$A$ body of mass $m_1$ moving with an unknown velocity $v_1 \hat{i}$ undergoes a one-dimensional collision with a body of mass $m_2$ moving with velocity $v_2 \hat{i}$. After the collision,the bodies $m_1$ and $m_2$ move with velocities $v_3 \hat{i}$ and $v_4 \hat{i}$ respectively. If $m_2 = 0.5\, m_1$ and $v_3 = 0.5\, v_1$,find $v_1$.

• A
  $v_4 - \frac{v_2}{2}$
• B
  $v_4 - \frac{v_2}{4}$
• C
  $v_4 - v_2$
• D
  $v_4 + v_2$