Chapter 9, Problem 83GP

Consider a one-dimensional. head-on elastic collision. One object has a mass m₁ and an initial velocity u₁; the other has a mass m₂ and an initial velocity u₂. Use momentum conservation and energy conservation to show that the final velocities of the two masses are

$\begin{array}{l}{v}_{1,\text{f}}=\left(\frac{m_1-m_2}{m_1+m_2}\right)v_1+\left(\frac{2m_2}{m_1+m_2}\right)v_2\\ v_{2,\text{f}}=\left(\frac{2m_1}{m_1+m_2}\right)v_1+\left(\frac{m_2-m_1}{m_1+m_2}\right)v_2\end{array}$