Derivation of Conservation of Linear Momentum
The principle of conservation of linear momentum states that, in an isolated system, the total linear momentum remains constant if no external forces act upon it. Here’s a detailed derivation:
1. Definition of Linear Momentum
- Linear momentum (p) of an object is defined as the product of its mass (m) and velocity (v):
p = mv
2. Consider Two Colliding Objects
- Let two objects, A and B, with masses mA and mB, move towards each other with velocities vA and vB before collision.
3. Initial Momentum Calculation
- The total initial momentum (p_initial) of the system before the collision is:
p_initial = mA * vA + mB * vB
4. Collision and Final Velocities
- After the collision, let their velocities change to vA' and vB'. The total final momentum (p_final) is:
p_final = mA * vA' + mB * vB'
5. Applying Newton's Third Law
- According to Newton's third law, the forces exerted during the collision are equal and opposite. Therefore, the impulse experienced by each object is equal in magnitude and opposite in direction.
6. Change in Momentum
- The change in momentum for each object can be expressed as:
Change in momentum for A = mA(vA' - vA)
Change in momentum for B = mB(vB' - vB)
7. Total Change in Momentum
- The total change in momentum of the system is:
Total change = Change in momentum of A + Change in momentum of B
Total change = mA(vA' - vA) + mB(vB' - vB)
8. Setting Total Change to Zero
- In the absence of external forces, the total change in momentum must be zero:
0 = mA(vA' - vA) + mB(vB' - vB)
9. Conclusion
- Thus, the initial and final momentum of the system are equal:
mA * vA + mB * vB = mA * vA' + mB * vB'
- This reflects the conservation of linear momentum in isolated systems, confirming that momentum is conserved during collisions.