Linear Momentum

What is Linear Momentum?

Linear momentum is defined as the product of mass of the particle and its velocity, it is a vector quantity.

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

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.