AP Calculus AB and BC
Average Velocity vs Average Speed
Average velocity is displacement divided by elapsed time, so it can be zero or negative. Average speed is total distance divided by elapsed time, so it is never negative. They are equal only when the particle never changes direction.
Average velocity
Use when: The question asks for net movement per unit time, or you are applying the Mean Value Theorem to position.
Average speed
Use when: The question asks how fast on average, which needs the whole path travelled rather than the net change.
Side by side
| Average velocity | Average speed | |
|---|---|---|
| Formula | ||
| Uses | Displacement | Total distance |
| Can be zero on a moving particle | Yes | No |
| Can be negative | Yes | No |
| Equal to each other when | The particle never turns around | Same condition |
A particle that leaves home and returns has displacement zero, so its average velocity is zero no matter how far it went. Its average speed is not, and the gap between the two numbers is exactly the amount of doubling back.
Average velocity is a secant slope
Displacement over elapsed time is the slope of the secant line on the position graph, which is why the Mean Value Theorem says some instant matched it exactly. Average speed has no such interpretation.
Frequently asked questions
Can average velocity be zero while the particle moves?
Yes. If it returns to its starting position the displacement is zero, so the average velocity is zero even though the average speed is not.
When are they equal?
Only when the particle never changes direction over the interval, so distance and displacement agree.
Is average velocity the average of the velocity function?
Yes. It equals the average value of over the interval, which is the same number as displacement over elapsed time.
In the CED: Unit 4: Contextual Applications, Unit 8: Applications of Integration