AP® Physics 1: Algebra-Based · Unit 1 · Topic 1.2
Average and Instantaneous Velocity
Average velocity is total displacement divided by total time elapsed; instantaneous velocity is how fast something is moving at one specific moment, found as the slope of the tangent line on a position-time graph.
A speedometer shows instantaneous velocity: your speed right now, at this exact tick of the clock. Average velocity is a different question entirely: over the whole trip, what single velocity would have gotten you the same displacement in the same total time? If you drive 100 km in 2 hours but spend 20 minutes stuck at a red light, your speedometer bounces between 0 and 100 km/h, but your average velocity is just 50 km/h, total displacement over total time. The speedometer and the average almost never agree because the average erases every speed-up, slow-down, and stop in between.
To get from average to instantaneous, shrink the time interval. Average velocity over a huge chunk of time can hide a lot of changing motion, but average velocity over a tiny sliver of time, a millisecond, barely has room to change at all. Keep shrinking that interval toward zero, and the average velocity converges on the instantaneous velocity at that single moment. On a position-time graph, average velocity over an interval is the slope of the secant line connecting the start and end points. As the interval shrinks, that secant line rotates and settles into the tangent line at one point, and its slope is the instantaneous velocity there.
This distinction sets up everything that follows in kinematics. Velocity-time graphs are built point by point from instantaneous velocities, and the area under them gives displacement, the same quantity average velocity is built from. Acceleration is defined the same way velocity was: average acceleration over an interval, instantaneous acceleration as the slope of a tangent to the velocity-time graph as the interval shrinks to zero.
Key terms
- Average velocity
- Total displacement divided by total elapsed time, regardless of how speed varied in between.
- Instantaneous velocity
- The velocity at one specific instant, found as the limit of average velocity as the time interval shrinks to zero.
- Secant line
- A line connecting two points on a position-time graph, whose slope equals average velocity over that interval.
- Tangent line
- A line touching a curve at a single point, whose slope equals the instantaneous velocity there.
Watch it explained
- How it worksHow does shrinking the time interval turn an average velocity into an instantaneous one?
- How it worksWalk me through reading instantaneous velocity as the slope of a tangent line.
- Worked exampleFind the average velocity of a car that covers 150 m in 12 s and then sits still for 8 s.
- Common mistakeWhy isn't average velocity just the starting speed and ending speed averaged together?
Check yourself
End of topic.