$${\begin{align*}v&= \sqrt{\frac{2}{m_a}\left(\frac{m_a }{2}\left(\frac{m_v}{m_a} \right)^2 v_v^2 + \frac{m_v v_v^2}{2}\right)} \\&= v_v\sqrt{\frac{m_v }{m_a} + \left(\frac{m_v }{m_a}\right)^2}\end{align*}}$$