# Linear Momentum

## What is Linear Momentum?

Linear momentum generally known as the momentum of a body is defined as the total quantity of motion possessed by the moving body and it is measured as the product of the mass of the particle and its velocity.

Momentum is a vector quantity. Its direction is in the direction of the velocity of the body. The S.I. unit of momentum is given by kgms-1. Linear momentum is denoted by $\vec{p}$ . If a body of mass m is moving with a velocity $\vec{v}$ then its momentum is

$\overrightarrow{p}=m\overrightarrow{v}$

## Newton’s Second Law of Motion-Momentum

When the same force acts on two bodies of different masses for the same interval of time, we can observe different effects on the objects. The lighter object moves with a higher velocity than the heavier object. However, the change in momentum of both bodies is the same. This leads to Newton’s second law of motion and momentum. According to Newton’s second law of motion,

” The time rate of change of momentum of the body is directly proportional to the impressed force and takes place in the direction of the force”.

From Newton’s second law of motion, for a fixed mass particle

$\overrightarrow{F}=m\overrightarrow{a}=m\frac{d\overrightarrow{v}}{dt}=\frac{d}{dt}\left( m\overrightarrow{v} \right)=\frac{d\overrightarrow{p}}{dt}$

For a system of n particles with masses ${{m}_{1}},{{m}_{2}},{{m}_{3}},…{{m}_{n}}$and velocities $\overrightarrow{{{v}_{1}}},\overrightarrow{{{v}_{2}}},\overrightarrow{{{v}_{3}}},…\overrightarrow{{{v}_{n}}}$ respectively, then the net momentum of the system is

$\overrightarrow{{{p}_{net}}}={{m}_{1}}\overrightarrow{{{v}_{1}}}+{{m}_{2}}\overrightarrow{{{v}_{2}}}+{{m}_{3}}\overrightarrow{{{v}_{3}}}+…+{{m}_{n}}\overrightarrow{{{v}_{n}}}=\overrightarrow{{{p}_{1}}}+\overrightarrow{{{p}_{2}}}+\overrightarrow{{{p}_{3}}}+…+\overrightarrow{{{p}_{n}}}$

$\overrightarrow{{{p}_{net}}}=M\overrightarrow{{{V}_{cm}}}$

Differentiating the above expression with respect to time

$\frac{d\overrightarrow{{{p}_{net}}}}{dt}=M\frac{d\overrightarrow{{{V}_{cm}}}}{dt}$

$\overrightarrow{{{F}_{net}}}=M\overrightarrow{{{a}_{cm}}}$

And also

$\overrightarrow{{{F}_{net}}}=\frac{d\overrightarrow{{{p}_{net}}}}{dt}$

The magnitude of linear momentum may be expressed in terms of kinetic energy as well

$p=mv$

${{p}^{2}}={{m}^{2}}{{v}^{2}}=2m\left( \frac{1}{2}m{{v}^{2}} \right)=2mK$

### Law of Conservation of Momentum

If the net force acting on a body is equal to zero, then the momentum of the body remains constant. This is known as law of conservation of momentum.

${{F}_{net}}=0$ $\frac{d{{p}_{net}}}{dt}=0$

Therefore, pnet = 0 or pnet = constant

If the velocity of centre of mass is equal to zero, $\left( {{v}_{cm}}=0 \right)$ then from $\overrightarrow{{{p}_{net}}}=M\overrightarrow{{{V}_{cm}}}$ we get, $\overrightarrow{{{p}_{net}}}=0.$

If the velocity of centre of mass is constant (vcm = constant) then we get $\overrightarrow{{{p}_{net}}}$ = constant.

This is known as the law of conservation of linear momentum of the system of particles.