Overview
This lecture explains the concepts of instantaneous velocity and speed, showing how they differ from average velocity and how to calculate them using calculus and position-time equations.
Instantaneous vs. Average Velocity
- Average velocity is total displacement divided by elapsed time between two positions.
- Instantaneous velocity is the velocity at a specific moment, found as the derivative of position with respect to time.
Calculating Instantaneous Velocity
- To compute instantaneous velocity, use ( v(t) = \frac{dx(t)}{dt} ).
- The instantaneous velocity at time ( t_0 ) is the slope of the position-time curve at that point.
- At maxima or minima of the position function, instantaneous velocity is zero.
Example: Position-Time and Velocity-Time Graphs
- The slope of a straight line segment on a position-time graph gives the constant velocity during that interval.
- Positive slope = positive velocity, flat line = zero velocity, negative slope = negative velocity.
- When the position reverses direction, velocity changes sign.
Speed vs. Velocity
- Velocity is a vector (direction and magnitude); speed is a scalar (only magnitude).
- Average speed = total distance/elapsed time, not always equal to the magnitude of average velocity.
- Instantaneous speed is the magnitude of instantaneous velocity: ( |v(t)| ).
Example Calculations
- Use the power rule (( \frac{d}{dt}At^n = An t^{n-1} )) to differentiate polynomial position functions.
- Position ( x(t) = (3.0, m/s)t + (0.5, m/s^3)t^3 ): instantaneous velocity at ( t = 2.0, s ) is ( v(2.0, s) = 9.0, m/s ).
- Average velocity between 1.0 s and 3.0 s is ( 9.5, m/s ).
- Position ( x(t) = (3.0, m/s)t - (3, m/s^2)t^2 ): velocities and speeds at given times are found by differentiation and taking the absolute value.
Graphical Interpretation
- The slope of a position-time graph at a point gives the instantaneous velocity.
- Speed is always non-negative, while velocity can be positive or negative.
Key Terms & Definitions
- Average Velocity — Displacement divided by elapsed time.
- Instantaneous Velocity — The derivative of position with respect to time at a specific instant.
- Speed — The magnitude of velocity; a scalar quantity.
- Power Rule — A calculus rule to differentiate terms of the form ( At^n ).
Action Items / Next Steps
- Practice differentiating position functions to find velocity.
- Answer Check Your Understanding 3.2: For ( x(t) = (3, m/s^2)t^2 ), find velocity as a function of time, its sign, and values at ( t=1.0, s ).