Physics - Center of Mass Question with Solution | TestHub

Let the initial velocity of mass $m$ be $\vec{v}=v_0\hat{i}$ and its final velocity be ${\vec{v}}_1=\frac{v_0}{3}\hat{i}+\frac{2v_0}{3}\hat{j}$.

Let the initial velocity of mass $2m$ be $0$ and its final velocity be ${\vec{v}}_2$.

By conservation of momentum:

$m\vec{v}+2m(0)=m{\vec{v}}_1+2m{\vec{v}}_2$

$m(v_0\hat{i})=m\left(\frac{v_0}{3}\hat{i}+\frac{2v_0}{3}\hat{j}\right)+2m{\vec{v}}_2$

$v_0\hat{i}=\frac{v_0}{3}\hat{i}+\frac{2v_0}{3}\hat{j}+2{\vec{v}}_2$

$2{\vec{v}}_2=v_0\hat{i}-\frac{v_0}{3}\hat{i}-\frac{2v_0}{3}\hat{j}=\frac{2v_0}{3}\hat{i}-\frac{2v_0}{3}\hat{j}$

${\vec{v}}_2=\frac{v_0}{3}\hat{i}-\frac{v_0}{3}\hat{j}$

Hence option (A).