Conservation of Momentum

Introduction

Conservation momentum implies that the physical quantity momentum of a body which comes into play when body has got mass as well as velocity vector over a course of time remains conserved provided no external force acting on an isolated system. Moreover,mutual forces acting between the pairs of particles can cause change in momentum but the force acting between them are equal and opposite , therefore change in momentum cancels each other and has no impact on overall total momentum, which remains the same throughout the process.

To have insight into the topic ,let’s study some basic definitions and formulations of momentum and laws of Newton.

Momentum

Momentum of a body is defined as the product of its mass and velocity . It is a vector quantity denoted by as

Newton’s First Law

Every body continues to be in its state of rest or of uniform motion in a straight line unless compelled by some external force to act otherwise.

Newton’s Second law of Motion

The rate of change of momentum of a body is directly proportional to the applied force and

takes place in the direction in which the force acts.

If a body of mass undergoes a change in velocity from to under action of force F, its initial momentum is changes by . According to Second law-

, where proportionality constant.

Newton’s Third law of Motion

To every action, there is always an equal and opposite reaction.Consider a pair

of bodies A and B. According to the third law,

(force on A by B) = – (force on B by A)

Conservation of Momentum

According to the law of conservation of momentum, a system’s overall momentum is always conserved for an isolated system. Law of conservation states that-

The overall momentum of two or more bodies acting on one another in an isolated system stays constant unless an external force is introduced.

Or

The total momentum of an isolated system of interacting particles is conserved.

Let’s understand this principle by an example of a bullet fired from a gun.

According to the third law, if the force applied to the bullet by the gun is , the force applied to the gun by the bullet is .

The two forces operate for the same amount of time, . The second law states that is the change in bullet momentum and is the change in gun momentum. Since both are initially at rest, the momentum change is equal to the final momentum for both.

Therefore, , or , if is the momentum of the bullet after firing and is the momentum of the gun’s recoil. In other words, the (bullet + gun) system’s overall momentum is conserved.

Another important example of the application of the law of conservation of momentum is the collision of two bodies. Consider two bodies A and B, with initial momenta and with masses and with initial velocity and . After collision, body gets apart from each other with final velocity and , with final momenta and respectively. According to the Second Law of Newton’s-

By using Newton’s third law, , then we can write

further we can write as,

Above equation shows that the total final momentum of the isolated system equals its initial momentum.

It should be noted that total momentum of a system is conserved, whether the collision is elastic or inelastic. In case of elastic collision there is an additional condition that total initial kinetic energy is equal to total final kinetic energy.

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