Overview
This lecture reviews how derivatives of position and velocity functions relate to velocity and acceleration, using units and examples to clarify these concepts.
Position, Velocity, and Acceleration Functions
- If ( s(t) ) is a position function, then ( s'(t) ) is the velocity function.
- Velocity describes the rate of change of position; its units are distance per time (e.g., feet per second).
- The slope of the position function at any point gives the instantaneous velocity at that moment.
- Sometimes, velocity is denoted as ( v(t) = s'(t) ).
- The derivative of the velocity function (( v'(t) ) or ( s''(t) )) gives the acceleration function.
- Acceleration describes the rate of change of velocity; its units are velocity per time (e.g., feet per second per second).
Interpreting Velocity and Acceleration
- A positive change in velocity over time indicates acceleration (speeding up).
- A negative change in velocity (negative acceleration) is called deceleration (slowing down).
- Example: If ( v(1) = 10 ) ft/s and ( v(2) = 15 ) ft/s, velocity increased by 5 ft/s over 1 second.
- The acceleration in this example is ( 5 ) ft/s² (5 feet per second per second).
Key Terms & Definitions
- Position Function (( s(t) )) — Represents the location over time.
- Velocity Function (( v(t) ) or ( s'(t) )) — Rate of change of position; units are distance/time.
- Acceleration Function (( a(t) ) or ( v'(t) ) or ( s''(t) )) — Rate of change of velocity; units are distance/time².
- Deceleration — Negative acceleration (slowing down).
Action Items / Next Steps
- Review the definitions and relationships between position, velocity, and acceleration.
- Practice finding and interpreting derivatives of position and velocity functions.