Answer
$v_{1f}=\frac{m_1-m_2}{m_1+m_2}v_{1i}+\frac{2m_2}{m_1+m_2}v_{2i}$
Work Step by Step
We know the following equations from conservation of momentum and conservation of kinetic energy:
$\frac{1}{2}mv_{1i}^2+\frac{1}{2}mv_{2i}^2=\frac{1}{2}mv_{1f}^2+\frac{1}{2}mv_{2f}^2$
$m_{1}v_{1i}+m_2v_{2i}=m_{1}v_{1f}+m_2v_{2f}$
Using substitution and combining these equations gives:
$v_{1f}=\frac{m_1-m_2}{m_1+m_2}v_{1i}+\frac{2m_2}{m_1+m_2}v_{2i}$
