Elastic Collisions in One Dimension - Direct Impact and Newton's Law of Restitution

Key Concepts

Direct Impact

• Two particles directly impact each other, moving towards each other and then separating.
• Before the impact: Particles move towards each other with velocities u1 and u2.
• After the impact: Particles move away with velocities v1 and v2.
• The separation could result in particles moving in opposite directions or the same direction.
• Coefficient of restitution (e) describes elasticity of collision.

Coefficient of Restitution (e)

• Definition: e = (Speed of Separation) / (Speed of Approach)
  • Speed of Separation: Speed moving away from each other after collision.
  • Speed of Approach: Speed moving towards each other before collision.
• Range: 0 ≤ e ≤ 1
  • e = 1: Perfectly elastic collision (e.g., snooker balls).
  • e = 0: Perfectly inelastic collision (particles stick and stop moving).

Calculation

Speed of Separation and Approach

• Separation (v):
  • Opposite directions: v1 + v2
  • Same direction: v2 - v1
• Approach (u):
  • Opposite directions: u1 + u2
  • Same direction: u2 - u1 (u1 > u2 to catch up)

Examples

1. Scenario A:

   • v1 at rest, v2 moves away at 2 m/s
   • Speed of Separation = 2, Speed of Approach = 8
   • Coefficient of Restitution, e = 1/4

2. Scenario B:

   • v1 = 4, v2 = 5, Speed of Separation = 1 m/s, Speed of Approach = 3 m/s
   • Coefficient of Restitution, e = 1/3

3. Scenario C:

   • v1 at rest, u1 = 2, u2 = 7, Speed of Separation = 9, Speed of Approach = 18
   • Coefficient of Restitution, e = 1/2

Problem Example

• Given: Two particles traveling in the same direction with speeds 4 and 3 m/s, respectively, coefficient of restitution e = 1/3.
• Find: Speed v post-collision.
  • Speed of Separation = v - 2, Speed of Approach = 4 - 3 = 1
  • Equation: 1/3 = (v - 2) / 1
  • Solving: v = 2 1/3 m/s

Conservation of Linear Momentum

• Equation: Total momentum before collision = Total momentum after collision
• Example Calculations:
  • Before collision: (0.2 kg * 5 m/s) + (0.4 kg * -4 m/s)
  • After collision: 0.2*(-v1) + 0.4*v2

Practice Problems: Exercise 4A

• Coefficient of Restitution:
  • Speed of Separation (e.g., v1 + v2 for opposite directions)
  • Speed of Approach (e.g., v1 + v2 or v1 - v2 depending on direction)

Note: Practice using common sense to determine speed of separation and speed of approach for various scenarios to avoid confusion with formulae.

Additional Tips

• Always determine if particles are moving in the same or opposite directions to choose the correct calculation method.
• Use diagrams and clear notation to distinguish between the different cases of separation and approach.

Simplified Approach

• Separation: Add velocities for opposite directions or subtract for same directions.
• Approach: Add velocities for opposite directions or subtract for same directions.

Apply these principles to solve relevant exercises in the provided textbook sections.