Newton’s laws of motion are indeed the essence of all modern physics theories. And one such law of Newton is the law of conservation of momentum. This law talks about momentum and its conservation.

Therefore, to understand this law, you need to understand momentum and then apply its concept in terms of this law.

**What Is Momentum?**

Momentum can be defined as the product of the mass and velocity of a moving object. It is a vector quantity, and it can be expressed in terms of kilogram meters per second. The momentum of a body has the same direction as the direction of the velocity of the body.

This is also a relative quantity, and its value depends on the frame of reference used.

**Linear Momentum**

Linear Momentum is a term that refers to the product of mass and the velocity of a body that is traveling in one direction. It is also a vector quantity.

**Law Of Conservation Of Momentum Principle **

Conservation of momentum is the general law of physics. It is the consequence of Newton’s third law. This universal law of motion states that the total momentum of any object is always conserved, which means that “the total momentum of any two or more objects in an isolated system which is acting upon each other, remains constant until and unless no external force acts on the system”. Momentum can neither be created nor destroyed.

## **Derivation **

According to Newton’s third law of motion, when there are two bodies named object1 and object2, the force is applied on object2 by object1. Then the object2 always exerts back a force with the same magnitude but in the opposite direction.

By this experiment, Newton drew the idea of the law of conservation of momentum.

Let’s take any two objects X and Y, with the mass of m1 and m2, respectively.

When both the objects collide with one another.

The initial and final velocities of object X are U1 and V1 ;

The initial and final velocities of object Y are U2 and V2.

The force exerted by object X = mass x acceleration

= mass x final velocity – initial velocity

= m1 x (V1 – U1t)

The force exerted by object Y = mass x acceleration

= mass x final velocity – initial velocityt

= m2 x (V2 – U2t)

Where t is the time of contact between two objects

So, according to Newton’s third law, we get the equation,

= Force exerted by object X = Force exerted by Object Y

= m1 x (V1 – U1t) = m2 x (V2 – U2t)

Now, on cancelling t from both the sides of the equation, we get,

= m1 ( V1 – U1) = – m2 ( V2 – U2) { since, the direction of exerted force is opposite}

= m1V1 – m1V1 = – m2V2 + m2U2

= m1V1 + m2V2 = m2U2 + m1U1

= m1V1 + m2V2 = m1U1 + m2U2

Therefore, the final momentum of both the objects X and Y = the Original momentum of both the objects X and Y.

Hence, the momentum is conserved.

## **Examples Of Law Of Conservation Of Momentum**

### Kickback Or Recoiling of a Gun

A gun’s recoil is the gun’s sudden backward movement after firing a bullet. Before firing off a bullet, both gun and bullet are at rest, which means that the total momentum is zero. When a bullet is fired, the gun exerts a force on it, launching it forward.

As per the law of conservation of momentum, bullets will also exert the same force but in the opposite direction, i.e., in the backward direction. Let’s understand this using the law of conservation of momentum.

The mass of a bullet is m, and the mass of the gun is m2.

The velocity of the bullet in the forward direction is V,

The velocity of the bullet in the backward direction is V2.

According to the law of conservation of momentum, mV m2V2 = 0

The value of total momentum before firing is zero, and the value of total momentum after firing is zero, which satisfies the law of conservation of momentum.

= mV+m2V2 = 0

= V2 = – (m/m2)V

As the mass of the bullet (m) is much less than the gun (m2), we can negate their ratio. This V2 becomes equal to V.

### **Rocket**

When a rocket is launched, the fuel inside the rocket is propelled outside, which contributes to its velocity. The initial momentum of the whole system was zero. However, after the gas gets ejected, the rocket is pushed upwards due to the law of conservation of momentum.

Let’s understand this using the expression of the law of conservation of momentum.

Let the rocket’s mass be mR and the mass of fuel ejected mF and Ve as the velocity of fuel ejected and Vr as the velocity the rocket acquired after fuel ejection.

Ve mF = – (mR – mF)Vr

Since the mass of fuel and the velocity of fuel is small, we can ignore it.

Therefore, Vr = – Ve { mF / mR}

After integration, the final value is

V = Ve ln ( mR0/m)

**Solved Example**

Question. There are two trucks at rest of mass 5kg and 12 kg respectively. A truck of mass 12kg travels to the east with a velocity of 6 meters per second. Find the velocity of the truck, which has a mass of 5kg. There are two trucks at rest of mass 5kg and 12 kg respectively.

Answer. Given that,

Mass of first truck (m) = 5kg

Mass of second truck (M) = 12kg

The velocity of the first truck (v) =?

Velocity of second truck (V) = 6 m/s

According to the law of conservation of momentum,

Initial potential energy = 0 ( since the truck is at rest)

Final potential energy = potential energy of first truck + potential energy of second truck.

Final potential energy = m x v + M x V

= 5 x v + 12 x 6

= 5v + 72

Initial potential energy = final potential energy

0 = 5v + 72

v = 14.4 m/s

Hence, the velocity of the first truck is 14.4 meters per second.

### **Concluding Remarks**

The law of conservation of momentum is a very crucial concept. It is applied in solving questions where the energy of bodies in a system is either exchanged or transferred from one form to another.

This law is often used in various technologies, such as in the functioning of a gun and during the launch of a rocket into space.