Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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To find where the average velocity of an object is positive or negative, simply find slopes between the given points (these points will be the time at which an object is at a certain position). By calculating the change in position over the change in time, this slope will be positive or negative, which will tell you whether the average velocity is positive or negative. When average velocity is positive, it usually means that the object is moving forward from its starting point. When average velocity is negative, it usually means that the object is moving backward. To find where the average velocity is greatest, simply find between which two points the slope is the greatest. Note that positive average velocities are greater than negative average velocities.

Let’s do one more average velocity problem. Here I have a graph representing the position of an object over time. Time is given in seconds and the position is given in meters. Although you’ll notice that on the position axis I’ve got no numbers indicated. So I’ll have to use just relative positions of these points to figure out what the position of the object actually is.

Here are the questions about this object. A over which interval is the average velocity negative between 0 to 10, 10 to 20, 20 to 30, or 30 to 40? And b, over which intervals is the average velocity greatest? 0 to 10, 10 to 20,20 to 30 and 30 to 40?

The way I’m going to answer that question is, I’m going to draw secant lines connecting each of these endpoints. Remember that the slope of the secant line between t equals 0 and t equals 10, gives you the average velocity between 0 and 10.

So I’ll draw that now. And the slope of this line looks like it’s 0. So I would say that the average velocity between 0 and 10 is 0. And average between 10 and 20, I’ll draw a secant line, between 20 and 30, and between 30 and 40. They're just at a glance. The slope of this segment and this segment, those slopes are both positive. This is the slope that’s negative. And that indicates that the average velocity between t equals 30 and t equals 40 is negative. So that’s my answer.

So for this interval, that the average velocity is a negative number. Remember this one had an average velocity of 0, and these two were positive. Now for part of b, over which intervals is the average velocity the greatest?

Let’s keep in mind our answer to part A. The average velocity was 0 here, it was negative here. So these two intervals are candidates for the greatest average velocity. Let’s take look at the graph again. It’s between this interval, between 10 and 20, and this interval between 20 and 30. But if you notice, this line segment has a slightly greater slope than this one. So I would say that between 10 and 20 the average velocity is the greatest. And that’s my answer.