Law of conservation of momentum: Let us consider an isolated system. It is a system on which no external agency exerts any force. For example, the molecules of a gas enclosed in a glass vessel at constant temperature constitute an isolated system. The molecules can collide with one another because of their random motion but, being enclosed by glass vessel, no external agency can exert a force on them.

Consider an isolated system of two smooth hard interacting balls of masses m1 and m2, moving along the same straight line, in the same direction, with velocities v1 and v2 respectively. Both the balls collide and after collision, ball of mass m1 moves with velocity v’1 and m2 moves with velocity v’2 in the same direction as shown in fig 3.8.

Fig 3.8

To find the change in momentum of mass m1, using Eq 3.11we have,

Since the action force F is equal and opposite to the reaction force F’, we have F’  = -F, so the left hand side of the equation is zero. Hence,

In other words, change in momentum of 1st ball + change in momentum of the 2nd ball = 0

Which means that total initial momentum of the system before collision is equal to the total final momentum of the system after collision. Consequently, the total change in momentum of the isolated two ball system is zero.

For such a group of objects, if one object within the group experiences a force, there must exist an equal but opposite reaction force on some other object in the same group. As a result, the change in momentum of the group of objects as a whole is always zero. This can be expressed in the form of law of conservation of momentum which states that

The total linear momentum of an isolated system remains constant.

Point to ponder

“What is the effect on the speed of a fighter plane chasing another when it opens fire? What happens to the speed of pursued plane when it returns the fire?”

In applying the conservation law, we must notice that the momentum of a body is a vector quantity.

Example 3.3: two spherical balls of 2.0 kg and 3.0 kg masses are moving towards each other with velocities of 6.0ms-1 respectively. What must be the velocity of the smaller ball after collision, if the velocity of the bigger ball is 3.0 ms-1?

Solution: As both the balls are moving towards one another, so their velocities are of opposite sign. Let us suppose that the direction of motion of 2 kg ball is positive and that of the 3 kg is negative.

Do you wear seat belts?

When a moving car stops quickly, the passengers move forward towards the windshield. Seat belts change the forces of motion and prevent the passengers from moving. Thus the chance of injury is greatly reduced.

Do you know?

A motorcycle’s safety helmet is padded so as to extend the time of any collision to prevent serious injury.