Physics - Center of Mass Question with Solution | TestHub

Since there are no external forces, the center of mass of the system remains at rest. Let the initial position of $m_1$ be $x_1$ and $m_2$ be $x_2$. After interaction, their positions are $x_1^{\prime }$ and $x_2^{\prime }$. The displacement of $m_1$ is $\Delta x_1=x_1^{\prime }-x_1=d_1$. Since the COM does not move, $m_1\Delta x_1+m_2\Delta x_2=0$, so $m_2\Delta x_2=-m_1\Delta x_1=-m_1{d}_1$. Thus, $\Delta x_2=-\frac{m_1}{m_2}{d}_1$. The distance moved by $m_2$ relative to the center of mass is its absolute displacement, $\Delta x_2$, plus the displacement of the COM (which is zero). However, the question asks for distance moved by $m_2$ relative to the center of mass, which is $\Delta x_2-\Delta x_{CM}=\Delta x_2$. The relative displacement of $m_2$ with respect to $m_1$ is $\Delta x_2-\Delta x_1=-\frac{m_1}{m_2}{d}_1-d_1=-d_1(1+\frac{m_1}{m_2})$.