Lecture Notes: Conservation of Momentum Problem
Problem Overview
- Scenario: Two footballers running towards each other with different velocities and masses.
- Objective: Determine their combined velocity after they collide.
- Assumptions: Ignoring friction and other resistive forces.
Given Information
- Footballer 1:
- Mass = 100 kg
- Velocity = 5 m/s
- Direction = positive (arbitrary choice)
- Footballer 2:
- Mass = 70 kg
- Velocity = 4 m/s
- Direction = negative (opposite to Footballer 1)
Key Concept: Law of Conservation of Momentum
- Formula: ( m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 )
- Before collision: Total momentum of both players.
- After collision: Combined system momentum.
Solution Steps
- Identify Mass and Velocity:
- Footballer 1: ( m_1 = 100 ) kg, ( u_1 = 5 ) m/s.
- Footballer 2: ( m_2 = 70 ) kg, ( u_2 = -4 ) m/s (negative due to opposite direction).
- Apply Conservation of Momentum:
- Substitute values: ( 100 \times 5 + 70 \times (-4) = (100 + 70)v )
- Simplify: ( 500 - 280 = 170v )
- Solve for Combined Velocity (v):
- Simplify equation: ( 220 = 170v )
- Solve: ( v = \frac{220}{170} \approx 1.3 ) m/s
Conclusion
- Result: Combined velocity = 1.3 m/s
- Direction: Positive, indicating movement in the direction of the heavier player.
- Conceptual Understanding: The heavier and faster player dictates the final direction, making the positive result logical.
Key Takeaways
- Momentum and Velocity as Vectors: Both magnitude and direction must be considered.
- Practical Insight: Heavier/faster objects tend to influence the final direction in collisions.
Presented by: Paul from Physics High
Additional Notes:
- Always draw arrows to represent direction in vector problems.
- Review vector properties in momentum calculations for further understanding.