Ans.

Let us consider \( \vec{r_i} \) be the position vector of ith particle of a system of n particles with respect to the origin and \( \vec{v_i} \) be the velocity of that ith particle of mass \( m \).

So the momentum of the system is given by,

\( \vec{p}=\displaystyle{\sum_{i=1}^{n}}m_i\vec{v_i}\\=\displaystyle{\sum_{i=1}^{n}}m_i\dot{\vec{r_i}}\\=\frac{d}{dt}\displaystyle{\sum_{i=1}^{n}}m_i\vec{r_i}\\=\displaystyle{M\frac{d}{dt}\left[\frac{\sum_{i=1}^{n}m_i\vec{r_i}}{M}\right]} \\=M\frac{d\vec{R}}{dt}\)

where, \( \vec{R}=\frac{\displaystyle{\sum_{i=1}^{n}}m_i\vec{r_i}}{M} \) is the position vector of the centre of mass of the system with respect to the origin.

If momentum \( \vec{p} \) is constant then \( \frac{d\vec{R}}{dt} \) is also constant. This means that the velocity of the centre of mass of the system is also constant. So if the total momentum of a system is constant then the centre of mass of the system is either at rest or in motion with constant velocity.