Physics - Kinematics Question with Solution | TestHub

The acceleration of each block is $a=\frac{(m_1-m_2)g}{m_1+m_2}=\frac{(3-1)\times 10}{3+1}=\frac{2\times 10}{4}=5{\text{m/s}}^2$. Block $m_1$ moves downwards (say, $-y$ direction) and $m_2$ moves upwards (say, $+y$ direction). So, ${\vec{a}}_1=-a\hat{j}$ and ${\vec{a}}_2=a\hat{j}$. The acceleration of the center of mass is ${\vec{a}}_{CM}=\frac{m_1{\vec{a}}_1+m_2{\vec{a}}_2}{m_1+m_2}=\frac{m_1(-a\hat{j})+m_2(a\hat{j})}{m_1+m_2}=\frac{(m_2-m_1)a}{m_1+m_2}\hat{j}$. Substituting values, ${\vec{a}}_{CM}=\frac{(1-3)\times 5}{1+3}\hat{j}=\frac{-2\times 5}{4}\hat{j}=-2.5\hat{j}{\text{m/s}}^2$. The magnitude of the acceleration of the center of mass is $2.5{\text{m/s}}^2$.