Law of Conservation of Momentum

Let two particles A and B collide with each other. Let be the initial and final velocities of particle 1 and similarly, for particle 2.

Let the two particles be in contact for a time t.

Change in momentum of A and B are

During the collision, let A impart an average force equal to on B and let B exert an average on A. We know that

From (1) we get,

which is the law of conservation of momentum.

Generalising to n particles,

The law of conservation of momentum states that the total vector sum of momenta of bodies, in an isolated system, along any straight line remains conserved and remains unchanged due to reaction forces between the forces of the system.

Sub Topics

Applications / illustrations of conservation of momentum
Recoil of the gun
Explosion of a bomb
Conservation of momentum and centre of mass
Equilibrium of concurrent forces

Applications / illustrations of conservation of momentum

Recoil of the gun

When a bullet is fired from a gun, the gun moves backwards when the bullet leaves the barrel. This can be proved using the law of conservation of momentum.

Let m_g and m_b be the masses of the gun and the bullet respectively.

Firstly, we have to identify that the gun with the bullet constitute one isolated system.

By the conservation of momentum

The negative sign shows that the gun moves in a direction opposite to that of the bullet.

It also shows that heavier the gun, lesser will be the recoil.

Explosion of a bomb

When a bomb explodes into many fragments it splits up in such a way that the total final momentum is equal to the momentum of the bomb.

When a person jumps out of the boat, the boat moves in the opposite direction.

Conservation of momentum and centre of mass

as the vector of the linear momenta of all the bodies in the system.

We can conclude that when there is no external force, the velocity of the centre of mass of the system remains the same.

Equilibrium of concurrent forces

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A body on which two or more forces act is said to be in equilibrium if there is no change in its state of rest or of uniform motion.

The condition necessary for equilibrium is that the vector sum of forces acting on the body should be zero. i.e.,

This also implies that