Conservation of Linear Momentum and Its Principle

Suppose an object A has mass m₁ and its initial velocity is u₁. An object B has mass m₂​ and initial velocity u₂​. According to the formula for momentum, the initial momentum of A is m₁u₁_​ and that of B is m₂u₂.​
Suppose these two objects collide. Let the force on A due to B be F_1​. This force will cause acceleration in A and its velocity will become v₁​.
Momentum of A after collision = m₁v₁​
According to Newton’s third law of motion, A also exerts an equal force on B but in the opposite direction. This will cause a change in the momentum of B. If its velocity after collision is v₂​, the momentum of B after collision = m₂v₂​. If F₂ is the force that acts on object B,

F₂ = -F₁

m₁ a₁ = - m₁ a₁ ………… [F= ma]

`(m_2)×(v_2-u_2)/(t)=(m_1)×(v_1-u_1)/(t).......... [(a)=(v-u)/(t)] `

m₂(v₂-u₂)=-m₁(v₁-u₁)

m₂v₂-m₂u₂=-m₁v₁+m₁u₁

(m₂v₂+m₁v₁)=(m₁u₁+m₂u₂)

1. The magnitude of total final momentum = the magnitude of total initial momentum.

The total final momentum of a system is equal in magnitude to its total initial momentum. This implies that if no external force acts on two objects, their total initial momentum remains equal to their total final momentum. This principle holds true for any number of interacting objects.

2. ‘When no external force acts on two interacting objects, their total momentum remains constant. It does not change.’

According to this corollary of Newton's third law of motion, the total momentum of a system remains constant when no external forces are present. After a collision, the momentum is redistributed between the objects involved. While the momentum of one object decreases, the momentum of the other increases, ensuring that the total momentum remains unchanged. This can also be stated as, "The total momentum before a collision is equal to the total momentum after the collision."

3. ‘When two objects collide, the total momentum before collision is equal to the total momentum after collision.’

Consider the example of a gun firing a bullet. When a gun with mass m₂ fires a bullet with mass m₁, the bullet achieves a velocity of v₁, resulting in a momentum of m₁v₁._​​​ Before firing, both the bullet and the gun are at rest, resulting in a total initial momentum of zero. According to the law of conservation of momentum, the total final momentum must also be zero. Consequently, the forward motion of the bullet causes the gun to move backward with a recoil velocity v₂​. This backward motion, known as recoil, ensures that the total momentum of the system remains conserved.

m₁v₁ + m₂v₂ = 0 or v₂ = -`(m_1)/(m_2)×(v_1)`