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When a moving billiard ball strikes an identical stationary ball head-on, the moving ball suddenly stops dead in its tracks while the target ball shoots forward with the exact same initial speed. What fundamental laws force this exact exchange to happen every single time?
In physics, a collision is elastic when zero mechanical energy is lost to heat, sound, or permanent deformation. For any closed system with no external net force, both total momentum and total kinetic energy remain perfectly constant before and after the impact.
How do we translate these two physical principles into a mathematical system we can solve for any combination of masses and speeds?

The Governing Conservation Laws

Linear momentum is defined as mass multiplied by velocity (p=mvp = mv). Conservation of linear momentum in one dimension states that the sum of initial momenta equals the sum of final momenta:
m1u1+m2u2=m1v1+m2v2m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2
Because the collision is elastic, total kinetic energy (K=12mv2K = \frac{1}{2}mv^2) must also be conserved across the system:
12m1u12+12m2u22=12m1v12+12m2v22\frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2
In 1668, Dutch physicist Christiaan Huygens presented these exact rules to the Royal Society of London, successfully resolving the laws of elastic collisions decades before Isaac Newton published the Principia.
Solving a linear equation alongside a quadratic equation can get messy fast — is there an algebraic trick that makes finding final velocities much simpler?

Deriving the Relative Velocity Relation

First, rearrange both conservation equations by grouping terms for mass m1m_1 on the left and mass m2m_2 on the right. For momentum, this gives m1(u1−v1)=m2(v2−u2)m_1(u_1 - v_1) = m_2(v_2 - u_2).