# Prove the law of conservation of linear momentum in case of collision of two balls

Prove the law of conservation of linear momentum in case of collision of two balls.

Law of conservation of linear momentum states that total momentum of the system is always conserved if no external force acts on an object or system of objects.

Consider a collision between two balls wherein there occurs no energy losses during the collision.

Momentum of the two balls before collision,

p1i = m1u1
p2i = m2u2
Total momentum of the system of two balls before collision, pi = p1i + p2i = m1u1 + m2u2.
During the collision m1 exerts an action force F12 on m2.
In response, from Newton"s third law, m2 exerts a reaction on m1, that is, F21, such that F12 = -F21
Negative sign implies that the two forces are directed in opposite directions.

After the collision they undergo change in velocity and the corresponding change in momentum. Momenta of the two balls after collision p1f = m1v1
p2f = m2v2
Total momentum of the system of two balls after collision, pf = p1f +p2f = m1v1 + m2v2
Also from Newton"s Second Law: Total momentum before collision = Total momentum after collision
Hence this equation implies that if no external force acts on the system of two colliding balls, the total momentum is conserved.