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.