 # Law of Conservation of Momentum

##### Law of Conservation of Momentum

Statement

The momentum of an isolated system of two or more than two interacting bodies remains constant.

###### Description

Momentum of a system depends upon its mass and velocity. A system is a group of bodies within certain borders. An isolated system is a group of bodies interacting witch each other on which no external force is acting. If no unbalanced or net force acts upon a system, then its momentum remains consistent. Thus, the momentum of an isolated system is constantly conserved. ##### Mathematical Expression

Let consider an isolated system of two spheres of masses m1 and m2. They are moving in a straight line with initial velocities u1 and u2 respectively, such that u1 is greater than u2. The sphere of mass m1 approaches the sphere of mass m2 as they move.

• Initial momentum of mass m1 = m1u1
• Initial momentum of mass m2 = m2u2
• The overall initial momentum of the system before collision = m1u1 +m2u2.

After at some point mass m1 strikes m2 with some force. According to Newton’s third law of motion, m2 applies an equivalent and opposite reacting force on m1. Let their speeds end up being v1 and v2 respectively after a collision. Then

• Final momentum of mass m1 = m1v1.
• Final momentum of mass m2 = m2v2.
• Overall final momentum of the system after collision = m1v1 +m2v2.
##### According to the law of conservation of momentum

Total initial momentum of the system before collision = Total final momentum of the system after the collision.

m1u1 + m2u2= +m1v1 + m2v2

##### Example of Law of Conservation of Momentum

Consider a system of gun and a bullet. Before firing the gun, both the gun and bullet are at rest, so the total momentum of the system is absolutely zero. As the gun is fired, bullet shoots out of the gun and obtains momentum. To conserve the momentum of the system, the gun recoils.

According to the law of conservation of momentum, the total momentum of the gun and the bullet will likewise be absolutely zero after the gun is fired.

Let m be the mass of the bullet and v be its speed on firing the gun; M be the mass of the gun and V be the velocity with which it recoils. Therefore, the overall momentum of the gun and the bullet after the gun is fired will be; A negative mark indicates that the speed of the gun is opposite to the speed of the bullet, i.e., the gun recoils. Since the mass of the gun is much larger than the bullet, for that reason, the recoil is much smaller than the speed of the bullet.

Rockets and jet engines also work on the exact same principle. In these makers, hot gases produced by the burning of fuel rush out with large momentum. The machines acquire an equivalent and opposite momentum. This enables them to move with really high speeds.