Physics - Center of Mass Question with Solution | TestHub

For a perfectly inelastic collision, momentum is conserved. Let the final velocity of the combined mass $(m+m=2m)$ be $\vec{V}$. Initial momentum: ${\vec{P}}_i=m(v_0\hat{i})+m(v_0\hat{i}+v_0\hat{j})=m{v}_0\hat{i}+m{v}_0\hat{i}+m{v}_0\hat{j}=2m{v}_0\hat{i}+m{v}_0\hat{j}$. Final momentum: ${\vec{P}}_f=(2m)\vec{V}$. By conservation of momentum, ${\vec{P}}_i={\vec{P}}_f$: $2m{v}_0\hat{i}+m{v}_0\hat{j}=2m\vec{V}$.$$\vec{V}=\frac{2m{v}_0\hat{i}+m{v}_0\hat{j}}{2m}=v_0\hat{i}+\frac{v_0}{2}\hat{j}$$ The magnitude of the final velocity is $|\vec{V}|=\sqrt{(v_0{)}^2+(v_0/2{)}^2}=\sqrt{v_0^2+v_0^2/4}=\sqrt{\frac{5v_0^2}{4}}=\frac{\sqrt{5}{v}_0}{2}$.