Chapter 8, Problem 51PE

Starting with equations $m_1{v}_1=m_{1}v{\text{'}}_1\cos {\theta }_1+m_{2}v{\text{'}}_2\cos {\theta }_2$ and $0=m_{1}v{\text{'}}_1\cos {\theta }_1+m_{2}v{\text{'}}_2\sin {\theta }_2$ for conservation of momentum in the x- and y -directions and assuming that one object is originally stationary, prove that for an elastic collision of two objects of equal masses,

$\frac{1}{2}m{v}_1{}^2=\frac{1}{2}mv{\text{'}}_2{}^2+mv{\text{'}}_{1}v{\text{'}}_2\cos \left({\theta }_1-{\theta }_2\right)$ as discussed in the text.