Conservation of Linear Momentum

Consider a system $S$ composed of two interacting objects, $A$ and $B$. When these objects collide, they exert forces on one another. We classify these as internal forces because they act between components within our defined system boundary.

According to Newton's Third Law, the force exerted by object $A$ on object $B$, denoted $\vec{F}_{A \to B}$, is equal in magnitude and strictly opposite in direction to the force exerted by object $B$ on object $A$, denoted $\vec{F}_{B \to A}$.

$$\vec{F}_{A \to B} = -\vec{F}_{B \to A}$$

The collision or interaction occurs over a specific time interval $\Delta t$ common to both objects, therefore, the internal impulse exerted on $B$ is $\vec{J}_{A \to B} = \vec{F}_{A \to B} \cdot \Delta t$, and the internal impulse exerted on $A$ is $\vec{J}_{B \to A} = \vec{F}_{B \to A} \cdot \Delta t = -\vec{F}_{A \to B} \cdot \Delta t$. To find the net effect of these internal interactions on the system's total momentum, we calculate the vector sum of these impulses.

$$\sum \vec{J}_\text{int} = \vec{J}_{A \to B} + \vec{J}_{B \to A} = (\vec{F}_{A \to B}) \cdot \Delta t + (-\vec{F}_{A \to B}) \cdot \Delta t = \vec{0}$$

Newton's third law ensures the internal impulses always sum to zero, they cannot result in a net change to the total linear momentum of the system. This cancellation implies that the total momentum of an isolated system is conserved, it does not change.

This doesn't mean the individual objects within an isolated system can't change momentum, it just means that in any interaction between objects within the system, the changes to momentum between the objects must be equal and opposite:

Big Picture:

Momentum is conserved in every interaction between two objects (take the system as the two interacting objects), therefore momentum of a system of objects is conserved if all interactions are between objects within the system, or in other words, the system is isolated.