Elastic Collisions

In an isolated system, an elastic collision conserves both linear momentum and kinetic energy. This implies that none of the kinetic energy is dissipated into internal energy, heat, or sound during the collision.

Consider two objects with masses $m_1$ and $m_2$ moving in one dimension. Let $v_{1i}$ and $v_{2i}$ be the initial velocities, and $v_{1f}$ and $v_{2f}$ be the final velocities. The conservation of linear momentum states that the total momentum of the system remains constant:

$$P_\mathrm{total, i} = m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}=P_\mathrm{total, f}$$

When the collision is elastic, the total kinetic energy is also conserved. The sum of kinetic energies before the collision equals the sum after:

$$K_i=\frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2 = \frac{1}{2}m_1 v_{1f}^2 + \frac{1}{2}m_2 v_{2f}^2=K_f$$

To find the unknown final velocities, treat these two conservation laws as a system of simultaneous equations. The linear momentum equation constrains the vector motion, while the quadratic energy equation constrains the speeds.