Newton's third law of motion states that, " every action force has an equal and opposite reaction force which acts simultaneously."
According to law of conservation of momentum, " when no external force acts on two interacting objects, their total momentum remains constant. it does not change." In another way it also stated as, " when two objects collide, the total momentum before collision is equal to the total momentum after collision." i.e., the magnitude of total final momentum = the magnitude of total initial momentum.
Suppose object A has mass m1 & it's initial velocity u1. object B has mass m2 & initial velocity u2.
according to formula for momentum,
initial momentum of A p1=m1u1 & that of B p2=m2u2.
Suppose And B collide. Let force on A due to B be F1. this force will cause acceleration in A and it's velocity v1.
momentum of A after collision = m1 v1.
According to Newton's third law of motion, A also exerts an equal force on B but in the opposite direction. This will cause a change in the momentum of B. If it's velocity after collision is v2.
momentum of B after collision = m2 v2.
if F2 is the force that acts on object B,
F2 = - F1
since F = ma
m2a2 = - m1a1
therefore after solving,
m2(v2-u2) = -m1(v1- u1)
i.e. m2 v2 + m1v1 = m1u1 + m2u2
Thus, if no external force is acting on two objects, then their total initial momentum is equal to their total final momentum.
Law of conservation of momentum is corollary to Newton's law of motion.