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2 Rates of Change
- Johns Hopkins University Press
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CHAPTER 2 Rates of Change This may be the most intellectually demanding chapter in the entire book. You don’t have to become expert in this material now, but you do have to get the basics right away. And you certainly will have to become expert before long if you want to master physics. Expect to revisit this chapter several times. Change is at the heart of physics. That’s because physics is the science that seeks to explain why anything happens at all! The physicist wants to know what makes any given situation develop as time progresses. In order to begin to answer this question, it is important to be able to understand change, and rates of change, intuitively as well as quantitatively. An example of a quantity that changes over time is your own height over the course of your life. Figure 4 shows a rough graph 4 0 12 0 24 36 48 60 72 84 8 12 16 20 24 28 32 60 70 80 90 Height,in. Time,yr FIGURE 4. My height versus time. 7 8 PART I DESCRIBING MOTION of how my height has changed over time and how it should continue to change over the decades to come (we’ll make the hopeful assumption that I live as long as the graph indicates!). Notice the way the horizontal scale of the graph changes at age 36. To the left of this point, each interval represents one year; to the right, each interval represents 2.5 years. Rate of Growth Examine the curve at the left-hand side of Figure 4, in the age range from 2 years to 4 years. Looking at the curve there, how would you describe the rate of growth between ages 2 and 4? . You might use a word like “fast.” (This kid is shooting up like a weed!) How would you describe the rate of growth between the ages of 20 and 25? . You might say, “No growth” or “Zero growth.” How would you describe the rate of growth between the ages of 70 and 90? . You might say, “Shrinking”—or, even better, “Negative growth.” You might also note that the negative growth during this period is “gradual,” or “slow,” only a few inches over the course of 30 years. These are all good intuitions, and we want to build on them so that we can be more quantitative. . Here are the basic terms we use to describe rates of change. When the quantity is going up, we say that the quantity is increasing.When the quantity is going down, we say that the quantity is decreasing.When the quantity is staying the same, we say that the quantity is, uh, staying the same.» WORKED EXAMPLE 1 Figure 5 shows a graph of soil temperature versus time over a 24-hour period, with t = 0 representing the time when the measurements first began to be recorded. During which times is the temperatureincreasing?Duringwhichtimesisthetemperaturedecreasing ? During which times is the temperature staying the same? [54.172.162.78] Project MUSE (2024-03-29 08:29 GMT) Chapter 2 Rates of Change 9 2 0 4 6 8 10 12 14 16 18 20 22 24 Time, hr 17 15 19 21 23 25 27 29 31 33 35 Temperature, °C FIGURE 5. Worked Example 1. Soil temperature versus time.» SOLUTION 1 The temperature is increasing from the beginning of the graph (t = 0) until just before t = 6 hr. The temperature is also increasing from just after t = 18 hr through the end of the graph (t = 24 hr). The temperature is decreasing from just after t = 6 hr until just before t = 18 hr. The temperature is staying the same only momentarily at t = 6 hr and t = 18 hr. ♦ Rate of Change = Slope of the Graph There is a simple but profound relationship between the rate at which a quantity is changing and the appearance of its graph. . In places where the curve is steep uphill, the quantity is increasing rapidly. . In places where the curve is flat—horizontal—the quantity is not changing at all. . In places where the curve is steep downhill, the quantity is decreasing rapidly. In these places, the rate of change is a negative number with a large absolute value. The rate at which a quantity is changing at any given moment equals the slope of the curve at that moment. Toillustratetheideathattherateofchangeequalstheslope, let me show you a rough sketch of the rate...