# Law of Conservation of Linear Momentum

The linear momentum of a particle is defined as the product of the mass of the particle times the velocity of that particle. Conservation of momentum of a particle is a property exhibited by any particle where the total amount of momentum never changes. Linear momentum of a particle is a vector quantity and is denoted by

$$\begin{array}{l}\underset{p}{\rightarrow}\end{array}$$

## Conservation of Linear Momentum

According to the conservation of linear momentum,

If the net external force acting on a system of bodies is zero, then the momentum of the system remains constant.

We have to remember that the momentum of the system is conserved and not that of the individual particles. The momentum of the individual bodies in the system might increase or decrease according to the situation, but the momentum of the system will always be conserved, as long as there is no external net force acting on it.

## Conservation of Linear Momentum Formula

The principle of conservation of momentum states that if two objects collide, then the total momentum before and after the collision will be the same if there is no external force acting on the colliding objects.

The conservation of linear momentum formula mathematically expresses that the momentum of the system remains constant when the net external force is zero.
Initial momentum = Final momentum

 Pi = Pf

### Linear Momentum Formula

Linear momentum is mathematically expressed as:

$$\begin{array}{l}\vec{p}= m\vec{v}\end{array}$$

$$\begin{array}{l}\vec{p}\,is\,the\,linear\,momentum \end{array}$$

$$\begin{array}{l}\vec{v}\,is\,the\,linear\,velocity\end{array}$$

$$\begin{array}{l}m\,is\,the\,mass\,of\,the\,body\end{array}$$

## Conservation of Linear Momentum Equation

The law of conservation of momentum can be explained from the second law of motion. Newton’s second law of motion says that the rate of change of linear momentum of a body is equal to the net external force applied to it.

Mathematically it is expressed as:

$$\begin{array}{l}\frac{dP}{dt}=\frac{(mv)}{dt}=m\frac{dv}{dt}=ma=F_{net}\end{array}$$

If the net external force acting on a body is zero, then the rate of change of momentum is also zero, which means that there is no change in momentum.

Read More: Newton’s Second Law of Motion

## Conservation of Linear Momentum Example

Two bodies of mass M and m are moving in opposite directions with the velocities v. If they collide and move together after the collision, we have to find the velocity of the system.

Since there is no external force acting on the system of two bodies, momentum will be conserved.

Initial momentum = Final momentum

(Mv – mv) = (M+m)VFinal

From this equation, we can easily find the final velocity of the system.

## Conservation of Linear Momentum Applications

One of the applications of conservation of momentum is the launching of rockets. The rocket fuel burns are pushes the exhaust gases downwards, and due to this, the rocket gets pushed upwards. Motorboats also work on the same principle, it pushes the water backward and gets pushed forward in reaction to conserve momentum.

## Frequently Asked Questions – FAQs

Q1

### Discuss the law of conservation of momentum. State its unit.

The law of conservation of momentum states that when two objects collide in an isolated system, the total momentum before and after the collision remains equal. This is because the momentum lost by one object is equal to the momentum gained by the other. In other words, if no external force is acting on a system, its net momentum gets conserved. The unit of momentum in the S.I system is kgm/s or simply Newton Second(Ns).
Q2

### How does Newton’s second law of motion apply to rockets?

According to Newton’s second law of motion, we know that force is a product of mass and acceleration. When a force is applied to the rocket, the force is termed as thrust. The greater the thrust, the greater will be the acceleration. Acceleration is also dependent on the rocket’s mass, and the lighter the rocket faster is the acceleration.
Q3

### How does Newton’s second law apply to a car crash?

According to the definition of Newton’s second law of motion, force is the dot product of mass and acceleration. The force in a car crash is dependent either on or the acceleration of the car. As the acceleration or mass of the car increases, the force with which a car crash takes place will also increase.
Q4

### What is the other name for Newton’s second law?

The other name for Newton’s second law is the law of force and acceleration.
Q5

### What are some daily life examples of Newton’s second law of motion?

Newton’s second law of motion explains how force can change the acceleration of the object and how the acceleration and mass of the same object are related. Therefore, in daily life, if there is any change in the object’s acceleration due to the applied force, they are examples of Newton’s second law.
• Acceleration of the rocket is due to the force applied known as thrust and is an example of Newton’s second law of motion.
• Another example of Newton’s second law is when an object falls from a certain height, the acceleration increases because of the gravitational force.

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10. Conservation of linear momentum is equivalent to which laws of Newton?

• Conservation of linear momentum is equivalent to Newton’s Third Law of Motion.

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