---

What is the Law of conservation of momentum?

For the collision occurring between two objects, the sum of momenta of the two objects before the collision is equal to the sum of momenta after the collision, provided there is no external unbalanced force acting on them.

This law is the direct outcome of Newton’s third law of motion which states that every action has an equal and opposite reaction.

Equation for conservation of momentum

Consider two objects with masses, ‘m1’ and ‘m2’ moving towards each other at velocity, ‘u1’ and ‘u2’, respectively. When they collide, both objects apply equal and opposite forces on each other. Let the force applied by the first object be F1 and that applied by the second object be F2.

Collision of two objects

When a force is applied on an object, the object accelerates.

Let us assume that the forces cause acceleration of ‘a1’ and ‘a2’ on the objects, respectively and

there is no external unbalanced force acting here.

The time duration for which the objects apply a force on each other, ‘t1’ and ‘t2’ will be equal.

This is because the objects are in contact for the same period of time.

As per Newton’s third law,

F1 = – F2

Given that force is the product of mass and acceleration, the equation can also be represented as follows:

m1a1 = – m2a2

We know that acceleration is the rate of change of velocity. Therefore the above equation can be written as:

m1(𝑣1−𝑢1)/t1= – m(𝑣2−𝑢2)/𝑡2

As t1 and t2 are equal, the above equation can be written as:

m1(v1 – u1) = – m2(v2 – u2)

That is,

m1v1– m1u1 = – m2v2+ m2u2

Rearranging the terms gives us the equation for the law of conservation of momentum which is:

m1v1 + m2v2 = m1u1 + m2u2

Momentum is the product of the mass of an object and its velocity. This means the above equation is nothing but,

Total momentum of the objects after the collision

=

Total momentum of the objects before the collision