Instantaneous Velocity

We can define the instantanous velocity as a limit of an average velocity, as the time interval gets smaller and smaller. Let s (t) be the position of an object at time t. The instantaneous velocity at t = a is defined as .

This device cannot display Java animations. The above is a substitute static image
 See About the calculus applets for operating instructions.

## 1. Graph of height vs time

Initially, the applet shows a graph of height above the ground versus time, for the object that we examined in the previous page. Note that this graph is not a drawing of the path of the object, but is a graph of height versus time. The object was actually tossed straight up and fell straight back down.

## 2. Velocity at 1 second

Select the second example from the drop down menu. This just zooms in on the interval between 0 and 3 seconds. Notice the green line, which extends from the green dot at the point (2,142) to the red dot at the point (1,90). What is the slope of this green line? It's just rise over run, or But this is also the average velocity over the interval from 1 to 2 seconds. In other words, we can visualize the average velocity over an interval as the slope of the secant line between the endpoints of that interval. The slope of the green secant line is displayed in a small box on the graph.

What we want to find out is the instantaneous velocity at t = 1 second. We can approach this just like on the previous page by making the interval smaller. Click-drag the green dot closer to the red dot (zooming with the mouse is turned off on this applet so you don't accidentally zoom when you really meant to click on a dot). Notice that the slope is increasing as the green dot approaches the red dot.

In fact, the slope of the secant is approaching the slope of the red line, which is tangent to the curve at the point (1,90). The slope of this tangent line is 68, which is the instantaneous velocity at t = 1. We can think of instantaneous velocity as the slope of the tangent line at a point on our position curve, just like average velocity is the slope of the secant line.

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## Acknowledgements

Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.