Introduction
The principle of conservation of linear momentum is a fundamental law in physics. It states that for a closed system (where no external forces act), the total momentum of the system remains constant over time. This principle is a direct consequence of the symmetries of space and is deeply connected to Newton's laws.
The Principle Defined
The Principle: In an isolated system (where the vector sum of external forces is zero), the total linear momentum of the system remains conserved. That is, the initial momentum equals the final momentum.
Derivation from Newton's Second Law
To derive this, consider two bodies $A$ and $B$ with masses $m_1$ and $m_2$ moving with velocities $u_1$ and $u_2$. They collide, and after a short time $\Delta t$, their velocities become $v_1$ and $v_2$.
According to Newton's Second Law, the force exerted by body $A$ on $B$ ($F_{BA}$) is the rate of change of momentum of $B$: $$F_{BA} = \frac{m_2v_2 - m_2u_2}{\Delta t}$$
Similarly, the force exerted by body $B$ on $A$ ($F_{AB}$) is: $$F_{AB} = \frac{m_1v_1 - m_1u_1}{\Delta t}$$
By Newton's Third Law, every action has an equal and opposite reaction: $$F_{AB} = -F_{BA}$$
Substituting the expressions: $$\frac{m_1v_1 - m_1u_1}{\Delta t} = -\frac{m_2v_2 - m_2u_2}{\Delta t}$$
Canceling the $\Delta t$ and rearranging the terms: $$m_1v_1 + m_2v_2 = m_1u_1 + m_2u_2$$
Thus, the total final momentum equals the total initial momentum.
Intuition
Think of a collision as an exchange of 'motion'. When two pool balls hit, they exert forces on each other. Because these forces are internal to the 'two-ball system', they don't change the system's overall 'amount of motion' (momentum), only how it is distributed between the two objects.