AP® Physics C: Mechanics · Unit 1 · Topic 1.2
Average Velocity and Average Acceleration
Average velocity is total displacement divided by total time, and average acceleration is total change in velocity divided by total time — both are overall rates, not a snapshot of any single instant.
A road trip that averages 60 km/h almost never sits at exactly 60 km/h because that number only describes the whole journey, not any moment inside it. You might cruise at 110 on the highway, then crawl at 20 through town. Average velocity throws away all that detail and just asks: where did you end up, where did you start, and how long did it take? Average velocity = displacement / elapsed time. Average acceleration works the same way: it's the total change in velocity divided by the total time, regardless of how jerky or smooth the speeding-up was in between.
On a position-versus-time graph, average velocity over an interval is the slope of the straight line (the secant line) connecting the starting point and the ending point. It doesn't matter if the actual path curved wildly between those points — the average only cares about the two endpoints. The same idea applies to a velocity-versus-time graph for average acceleration: draw the secant line between two points and its slope is the average acceleration over that stretch.
This sets up the next idea directly: if you slide the two endpoints closer and closer together, the secant line rotates until it becomes the tangent line at a single instant. That tangent slope is the instantaneous velocity (or acceleration) — the derivative. So average velocity and average acceleration are the coarse, two-point version of the same concept that instantaneous velocity and acceleration refine into a single moment, which is exactly where calculus enters kinematics.
Key terms
- Average velocity
- Total displacement divided by total elapsed time over an interval.
- Average acceleration
- Total change in velocity divided by total elapsed time over an interval.
- Secant line
- A straight line connecting two points on a graph, whose slope gives the average rate between them.
- Tangent line
- The line touching a curve at a single point, whose slope gives the instantaneous rate at that point.
Watch it explained
- How it worksHow does drawing a straight line between two points on a position graph give you an average velocity?
- Worked exampleFind the average acceleration of a car that speeds up from 8 m/s to 26 m/s in 4.5 seconds.
- Common mistakeWhy doesn't average speed equal the plain average of your starting speed and your ending speed?
- Going furtherWhat happens to an average velocity as the two endpoints slide closer and closer together?
Check yourself
End of topic.