Lecture Notes: Elastic Collision Problem

Problem Statement

• A 4 kg ball moving east at 5 m/s strikes a 2 kg ball at rest.
• Objective: Calculate the velocities of both balls post-collision assuming a perfectly elastic collision.

Key Concepts

• Conservation of Momentum: Momentum is always conserved in collisions.
• Conservation of Kinetic Energy: In perfectly elastic collisions, kinetic energy is conserved.

Equations Needed

1. Conservation of Momentum: [ m_1 \cdot v_1 + m_2 \cdot v_2 = m_1 \cdot v_1' + m_2 \cdot v_2' ]

   • (v_1): Initial velocity of the first ball
   • (v_2): Initial velocity of the second ball (0 since it's at rest)
   • (v_1') & (v_2'): Final velocities of the balls

2. Conservation of Kinetic Energy (simplified for elastic collisions): [ v_1 + v_1' = v_2 + v_2' ]

Solution Steps

Step 1: Apply Conservation of Momentum

• Initial condition: (m_1 = 4), (v_1 = 5), (m_2 = 2), (v_2 = 0)
• Momentum equation: [ 4 \times 5 = 4v_1' + 2v_2' ]
• Simplifies to: [ 20 = 4v_1' + 2v_2' ]

Step 2: Apply Conservation of Kinetic Energy

• Energy equation: [ 5 = -v_1' + v_2' ]

Step 3: Solve the System of Equations

• Combine momentum and energy equations:
  • (4v_1' + 2v_2' = 20)
  • (-v_1' + v_2' = 5)
• Use elimination method:
  • Multiply second equation by 4: (-4v_1' + 4v_2' = 20)
  • Add to first equation to solve for (v_2'): (6v_2' = 40)
  • (v_2' = 6.67) m/s
• Solve for (v_1'):
  • Substitute (v_2') back into energy equation
  • (v_1' = 6.67 - 5 = 1.67) m/s

Verification

Check Momentum Conservation

• Initial momentum: (4 \cdot 5 = 20)
• Final momentum: (4 \cdot 1.67 + 2 \cdot 6.67 = 20.02)
• Approximation confirms momentum conservation.

Check Kinetic Energy Conservation

• Initial kinetic energy: (\frac{1}{2} \cdot 4 \cdot 5^2 = 50)
• Final kinetic energy: (\frac{1}{2} \cdot 4 \cdot 1.67^2 + \frac{1}{2} \cdot 2 \cdot 6.67^2 = 50.06)
• Approximation confirms kinetic energy conservation.

Conclusion

• Final Velocities:
  • (v_1' = 1.67) m/s
  • (v_2' = 6.67) m/s
• Both momentum and kinetic energy are conserved, verifying the correctness of the solution.