2.1.3: Average and Instantaneous Velocity
- Page ID
- 58859
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Illustration 3: Average and Instantaneous Velocity
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, \(v_{\text{avg}}\), during the time interval \((5\text{ s},\: 10\text{ s})\). What is the Lamborghini's average velocity during the time interval \((6\text{ s},\: 9\text{ s})\)? It is the slope of the new line segment shown when you enter in \(6\text{ s}\) for the start and \(9\text{ s}\) 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, \(v_{\text{avg}}\), during the time interval \((7\text{ s},\: 8\text{ s})\)? How about the average velocity during the time interval \((7.4\text{ s},\: 7.6\text{ s})\)? 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: \(x(t) = 1.0\ast t^{2}\), and therefore \(v(t) = 2\ast t\), which is the slope depicted in the Instantaneous Velocity Animation.