# Conservation of Linear Momentum

## Introduction

A particle's linear momentum is defined as the product of its mass and velocity. A particle's conservation of momentum is a property that any particle has in which the total amount of momentum never changes. Linear momentum of a particle is a vector quantity denoted by $\mathrm{\vec{p}}$

## 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.

It is important to remember that the momentum of the system, not the individual particles, is conserved. The momentum of the individual bodies in the system may increase or decrease depending on 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

If two objects collide, the total momentum before and after the collision will be the same if no external force acts on the colliding objects, according to the principle of conservation of momentum. Linear momentum conservation When the net external force is zero, the momentum of the system remains constant, as expressed mathematically by the formula.

Final momentum = Initial momentum

$$\mathrm{P_i=P_f}$$

## Linear Momentum Formula

Linear momentum is mathematically expressed as:

$\mathrm{\vec{p}=m\:\vec{ u}}$

$\mathrm{\vec{p}}$ is the linear momentum

$\mathrm{\vec{ u}}$ is the linear velocity

m is the mass of body

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## Conservation of Linear Momentum Equation

The second law of motion can be used to explain the law of conservation of momentum. According to Newton's second law of motion, the rate of change of a body's linear momentum is equal to the net external force applied to it.

Mathematically it is expressed as:

$$\mathrm{\frac{dP}{dT}}$$

$$\mathrm{=\frac{m\: u}{dt}}$$

$$\mathrm{=m\frac{d u}{dt}}$$

$$\mathrm{=ma}$$

$$\mathrm{=F_{net}}$$

If a body's net external force is zero, the rate of change of momentum is also zero, implying that there is no change in momentum.

## Example of Linear Momentum Conservation

Two masses M and m are moving in opposite directions at velocities v. If they collide and move together after the collision, we must calculate the system's velocity. Momentum will be conserved because there is no external force acting on the system of two bodies.

Initial momentum = Final momentum

$$\mathrm{(Mv – mv) = (M+m)V_{Final}}$$

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

### Principle of Conservation of Momentum

• When there is no external force acting on the isolated system. In that case, the rate of change of total Momentum remains constant. This indicates that the quantity is said to be constant.

• The above explanation is the correct derivation of the principle of Linear Momentum Conservation.

• We can say that regardless of the characteristic or property of any system's interaction, the total Momentum will remain constant.

### Applying the Principle of Conservation of Linear Momentum

• We must consider the object that is part of the system.

• The bodies in relation to the system identify external and internal forces.

• Verification of the isolated position of the system.

• It should be ensured that the initial and final Momentums are equal.

Here, the Momentum is a vector quantity.

## Ice Skaters

Consider two skaters who began at rest and then pushed off against each other on the ice where friction is less. The woman here weighs 54 kg, while the man weighs 88 kg. The woman flees at a speed of 2.5 metres per second. What is man's recoil velocity?

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Now given,

$$\mathrm{m_1v+m_2v_2=0}$$

$$\mathrm{vf_2=\frac{m_1\:v\:f_1}{m_2}}$$

$$\mathrm{vf_2=-\frac{(54\:kg)(\frac{2,5 m}{s})}{88\:kg}}$$

$$\mathrm{vf_2=-1.5m/s}$$

### Linear Momentum Dimensional Formula

Momentum is defined as the sum of mass and velocity. Otherwise, it is the quantity of motion of a moving body.

Now, Linear Momentum = Mass * Velocity – (1)

The Mass and Velocity Dimensional Formula is as follows:

$\mathrm{Mass =(M^1L^0T^0)\:–\: (2)}$

Velocity = $\mathrm{(M^0L^1T^{-1})\:–\: (3)}$

Substituting equations (2) and (3) into equation (1) yields, p=mv or

$\mathrm{L =(M^1L^0T^0)*(M^0L^1T^{-1})=(M^1L^1T^{-1})}$

So, the dimensions of Linear Momentum is represented by, $\mathrm(M^1L^1T^{-1})$

## Conservation of Linear Momentum Applications

The launch of rockets is one application of momentum conservation. As the rocket fuel burns, the exhaust gases are pushed downwards, pushing the rocket upwards. Motorboats operate on the same principle, they push the water backward and are pushed forward in response to conserve momentum.

Example

The following are some of the most well-known applications of Conservation of Linear Momentum:

• The launching of rockets

• Motorboats: It pushes the water backward and gets pushed forward so as to conserve the Momentum.

• The recoil of a gun and the escape of a balloon in the air are two common examples of this phenomenon.

• Cars experience resistance as a result of Linear Momentum Conservation.

• A professional swimmer will frequently dive into the water in a diving pose rather than a belly flop. The diving pose allows the swimmer to reach greater depths in the water with minimal effort.

• Because there is Conservation of Linear Momentum in the case of a projectile motion, the horizontal forces acting on the projectile are zero.

• Because subatomic particles can only be produced and analysed through collision experiments, our understanding of Momentum has helped us decipher their nature over the years.

• One significant application has been in space science. This is the underlying principle of rocket launch. Because the Linear Momentum is conserved, the rocket is propelled by the burning of fuel, which exerts downward force, allowing the rocket to launch upwards into space, into Earth's orbit.

## Significance of Linear Momentum

The characteristic feature of motion of objects in the 2D plane is linear momentum.

• Linear Momentum components are used as coordinates in Hamiltonian mechanics to study the motion of objects.

• It represents translational symmetry because it is a conserved quantity of a moving system.

• It aids our understanding of linear motion in Newton's mechanics by emphasising not only the velocity but also the mass of the moving object.

## FAQs

Q1. What is momentum?

Ans. The product of mass and velocity is momentum. It is the quantity used to calculate the object's mass and velocity.

Q2. Give the formula for law of conservation of momentum.

Ans. The formula for law of conservation of momentum is:

$\mathrm{m_1u_1+m_2u_2\:=\:m_1v_1+m_2v_2}$

Q3. List some examples of law of conservation of momentum.

Ans. Examples of law of conservation of momentum are:

• Motion of rockets
• Air-filled balloons
• System of gun and bullet

Q4. Is momentum a scalar or a vector quantity?

Ans. Momentum is a vector quantity since it has both magnitude and direction.

Q5. Mention True/False: Momentum decreases as friction increases.

Ans. True