Answer
$\frac{m_1}{m_2}=\frac{1-\sqrt{\frac{1}{2}}}{\sqrt{\frac{1}{2}}+1}$
Work Step by Step
We first use the fact that half of the kinetic energy is lost to find:
$\frac{1}{2}m_1v^2 + \frac{1}{2}m_2v^2=(m_1+m_2)v_f^2$
$v_f = \sqrt{\frac{\frac{1}{2}(m_1+m_2)v^2}{m_1+m_2}}$
We use conservation of momentum to find:
$m_1v-m_2v=(m_1+m_2)v_f$
$m_1v-m_2v=(m_1+m_2)\sqrt{\frac{\frac{1}{2}(m_1+m_2)v^2}{m_1+m_2}}$
$(m_1-m_2)v=(m_1+m_2)\sqrt{\frac{1}{2}}v$
$(m_1-m_2)=\sqrt{\frac{1}{2}}m_1+m_2\sqrt{\frac{1}{2}}$
$\frac{m_1}{m_2}=\frac{1-\sqrt{\frac{1}{2}}}{\sqrt{\frac{1}{2}}+1}$
