Explain the conservation of linear momentum with a suitable example.

(N/A) The law of conservation of linear momentum can be derived using Newton's second and third laws of motion.
When a bullet is fired from a gun,the bullet moves in the forward direction and the gun moves in the backward direction (recoil).
Let the force applied by the gun on the bullet be $\overrightarrow{F}$. Then,the force applied by the bullet on the gun is $-\overrightarrow{F}$. These two forces act for an equal time interval $\Delta t$.
According to Newton's second law of motion,the change in momentum of the bullet is $\overrightarrow{F} \Delta t$ and the change in momentum of the rifle is $-\overrightarrow{F} \Delta t$.
Initially,both are at rest,so the change in momentum of both is equal to their final momentum. Let the final momentum of the gun be $\overrightarrow{p_{g}}$ and the bullet be $\overrightarrow{p_{b}}$. Then,$\overrightarrow{p_{g}} = -\overrightarrow{p_{b}}$,which implies $\overrightarrow{p_{g}} + \overrightarrow{p_{b}} = 0$.
Thus,the total linear momentum of the system (bullet and gun) is conserved.
Conservation of momentum: For an isolated system,the total linear momentum remains constant.
Example: Consider two objects $A$ and $B$. Let their initial momenta be $\overrightarrow{p_{A}}$ and $\overrightarrow{p_{B}}$. After collision,let their momenta be $\overrightarrow{p_{A}^{\prime}}$ and $\overrightarrow{p_{B}^{\prime}}$.
By the second law of motion,$\overrightarrow{F_{AB}} \Delta t = \overrightarrow{p_{A}^{\prime}} - \overrightarrow{p_{A}}$ and $\overrightarrow{F_{BA}} \Delta t = \overrightarrow{p_{B}^{\prime}} - \overrightarrow{p_{B}}$.
From Newton's third law of motion,$\overrightarrow{F_{AB}} = -\overrightarrow{F_{BA}}$,so $\overrightarrow{F_{AB}} \Delta t = -\overrightarrow{F_{BA}} \Delta t$.
Therefore,$\overrightarrow{p_{A}^{\prime}} - \overrightarrow{p_{A}} = -(\overrightarrow{p_{B}^{\prime}} - \overrightarrow{p_{B}})$,which simplifies to $\overrightarrow{p_{A}^{\prime}} + \overrightarrow{p_{B}^{\prime}} = \overrightarrow{p_{A}} + \overrightarrow{p_{B}}$.
Thus,for an isolated system,the total final momentum is equal to the total initial momentum. Momentum is conserved in both elastic and inelastic collisions. The law of conservation of linear momentum is universal and fundamental,holding true for systems of subatomic particles as well as macroscopic objects.