Example 4.4.14

Effie, Ivan and Cleo are in a foot race. Figure 4.4.15 models the distance each has traveled in the first few seconds. Each runner takes a second to accelerate up to their running speed, but then runs at a constant speed. So they are then traveling with a constant rate of change, and the straight line portions of their graphs have a slope. Find each line's slope, and interpret its meaning in this context. What comparisons can you make with these runners?

Figure 4.4.15 A three-way foot race

We will draw slope triangles to find each line's slope.

Figure 4.4.16 Find the Slope of Each Line

Using Formula (4.4.1), we have:

In a time-distance graph, the slope of a line represents speed. The slopes in these examples and the running speeds of these runners are both measured in ms. Another important relationship we can see is that, the more sharply a line is slanted, the bigger the slope is. This should make sense because for each passing second, the faster person travels longer, making a slope triangle's height taller. This means that, numerically, we can tell that Cleo is the fastest runner (and Effie is the slowest) just by comparing the slopes \(4>3.5>2.666\text{.}\)