Average Velocity
Document Actions
Let's take a look at average velocity. If you recall from earlier mathematics studies, average velocity is just net distance traveled divided by time. For example, if an object is tossed into the air we might find the following data for the height in feet, y, of the object as a function of the time in seconds, t, where t = 0 is when the object is released upward.
This table says that the object was 6 feet above the ground when released,
90 feet after 1 second, etc. How fast was it traveling? We can divide the net
distanced traveled by the time to compute the velocity. For example, over the
first second the average velocity is 
. Over the last second, the
average velocity is 
. Note that velocity can be
negative to indicate downward motion (positive velocity is upward motion in
this problem). Speed is just the magnitude of the velocity (i.e.,
the absolute value, in our one-dimensional example). We can also compute the
average velocity over the entire interval as 
.
Now, what if we wanted to know the velocity at a specific time, say at t = 1? Clearly there is some velocity, as the object is moving, but to
compute velocity we need a ratio of distance traveled to time spent
traveling. We could find the average velocity over the interval from 1 to 2
seconds, which is just 
. But this isn't quite right, as
the object does seem to be speeding up. If we had another measurement of
height at a time closer to 1 than 2 seconds, we could get a better estimate
of the instantaneous velocity at t = 1. The following applet
illustrates this.
Try the following:
- The applet shows a subset of the table from above. At the bottom is a
value, h, with an input box and a slider to control it. The first
row of the table show the time at 1 second and the time at 1 + h seconds. The second row shows the height at these two times, while the
third row computes the average velocity over this interval. Note that the
initial view of the applet, with h = 1, just shows the average
velocity between 1 and 2 seconds, as we computed above.
- Move the slider to make h smaller. You are essentially picking a
smaller interval, so the average velocity should be a better
approximation to the instantaneous velocity at t = 1. Keep making h smaller and smaller. What is happening to the velocity? Is it
approaching some number?
- Make the velocity 0 by moving the slider all the way to the left. What
happens to the velocity? Why?
- Type in 0.001 for h and press Enter. Then try 0.0001, etc. What
number is the velocity approaching as h approaches 0?
- Note that if you set h less than or equal to 0.000000001, round off errors start causing problems
For more information on rights and downloading, refer to http://www.calculusapplets.com/download.html.
© Copyright 2001 David J. Eck
© Copyright 2007 Thomas S. Downey
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
You must cause any files that you modify to carry prominent notices stating that you changed the files and the date of any change, and modified files must be put into a Java package different from edu.hws.
THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
