Law of conservation of momentum is derived from Newton's laws of motion.
Suppose a body of mass #m_1# traveling with a velocity #u_1# collides with another body of mass #m_2# traveling with a velocity #u_2#. Let velocities of these bodies after the collision be #v_1and v_2# respectively.
We know that change of velocity implies change of momentum.
Therefore change of momentum #Deltap_1# for first body
#Deltap_1=m_1(v_1-u_1)#
Since it is an isolated system, no external force is applied on ny of the two bodies during the collision. As such only force applied on body 1 during collision is due to body 2. Let it be #F_12#. Suppose this force acts for time #Deltat#. Also by Newton's second law, this force is equal to the product of acceleration and mass. We know that acceleration is rate of change of velocity. Thus
#F_12=m_1((v_1-u_1))/(Deltat)# .......(1)
Similarly, force #F_21# applied upon body 2 by body 1 is
#F_21=m_2((v_2-u_2))/(Deltat)# ..........(2)
Newton's third law states that for every action there is equal and opposite reaction . Therefore, force #F_12# equal and opposite to #F_21#
#F_12=-F_21# ......(3)
Inserting value of forces from (1) and (2) in equation (3) we get
#m_1((v_1-u_1))/(Deltat)=-m_2((v_2-u_2))/(Deltat)#
#=>m_1(v_1-u_1)=-m_2(v_2-u_2)#
#=>m_1v_1-m_1u_1=-m_2v_2+m_2u_2#
Rearranging we get
#m_1u_1+m_2u_2=m_1v_1+m_2v_2#
Initial momentum#=# Final momentum
