**Law of conservation of momentum states that total momentum of the system remains conserved in the absence of external force**.

**Proof**: Consider two bodies of mass m_{1} and m_{2} moving with initial velocity u_{1} and u_{2 }respectively. The two bodies collide with each other for a time interval ‘t’. The velocity after collision be v_{1} & v_{2 }respectively. Let F_{12} be the force applied by m_{1 }on m_{2} and F_{21} be the force applied by m_{2} on m_{1.}

Momentum of mass m_{1} before collision = m_{1 }u_{1}

Momentum of mass m_{2} before collision = m_{2} u_{2}

Momentum of mass m_{1 }after collision = m_{1} v_{1}

Momentum of mass m_{2} after collision = m_{2} v_{2}

Impulse = force × time = change in momentum

For mass m_{1} :

F_{12} t = m_{1} v_{1} - m_{1} u_{1} …………….(1)

For mass m_{2}:

F_{21}t = m_{2} v_{2} – m_{2} u_{2} ……………..(2)

Adding equation (1) & (2)

F_{12} t+ F_{21}t = (m_{1} v_{1} - m_{1} u_{1}) + (m_{2} v_{2} – m_{2} u_{2})

(F_{12 }+ F_{21} ) t = (m_{1} v_{1} + m_{2} v_{2}) - (m_{1} u_{1} + m_{2 }u_{2})

According to Newton’s third law:

F_{12} = - F_{21}

F_{12} + F_{21} = 0

(m_{1} v_{1} + m_{2} v_{2}) = (m_{1} u_{1 }+ m_{2} u_{2})

Final momentum = Initial momentum

Hence momentum is conserved.