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# Linear Momentum Formula

Any moving object always has the momentum for its motion. Momentum is an important property of a moving body. But this momentum always depending on the mass and velocity of the body. Why? In this topic, we will find the answer to this question. Here we will see the Linear Momentum Formula with some examples. Linear momentum is much common due to simple linear motion. Let us learn the concept from Physics!

## Linear Momentum Formula

### What is Linear Momentum?

If we are standing at the bottom of a hill and we faced with the option of stopping a bike or a bicycle, then we will probably choose to stop the bicycle. The reasoning behind this is that the bike has more momentum than the bicycle. Here, momentum simply means the mass in a moving body.

The bike has a large momentum because it is very massive. Also, it has more speed, which influences the momentum, as well. The bicycle will also have the momentum due to its speed, but due to its less mass, its momentum is also less.

### The formula for Linear Momentum:

Linear momentum is defined as the product of the mass (m) of an object and the velocity (v) of the object. This relationship can be described in the form of an equation. It is given as:

$$Momentum = mass of the body \times its velocity$$.

i.e. $$P = m \times v$$.

We know that velocity is the speed with direction. Therefore, if an object has a large speed, it also has a large velocity.

The units of linear momentum are $$kg ms^{-1}$$.

Note that the momentum will always be in the same direction as its velocity. It is a conserved quantity i.e. the sum total of the momentum of a system will always be constant. It is the law of conservation of momentum.

### Linear Momentum of a System of Particles:

The linear momentum of the particle is:

P = mv

For ‘ n ‘ no. of particles total linear  momentum is,

$$P = P_1 + P_2+ P_3 + ….+ P_n$$

Each of momentum is written as

$$m_1 v_1 + m_2v_2 + ………..+m_nv_n$$.

We know that the velocity of the center of mass is:

$$MV = Sum m_iv_i$$ ,

So comparing these equations we get,

P = M V

Where M is the center of mass, and V is the velocity of its center of mass.

### Law of Conservation of Linear momentum:

Since P = MV

Differentiating the above equation we get,

$$\frac{d}{dt} P = M \frac{d}{dt} V$$

$$\frac{d}{dt} P = ma , Newton’s second law of motion F = ma$$

Hence,  $$\frac{d}{dt} P = F$$

Thus, acting force is the rate of change of the momentum of the object.

Thus, if the total external force acting on the system is zero, then

$$\frac{d}{dt} P = 0$$

This means that P will be constant. Therefore, whenever the total force acting on the system of a particle is equal to zero then the total linear momentum of the system is constant or conserved. This is the law of conservation of total linear momentum of a system of particles

## Solved Examples

Q.1: Determine the linear momentum of a moving body whose mass is 10 kg and speed is $$30 ms^{-1}$$.

Solution:

Given parameters are,

m = 10 kg

$$v = 30 ms^{-1}$$

Linear momentum formula is:

P = mv

= 10× 30

Linear momentum  = $$300 kgms^{-1}$$.

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