikes, head-on, another billiard ball that is at rest. Assuming the collision is completely elastic, show that
$$
v_{1 i}^{2}=v_{\mathrm{lf}}^{2}+v_{2 f}^{2}
$$

Key Concepts

Head-on Collision Geometry

A head-on collision implies that the movement and impact occur along a single straight line. This one-dimensional nature of the collision simplifies the analysis because vector calculations reduce to scalar equations, directly correlating the speeds along the same line without needing to decompose into multiple dimensions.

Elastic Collision

An elastic collision is a type of collision in which there is no net loss in kinetic energy in the system. Both momentum and kinetic energy conservation are valid in such collisions, making it possible to derive specific analytic relationships between the velocities of the objects involved.

Conservation of Momentum

Momentum conservation states that the total momentum of a closed system remains constant before and after a collision. In the context of a head-on collision, this principle allows us to set up an equation relating the product of mass and velocity of the moving ball before impact to the combined momenta of both balls after the collision.

Conservation of Kinetic Energy

In a completely elastic collision, kinetic energy is conserved. This means that the total kinetic energy of the system (expressed as one half mass times velocity squared for each object) remains the same before and after the collision. This principle is essential for deriving relationships that involve the squares of the velocities, as seen in the equation provided.

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