# Derivation of Law of Conservation of Momentum

### What is the law of conservation of momentum?/State the law of conservation of momentum class 9/ State and prove law of conservation of momentum class 9

Derivation of Law of Conservation of Momentum - The product of an object's velocity and mass is the object's momentum. It's a quantity with a vector. The overall momentum of an isolated system is conserved, according to conservation of momentum, a fundamental law of physics. In other words, if no external force acts on a system of objects, their overall momentum remains constant during any interaction. The vector sum of individual momentum is the overall momentum. In any physical process, momentum is conserved.

Under the reasonable assumption that space is uniform, the law of conservation of momentum is established both empirically and theoretically. The law of momentum is found in nature, and the assertion is merely a theoretical statement based on the results of the studies.

State law of conservation of linear momentum

If there is no external force acting on an isolated system, its overall momentum remains constant. As a result, if a system's total linear momentum remains constant, the resultant force exerted on it is zero. In the absence of external torque, angular momentum is conserved as well.

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## Conservation of Momentum Derivation (conservation of momentum formula)

Newton's third law states that when object A produces a force on object B, object B responds with a force of the same magnitude but opposite direction. Newton derived the law of conservation of momentum from this concept.

Consider two colliding particles A and B with masses of m1 and m2 with starting and ultimate velocities of u1 and v1 for A and u2 and v2 for B, respectively. The contact time between two particles is denoted by the letter t.

Change in momentum of particle A is

A=m1 (v1-u1)

Change in momentum of particle B is

B=m2 (v2-u2)

From third law of motion we can write,

FBA=-FAB

FBA=m2×a2=m2(v2-u2)/t

FBA=m1×a1=m1(v1-u1)/t

m2(v2-u2)/t=-m1(v1-u1)/t

m1u1+m2u2=m1v1+m2v2

As a result, if no external force is exerted on the system, the momentum after the collision is identical to the momentum before the collision.

As a result, the equation of the law of conservation of momentum is as follows: m1u1+m2u2 represents the total momentum of particles A and B before the collision, and m1v1+m2v2 represents the total momentum of particles A and B after the collision.

Derivation of momentum or law of conservation of momentum in One-dimensional

A one-dimensional collision of two objects can be used to explain momentum conservation. Two objects with masses of m1 and m2 collide while moving in a straight line at velocities of u1 and u2, respectively. They gain velocities v1 and v2 in the same direction after colliding.

Before the impact, the total momentum

pi=m1u1+m2u2

After the impact, the total momentum

pf=m1v1+m2v2

Total momentum is conserved if no other force acts on the system of two objects. Therefore,

pi=pf

m1u1+m2u2=m1v1+m2v2

Derivation of law of conservation of momentum in Two-Dimensional

The overall momentum is pix=p1=m1u1 along the X-axis and piy=m2u2 along the Y-axis before the collision. The overall momentum after the collision is pfx=(m+M)ucosθ along the X-axis and pfy=(m+M)usinθ along the Y-axis

where m is mass and (m+M) is the resultant mass when particles get trapped inside it.

pix=pfx

m1v1=(m+M)ucosθ (1)

piy=pfy

m2v2=(m+M)usinθ (2)

As a result of squaring and adding equations (1) and (2),

(m1v1)2+(m2v2)2=(m+M)2u2(cos2θ+sin2θ)

It is the combined object's speed.

This determines the direction of the velocity.

## Derivation of Law of Conservation of Momentum Examples

1. A Gun's Recoil: When a bullet is shot from a gun, both the bullet and the gun are initially at rest, with zero total momentum. When a bullet is fired, it gains forward motion. The cannon obtains backward momentum as a result of momentum conservation. With a forward velocity of v, a bullet with mass m is discharged. The mass M gun achieves a rearward velocity u. The overall momentum before firing is zero, and the total momentum after firing is also zero.
2. Rocket propulsion involves a gas chamber at one end from which gas is expelled at high velocity. The total momentum is zero before the ejection. The rocket obtains a rebound velocity and acceleration in the opposite direction due to the ejection of gas. This occurs as a result of momentum conservation.
3. Motorboats- In order to conserve momentum, they push the water backwards and forwards.

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