Lecture on Average and Instantaneous Acceleration

Key Concepts

Average Acceleration

• Definition: Change in velocity over a given time interval.
  • Formula: ( \bar{a} = \frac{v_{f} - v_{i}}{t_{f} - t_{i}} )
  • Vector quantity (has both magnitude and direction).
  • Units: meters per second squared (m/s²).
  • Measured between two points in time.

Instantaneous Acceleration

• Definition: Acceleration at a specific point in time or position.
  • Measured as the first derivative of velocity with respect to time, ( a = \frac{dv(t)}{dt} ).
  • Alternatively, the second derivative of position.
  • Also a vector quantity with units of m/s².
  • Not dependent on time since it is measured at a specific moment.

Examples

Example 1

• Velocity Equation: ( v(t) = 5 - 10t )
• Average Acceleration:
  • Calculate between ( t = 2 ) s and ( t = 5 ) s.
  • Final velocity at ( t = 5 ) s: ( -45 ) m/s.
  • Initial velocity at ( t = 2 ) s: ( -15 ) m/s.
  • ( \bar{a} = \frac{-45 - (-15)}{5 - 2} = \frac{-30}{3} = -10 ) m/s².
• Instantaneous Acceleration:
  • ( a = \frac{dv(t)}{dt} = -10 ) m/s² (constant across time).

Example 2

• Velocity Equation: ( v(t) = 5 - 10t^2 )
• Average Acceleration:
  • Between ( t = 2 ) s and ( t = 5 ) s.
  • Final velocity at ( t = 5 ) s: ( -245 ) m/s.
  • Initial velocity at ( t = 2 ) s: ( -35 ) m/s.
  • ( \bar{a} = \frac{-245 - (-35)}{5 - 2} = \frac{-210}{3} = -70 ) m/s².
• Instantaneous Acceleration:
  • ( a = \frac{dv(t)}{dt} = -20t ).
  • At ( t = 2 ) s: ( -40 ) m/s².
  • At ( t = 5 ) s: ( -100 ) m/s².

Summary

• Average Acceleration is calculated over a time interval and provides a general rate of change.
• Instantaneous Acceleration measures the rate of change at a specific moment and can vary at different times.
• Both are essential for understanding motion dynamics.