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LESSON NOTES
CONSERVATION OF LINEAR MOMENTUM

The second and third laws of motion lead to one of the most important and fundamental
principles of physics, called the law of conservation of linear momentum. It can be
stated as follows:

When no external force acts on a system of several interacting particles, the total linear
momentum of the system is conserved. The total linear momentum is the vector sum of
the linear momenta of all the particles of the system.
Derivation of the law of conservation of linear momentum

As shown in fig, consider two bodies A and B of masses m 1 and m 2 moving in the same
direction along a straight line with velocities u 1 and u 2 respectively (u 1> u 2) They collide
for time At. After collision, let their velocities be v 1 and v 2.

nd
From 2 law of motion,

Force applied by A on B = Rate of change of momentum of B
F AB = (m 2v 2-m 2u 2)/∆t
Similarly,
Force applied by B on A = Rate of change of momentum of A
F BA = (m 1v 1-m 1u 1)/∆t
rd
From Newton’s 3 law of motion,
F AB = -F BA
Or, (m 2v 2-m 2u 2)/∆t = -(m 1v 1-m 1u 1)/∆t
Or, m 2v 2-m 2u 2 = -m1v1+m1u1

Or, m 1u 1 + m 2u 2 = m 1v 1 + m 2v 2

i.e., Total linear momentum before collision = Total linear momentum after collision
This proves the law of conservation of linear momentum.

Practical applications based on the law of conservation of linear momentum:
(i) Recoil of a gun. Let 'M' be the mass of the gun and 'm’ be the mass of the bullet.
Before firing, both the gun and the bullet are at rest. After firing, the bullet moves with