LAW OF CONSERVATION OF MOMENTUM DERIVATION

Introduction:
The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. This principle is based on the fact that momentum is a vector quantity and can only be transferred between objects through interactions.

Derivation:
To understand the derivation of the law of conservation of momentum, let's consider a simple scenario of two objects colliding with each other.

1. Initial Momentum:
- Let the first object have a mass of m1 and initial velocity of u1.
- Similarly, let the second object have a mass of m2 and initial velocity of u2.
- The initial momentum of the first object is given by p1 = m1 * u1.
- Similarly, the initial momentum of the second object is given by p2 = m2 * u2.

2. Conservation of Momentum:
- When the two objects collide, they exert equal and opposite forces on each other according to Newton's third law of motion.
- The duration of the collision is very small, so the external forces can be neglected.
- Therefore, the total momentum of the system before the collision is equal to the total momentum after the collision.

3. Final Momentum:
- Let the first object have a final velocity of v1 after the collision.
- Similarly, let the second object have a final velocity of v2 after the collision.
- The final momentum of the first object is given by p1' = m1 * v1.
- The final momentum of the second object is given by p2' = m2 * v2.

4. Conservation Equation:
According to the law of conservation of momentum, the total momentum before the collision (p1 + p2) is equal to the total momentum after the collision (p1' + p2').

5. Mathematical Equation:
Mathematically, the conservation equation can be represented as:
m1 * u1 + m2 * u2 = m1 * v1 + m2 * v2

6. Conclusion:
From the above equation, we can observe that the total momentum of the system is conserved during the collision. This principle holds true for any number of objects involved in the collision, making the law of conservation of momentum a fundamental principle in physics.