Definition of Momentum
Momentum is a physical quantity defined as the product of a body's mass and its velocity. It is a vector quantity, possessing both magnitude and direction. Mathematically, momentum (p) is expressed as:
- p = m * v
where m is mass and v is velocity.
Conservation of Momentum During Collision
In a collision between two bodies, the principle of conservation of momentum states that the total momentum of the system before the collision equals the total momentum after the collision, provided no external forces act on it. This can be understood through Newton's Third Law of Motion.
Newton's Third Law of Motion
- For every action, there is an equal and opposite reaction.
This law implies that when two bodies collide, they exert forces on each other that are equal in magnitude and opposite in direction.
Proof of Momentum Conservation
1. Before Collision
- Let the masses of the two bodies be m1 and m2.
- Let their velocities before collision be u1 and u2.
- The total initial momentum (P_initial) is:
- P_initial = m1 * u1 + m2 * u2
2. During Collision
- When the bodies collide, they exert forces on each other.
- The force exerted by body 1 on body 2 (F12) and the force exerted by body 2 on body 1 (F21) are equal and opposite:
- F12 = -F21
3. After Collision
- Let their velocities after collision be v1 and v2.
- The total final momentum (P_final) is:
- P_final = m1 * v1 + m2 * v2
4. Applying Impulses
- The change in momentum is equal to the impulse:
- m1 * (v1 - u1) = F21 * Δt
- m2 * (v2 - u2) = F12 * Δt
5. Conclusion
- Since F12 = -F21, the changes in momentum of the two bodies are equal and opposite, leading to:
- P_initial = P_final
- Thus, the total momentum of the system is conserved during the collision.
In summary, the conservation of momentum during collisions is a direct consequence of Newton's Third Law, demonstrating the mutual interactions between colliding bodies.