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To address the thinking question about the Law of Conservation of Momentum, let’s break down the concept and apply it to a practical example:

What is the Law of Conservation of Momentum?

The law states that in a closed system (no external forces acting on the system, e.g., no friction, air resistance, or external pushes/pulls), the total momentum of the system remains constant over time.

Mathematically:
[ \text{Total initial momentum} = \text{Total final momentum} ]
[ m1v{1i} + m2v{2i} + ... = m1v{1f} + m2v{2f} + ... ]

Key note: Momentum is a vector, so direction matters (use positive/negative signs for opposite directions).

Example Thinking Question & Solution

Problem:
Two carts collide on a frictionless track. Cart A (mass = 2 kg) moves right at 5 m/s. Cart B (mass = 3 kg) moves left at 2 m/s. After collision, Cart A moves left at 1 m/s. What is the final velocity of Cart B?

Step-by-Step Solution:

1. Define direction: Let right = positive, left = negative.
2. Calculate initial momentum:
   [ p_{\text{initial}} = mA v{Ai} + mB v{Bi} ]
   [ = (2 \text{kg})(5 \text{m/s}) + (3 \text{kg})(-2 \text{m/s}) ]
   [ = 10 -6 = 4 \text{kg·m/s} ]
3. Calculate final momentum (let (v{Bf}) = final velocity of Cart B):
   [ p{\text{final}} = mA v{Af} + mB v{Bf} ]
   [ = (2 \text{kg})(-1 \text{m/s}) + (3 \text{kg})(v{Bf}) ]
   [ = -2 +3v{Bf} ]
4. Apply conservation of momentum:
   [ 4 = -2 +3v{Bf} ]
   [ 3v{Bf} =6 → v_{Bf}=2 \text{m/s} ]

Conclusion: Cart B moves right at 2 m/s (positive sign indicates direction).

Key Takeaway

Conservation of momentum helps predict motion after collisions or explosions (e.g., gun recoil, rocket propulsion) by focusing on the total momentum of the system, ignoring internal forces between objects.

If you had a specific question in mind, feel free to share it for a tailored explanation!

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