Physics Lecture: Elastic Collision Problem

Problem Statement

• A 4 kg ball moving east at 5 m/s collides with a 2 kg ball at rest.
• Objective: Calculate velocities of both balls after a perfectly elastic collision.

Key Concepts

Conservation of Momentum

• Equation: [ m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2' ]
  • ( v_1 ) is the initial velocity of the first ball.
  • ( v_1' ) is the final velocity of the first ball.
  • ( v_2 ) is the initial velocity of the second ball.
  • ( v_2' ) is the final velocity of the second ball.
• Momentum before collision equals momentum after collision.

Conservation of Kinetic Energy

• Applicable for perfectly elastic collisions.
• Equation (Simplified Form): [ v_1 + v_1' = v_2 + v_2' ]

Solution Steps

Information Given

• Masses:
  • ( m_1 = 4 ) kg
  • ( m_2 = 2 ) kg
• Initial Velocities:
  • ( v_1 = 5 ) m/s (east)
  • ( v_2 = 0 ) m/s (at rest)

Setting up Equations

1. Momentum Equation:

   • Substitute known values: [ 4 \times 5 + 2 \times 0 = 4v_1' + 2v_2' ]
     • Simplified: ( 20 = 4v_1' + 2v_2' )

2. Kinetic Energy Equation (Simplified):

   • [ 5 + v_1' = 0 + v_2' ]
   • Simplified: ( 5 = -v_1' + v_2' )

Solving the System of Equations

• Equations: [ 4v_1' + 2v_2' = 20 ] [ -v_1' + v_2' = 5 ]
• Use elimination method:
  • Adjust the second equation: Multiply by 4
  • Add the two equations to eliminate ( v_1' )
  • Solve for ( v_2' )
  • Result: ( v_2' = 6.67 ) m/s
  • Substitute back to find ( v_1' ):
    • ( v_1' = 1.67 ) m/s

Verification

Conservation Check

1. Momentum Verification:

   • Initial: ( 20 )
   • Final: ( 4 \times 1.67 + 2 \times 6.67 \approx 20 )

2. Kinetic Energy Verification:

   • Simplified Equation Verification: [ 5 + 1.67 = 6.67 ]
   • Detailed Kinetic Energy Calculation:
     • Initial KE: 50
     • Final KE: ( 5.5778 + 44.4889 \approx 50 )

Conclusion

• Both momentum and kinetic energy are conserved.
• Calculated velocities:
  • ( v_1' = 1.67 ) m/s
  • ( v_2' = 6.67 ) m/s