Law of conservation of linear momentum can be obtained by using Newton's second law and third law of motion.

When bullet is fired from gun, bullet moves in forward direction and gun moves in backward direction (recoil).

Force applied by gun on bullet be $\overrightarrow{\mathrm{F}}$, then force applied bullet on the gun be $-\overrightarrow{\mathrm{F}}$. These two force act for equal time interval $\Delta t$.

$\therefore$ By second law of motion change in momentum of bullet $=\overrightarrow{\mathrm{F}} \Delta t$ and change in momentum of rifle $=-\overrightarrow{\mathrm{F}} \Delta t$

Initially, both are at rest hence change in momentum of both equal to final momentum. Let final momentum of gun be $\overrightarrow{p_{g}}$ and bullet be $\overrightarrow{p_{b}}$, then

$\overrightarrow{p_{g}}=-\overrightarrow{p_{b}}$

$\therefore \overrightarrow{p_{g}}+\overrightarrow{p_{b}}=0$

Thus, total linear momentum of system (bullet and gun) is conserved.

Conservation of momentum : For isolated system total linear momentum of system remains constant.

Example : Consider two objects $\mathrm{A}$ and $\mathrm{B}$. Let their initial momentum be $\overrightarrow{p_{\mathrm{A}}}$ and $\overrightarrow{p_{\mathrm{B}}}$. Let after collision their momentum be $\overrightarrow{p_{\mathrm{A}}^{\prime}}$ and $\overrightarrow{p_{\mathrm{B}}^{\prime}}$

By second law of motion,

$\overrightarrow{\mathrm{F}_{\mathrm{AB}}} \Delta t=\overrightarrow{p_{\mathrm{A}}^{\prime}}-\vec{p}_{\mathrm{A}} \text { and } \overrightarrow{\mathrm{F}}_{\mathrm{BA}} \Delta t=\vec{p}_{\mathrm{B}}^{\prime}-\vec{p}_{\mathrm{B}}^{\prime}$

From Newton's third law of motion,

$\overrightarrow{\mathrm{F}}_{\mathrm{AB}}=-\overrightarrow{\mathrm{F}}_{\mathrm{BA}}, \overrightarrow{\mathrm{F}}_{\mathrm{AB}} \Delta t=-\overrightarrow{\mathrm{F}}_{\mathrm{BA}} \Delta t$

$\vec{p}_{\mathrm{A}}^{\prime}-\vec{p}_{\mathrm{A}}=-\left(\vec{p}_{\mathrm{B}}^{\prime}-\vec{p}_{\mathrm{B}}\right)$

$\therefore \vec{p}_{\mathrm{A}}^{\prime}+\vec{p}_{\mathrm{B}}^{\prime}=\vec{p}_{\mathrm{A}}+\vec{p}_{\mathrm{B}}$

Thus, for isolated system total final momentum is equal to total initial momentum. In both elastic and non-elastic collision momentum is conserved. But in elastic collision both momentum and kinetic energy is conserved.

Law of conservation of linear momentum is universal and fundamental. It equally hold true of system of smaller particles like electron, proton, neutron as well as larger system made up of electron, proton, neutron.