Lecture on Average Speed and Average Velocity

Key Definitions

Average Speed

• Definition: Total distance divided by total time.
• Units: Meters per second (m/s).
• Nature: Scalar quantity (only magnitude, no direction).
• Always Positive: Since distance is a scalar, average speed cannot be negative.

Average Velocity

• Definition: Total displacement divided by total time.
• Notation: Denoted by ( \overline{v} ).
• Mathematical Formula:
  • Change in position divided by change in time.
  • ( \frac{\text{Final Position} - \text{Initial Position}}{\text{Final Time} - \text{Initial Time}} )
• Nature: Vector quantity (has both magnitude and direction).
• Units: Meters per second (m/s).

Differences Between Average Speed and Average Velocity

• Average Speed: Concerned with total path traveled.
• Average Velocity: Concerned with the straight-line displacement between initial and final position.

Examples and Calculations

Example 1: Particle Moving in a Rectangle

• Scenario: Starts at Point A, returns to Point A after moving in a rectangle.
• Total Distance: 4m + 3m + 4m + 3m = 14m.
• Total Time: 7 seconds.
• Average Speed: ( \frac{14m}{7s} = 2 , \text{m/s} ).
• Average Velocity:
  • Displacement is 0 (returns to start).
  • ( \frac{0}{7s} = 0 , \text{m/s} ).

Example 2: Two Persons Moving in Opposite Directions

• Person 1:
  • Distance: 5m, Time: 2s, Average Speed: 2.5 m/s.
  • Displacement: 5m, Average Velocity: 2.5 m/s.
• Person 2:
  • Distance: 5m, Time: 2s, Average Speed: 2.5 m/s.
  • Displacement: -5m (opposite direction), Average Velocity: -2.5 m/s.
• Conclusion: Average speed is the same; average velocity differs by direction.

Example 3: Movement Between Two Points

• Scenario: Movement from Point A to Point B.
• Total Distance: 3m + 4m = 7m.
• Total Time: 2s + 3s = 5s.
• Average Speed: ( \frac{7m}{5s} = 1.4 , \text{m/s} ).
• Average Velocity:
  • Displacement: ( \sqrt{3^2 + 4^2} = 5m ).
  • ( \frac{5m}{5s} = 1 , \text{m/s} ).
• Direction:
  • Using triangle, ( \sin(\theta) = \frac{4}{5} ).
  • Angle ( \theta = 53.13^\circ ).
  • This indicates direction of travel at the angle from horizontal axis.

Conclusion

• Average Speed: Always consider the total path and is always positive.
• Average Velocity: Considers straight-line distance from start to end and includes direction.
• Vector vs Scalar: Understanding the difference is crucial for solving physics problems involving motion.