## Illustration 2.3: Average and Instantaneous Velocity

*Please wait for the animation to completely load.*

When an object's velocity is changing, it is said to be accelerating.
In this case, the average velocity over a time interval is (in general) not equal to
the instantaneous velocity at each instant in that time interval. So how do we
determine the instantaneous velocity? Play the first animation where the toy Lamborghini's velocity is changing (increasing) with time
**(position is given in centimeters and time is given in seconds)**.
Restart.

Click the "show rise, run, and slope" button. The slope of the blue-line segment represents
the Lamborghini's average velocity, **v**_{avg}, during the time interval
(5 s, 10 s). What is
the Lamborghini's average velocity during the time interval (6 s, 9 s)? It is the slope of the new
line segment shown when you enter in 6 s for the start and 9 s for
the end and click the "show rise, run, and slope*" *button.

*When you get a good-looking graph, right-click on it to
clone the graph and resize it for a better view.*

What is the
Lamborghini's average velocity, **v**_{avg}, during the time interval
(7 s, 8 s)? How about the average velocity during the time interval (7.4
s, 7.6 s)? As the time interval gets smaller and smaller, the average
velocity approaches the instantaneous velocity as shown by the following Instantaneous Velocity Animation.

The instantaneous velocity therefore is the slope of the
position vs. time graph at any time. If you have taken calculus, you
know that this slope is also the derivative of the function shown, here x(t).
The Lamborghini moves according to the function: x(t) = 1.0*t^{2}, and
therefore v(t) = 2*t, which is the slope depicted in the Instantaneous Velocity Animation.

© 2004 by Prentice-Hall, Inc. A Pearson
Company