Linear momentum of a system of particles:

The position vector \( \vec{R} \) of the centre of mass of a system of n particles relative to the origin O is given by,

\( \displaystyle{\vec{R}=\frac{1}{M}\sum_{i=1}^{n}m_i\vec{r_i}}\tag{1} \)

where, \( \vec{r_i} \) is the position vector of the its particle of mass \( m_i \) with respect to the origin O,

\( \displaystyle{M=\sum_{i=1}^{n}m_i} \) is the total mass of the system of n particles.

If, \( \vec{v_i}=\frac{d\vec{r_i}}{dt} \) be the velocity of the its particle, then the total linear momentum of the system with respect to the origin is given by,

\( \displaystyle{\vec{p}=\sum_{i=1}^{n}m_i\vec{v_i}} \)

or, \( \displaystyle{\vec{p}=\sum_{i=1}^{n}m_i\frac{d\vec{r_i}}{dt}}\tag{2} \)

Now, differentiating equation (1) with respect to time we get,

\( \displaystyle{\frac{d\vec{R}}{dt}=\frac{1}{M}\sum_{i=1}^{n}m_i\frac{d\vec{r_i}}{dt}} \)

or, \( \displaystyle{\frac{d\vec{R}}{dt}=\frac{1}{M}\vec{p}} \)
[ using equation (2) ]

or, \( \displaystyle{\vec{p}=M\frac{d\vec{R}}{dt}} \)

or, \( \displaystyle{\vec{p}=M\vec{V}} \)

where, \( \vec{V}=\frac{d\vec{R}}{dt} \) is the velocity of the centre of mass.

So the total linear momentum of a system of particles is equal to the linear momentum of the centre of mass of that system.