Overview
This lecture explains the difference between instantaneous and average velocity, discusses methods to determine instantaneous velocity, and introduces the concept of calculus in motion.
Instantaneous vs. Average Velocity
- Instantaneous speed is the speed of an object at a specific moment in time.
- Instantaneous velocity includes both magnitude (speed) and direction at a given instant.
- Average velocity is the total displacement divided by the total time taken for a trip.
- Instantaneous velocities at different moments can differ from the average velocity.
Calculating Instantaneous Velocity
- To approximate instantaneous velocity, examine very small displacements over very short time intervals centered on the point of interest.
- The perfect instantaneous velocity requires dividing an infinitesimally-small displacement by an infinitesimally-small time, leading to an indeterminate form (zero divided by zero).
- Calculus, invented by Newton, is used to rigorously solve for instantaneous velocity.
Methods Without Calculus
- If velocity is constant, average velocity equals instantaneous velocity at any time.
- On a position (x) vs. time (t) graph, the slope at any point equals the instantaneous velocity at that point.
- With constant acceleration, kinematic formulas can be used to find instantaneous velocity at any time.
Key Terms & Definitions
- Instantaneous Speed — Speed of an object at a precise moment.
- Instantaneous Velocity — Speed with direction at a specific instant.
- Average Velocity — Total displacement divided by total time taken.
- Slope (on x-t graph) — Represents instantaneous velocity at a given time.
- Calculus — Mathematical tool developed to handle infinitesimal changes; essential for exact instantaneous velocity.
- Kinematic Formulas — Equations used to describe motion with constant acceleration.
Action Items / Next Steps
- Practice distinguishing between average and instantaneous velocity in sample problems.
- Review how to interpret slopes on position-time graphs.
- Prepare to learn basic calculus concepts relevant to motion.