Lecture Notes: Average Velocity vs. Instantaneous Velocity

Key Concepts

• Average Velocity:

  • Formula: Total displacement divided by the total time.
  • Requires knowledge of the initial and final positions of the object.
  • Example Calculation:
    • Given position function: ( s(t) = 2 - 5t + 3t^2 )
    • To find average velocity between ( t = 1 ) second and ( t = 5 ) seconds:
      • Calculate position at ( t = 1 ):
        • ( s(1) = 2 - 5 \times 1 + 3 \times 1^2 = 0 ) meters
      • Calculate position at ( t = 5 ):
        • ( s(5) = 2 - 5 \times 5 + 3 \times 5^2 = 52 ) meters
      • Average velocity ( \bar{v} = \frac{s(5) - s(1)}{5 - 1} = \frac{52 - 0}{4} = 13.5 ) m/s

• Instantaneous Velocity:

  • Defined as the rate of change of position with respect to time.
  • Calculated using the derivative of the position function.
  • Example Calculation:
    • Derivative of the position function ( s(t) = 2 - 5t + 3t^2 ) is (-5 + 6t).
    • At ( t = 1 ):
      • ( v(1) = -5 + 6 \times 1 = 1 ) m/s
    • At ( t = 5 ):
      • ( v(5) = -5 + 6 \times 5 = 25 ) m/s

Summary

• Average Velocity is calculated by dividing the difference in position by the time interval.
• Instantaneous Velocity is obtained by taking the derivative of the position function relative to time.
• Understanding both concepts is crucial for analyzing the motion of objects, especially when considering varying speeds over time.

Additional Notes:

• For further understanding of derivatives, refer to additional resources or instructional videos on calculus.
• Remember the different applications of average and instantaneous velocities in physical problems.