Derivation of Law of Conservation of Momentum

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What is the Law of Conservation of Momentum?

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.

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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 u₁and v₁for 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,

F_BA=-F_AB

F_BA=m₂×a₂=m₂(v₂-u₂)/t

F_BA=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.

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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 u₁and u₂, respectively. They gain velocities v₁and v₂in the same direction after colliding.

Before the impact, the total momentum

p_i=m₁u₁+m₂u₂

After the impact, the total momentum

p_f=m₁v₁+m₂v₂

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

p_i=p_f

m₁u₁+m₂u₂=m₁v₁+m₂v₂

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Derivation of law of conservation of momentum in Two-Dimensional

The overall momentum is p_ix=p₁=m₁u₁ along the X-axis and p_iy=m₂u₂along the Y-axis before the collision. The overall momentum after the collision is p_fx=(m+M)ucosθ along the X-axis and p_fy=(m+M)usinθ along the Y-axis

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

p_ix=p_fx

m₁v₁=(m+M)ucosθ (1)

p_iy=p_fy

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)

$u=\frac{\sqrt{m_1^2v_1^2+m_2^2v_2^2}}{m+M}$

It is the combined object's speed.

$tan\theta=\frac{m_2v_2}{m_1v_1}$

This determines the direction of the velocity.

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Derivation of Law of Conservation of Momentum Examples

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.
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.
Motorboats- In order to conserve momentum, they push the water backwards and forwards.

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Frequently Asked Questions (FAQs)

1. Describe the circumstances that must be met in order for conservation of momentum to be applied.

The law of conservation of momentum is one of the most essential laws in physics. The following is how the law of energy conservation is represented. When objects 1 and 2 collide in a restricted space, the energy of the two things before the collision is equal to the force of the two articles after the collision. That is, the force picked up by item 2 is equal to the energy expended by object 1. This shows that the absolute force of the objects is monitored in a Framework, i.e., the total energy is constant and equal.

2. What is the concept of conservation of momentum?

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

3. Give examples of momentum conservation.

The law of conservation of momentum applies to all physical processes. Here are several examples:

Collision: The conservation of momentum and energy governs the collision of things.
The momentum of the propellant gas causes the rocket to go in the opposite direction. This is due to the law of conservation of momentum.
When a bullet is ejected from a gun, the gun suffers recoil momentum.

4. What Does Linear Momentum Mean?

The importance of linear momentum conservation in a system or body in motion is that it maintains total momentum and is equal to the product of mass and vector velocity when an external force is applied.

5. Determine the linear momentum conservation unit.

Linear momentum p is expressed symbolically as:

p=mv

Where,

The mass of the system is m, and its velocity is v.

As a result, the S.I unit of momentum is kg.m/sec.

The net external force is equal to the change in momentum of a system divided by the rate of change of time, according to Newton's 2nd law of motion.