JEE Main 2024 Question Paper Solution Discussion Live JEE Main 2024 Question Paper Solution Discussion Live

# Linear Momentum

Momentum is the product of a particle’s mass and its velocity. As momentum has both direction and magnitude, it is a vector quantity. The second law of motion (proposed by Issac Newton) shows that the change rate (time) of momentum equals the force exerted on the particle. There are two types of momentum: linear momentum and angular 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.

$$\begin{array}{l}\text{Linear momentum is denoted by}\ \vec{p}.\end{array}$$
$$\begin{array}{l}\text{If a body of mass m is moving with a velocity}\ \vec{v}\ \text{then its momentum is}\ \overrightarrow{p}=m\overrightarrow{v}\end{array}$$

## 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

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

For a system of n particles with masses m1, m2, m3,…, mn and velocities

$$\begin{array}{l}\overrightarrow{{{v}_{1}}},\overrightarrow{{{v}_{2}}},\overrightarrow{{{v}_{3}}},…\overrightarrow{{{v}_{n}}}\end{array}$$
respectively, then the net momentum of the system is

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

$$\begin{array}{l}\overrightarrow{{{p}_{net}}}=M\overrightarrow{{{V}_{cm}}}\end{array}$$

Differentiating the above expression with respect to time

$$\begin{array}{l}\frac{d\overrightarrow{{{p}_{net}}}}{dt}=M\frac{d\overrightarrow{{{V}_{cm}}}}{dt}\end{array}$$

$$\begin{array}{l}\overrightarrow{{{F}_{net}}}=M\overrightarrow{{{a}_{cm}}}\end{array}$$

And also

$$\begin{array}{l}\overrightarrow{{{F}_{net}}}=\frac{d\overrightarrow{{{p}_{net}}}}{dt}\end{array}$$

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

$$\begin{array}{l}p=mv\end{array}$$

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

### 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 the law of conservation of momentum.

$$\begin{array}{l}{{F}_{net}}=0\end{array}$$
$$\begin{array}{l}\frac{d{{p}_{net}}}{dt}=0\end{array}$$

Therefore, pnet = 0 or pnet = constant

If the velocity of the centre of mass is equal to zero, (vcm = 0), then from

$$\begin{array}{l}\overrightarrow{{{p}_{net}}}=M\overrightarrow{{{V}_{cm}}}\end{array}$$
we get,
$$\begin{array}{l}\overrightarrow{{{p}_{net}}}=0.\end{array}$$

If the velocity of the centre of mass is constant (vcm = constant), then we get

$$\begin{array}{l}\overrightarrow{{{p}_{net}}}= \text{constant.}\end{array}$$

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

## Frequently Asked Questions on Momentum

Q1

### Define the term linear momentum of a body.

The quantity of motion contained in a body is called the linear momentum of the body. Linear momentum is defined as the product of mass and velocity.

Q2

### The rate of change of momentum of the body is 2 kgm/s. What is the force acting on the body?

Force = Rate of change of momentum
Therefore, Force = 2 N

Q3

### The total momentum of the universe remains constant. Is this statement true?

The law of conservation of momentum is a general law and is applicable to all isolated systems. Hence, the given statement is true.

Q4

### Is linear momentum a vector or scalar quantity?

Linear momentum is a vector quantity.