Obtain an expression for the velocity of the centre of mass for a system of $n$ particles.

Consider a system of $n$ particles. Let $\overrightarrow{r_{1}}, \overrightarrow{r_{2}}, \overrightarrow{r_{3}}, \ldots, \overrightarrow{r_{n}}$ be the position vectors of the particles of masses $m_{1}, m_{2}, m_{3}, \ldots, m_{n}$ respectively with respect to the origin of a coordinate system.
If $\overrightarrow{R}$ is the position vector of the centre of mass,then:
$\overrightarrow{R} = \frac{m_{1} \overrightarrow{r_{1}} + m_{2} \overrightarrow{r_{2}} + \ldots + m_{n} \overrightarrow{r_{n}}}{m_{1} + m_{2} + \ldots + m_{n}}$
Let $M = \sum_{i=1}^{n} m_{i}$ be the total mass of the system. Then:
$M \overrightarrow{R} = m_{1} \overrightarrow{r_{1}} + m_{2} \overrightarrow{r_{2}} + \ldots + m_{n} \overrightarrow{r_{n}} \quad \ldots (1)$
Assuming the mass of the system does not change with time,we differentiate equation $(1)$ with respect to time $t$:
$M \frac{d \overrightarrow{R}}{d t} = m_{1} \frac{d \overrightarrow{r_{1}}}{d t} + m_{2} \frac{d \overrightarrow{r_{2}}}{d t} + \ldots + m_{n} \frac{d \overrightarrow{r_{n}}}{d t}$
Since $\frac{d \overrightarrow{R}}{d t} = \overrightarrow{V}$ (velocity of the centre of mass) and $\frac{d \overrightarrow{r_{i}}}{d t} = \overrightarrow{v_{i}}$ (velocity of the $i$-th particle),we get:
$M \overrightarrow{V} = m_{1} \overrightarrow{v_{1}} + m_{2} \overrightarrow{v_{2}} + \ldots + m_{n} \overrightarrow{v_{n}}$
Therefore,the velocity of the centre of mass is:
$\overrightarrow{V} = \frac{m_{1} \overrightarrow{v_{1}} + m_{2} \overrightarrow{v_{2}} + \ldots + m_{n} \overrightarrow{v_{n}}}{M} = \frac{\sum_{i=1}^{n} m_{i} \overrightarrow{v_{i}}}{\sum_{i=1}^{n} m_{i}}$