Answer
$2m$
Work Step by Step
The required mass can be determined as follows:
Mass of the cart toward left $m_L=m$
Initial velocity of the left cart $v_L=v_{\circ}$
Initial velocity of the right cart $v_r=\frac{v_{\circ}}{2}$
Velocity of the center of mass $v_{cm}=\frac{2v_{\circ}}{3}$
Velocity of the center of mass $v_{cm}=\frac{m_1v_1+m_2v_2}{m_1+m_2}$
$\implies v_{cm}=\frac{mv_{\circ}+m_2\frac{v_{\circ}}{2}}{m+m_r}$
$\implies \frac{2v_{\circ}}{3}=\frac{v_{\circ}(m+\frac{m_r}{2})}{m+m_r}$
$\implies \frac{2}{3}(m+m_r)=m+\frac{1}{2}m_r$
This simplifies to:
$m_r=2m$
