Linear impulse:

If $\vec{F}$ be the total external force acting on a system of particles, then the total linear impulse will be $\displaystyle{\int_{t_1}^{t_2}}\vec{F}dt$. This linear impulse is equal to the change in linear momentum.

If $M$ be the total mass of he system and $\vec{V}$ be the velocity of the centre of mass of the system then we can write,

$\displaystyle{\int_{t_1}^{t_2}}\vec{F}dt=\displaystyle{\int_{t_1}^{t_2}}M\frac{d\vec{V}}{dt}dt\\=\displaystyle{\int_{t_1}^{t_2}}M\ d\vec{V}\\=M(\vec{V_2}-\vec{V_1})\\=M\vec{V_2}-M\vec{V_1}\\=\vec{p_2}-\vec{p_1}$

where, $\vec{V_1}$ and $\vec{V_2}$ are the velocities of he centre of mass at times $t_1$ and $t_2$ respectively.

Here, $\vec{p_1}$ and $\vec{p_2}$ are the total linear momentum at times $t_1$ and $t_2$ respectively.