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Figure 3.7. Comparison of speed versus time and distance versus time graphs for uniform motion. Note that the distance traveled at any time is equal to the area under the left-hand graph up to that time.

Now consider the case of uniformly accelerated motion shown in Fig. 3.8. The speed versus time curve (Fig. 3.8a) is a straight line with slope. As discussed earlier, the speed is increasing at a uniform (constant) rate. The distance traveled in any time interval is, as discussed above, equal to the area under the line over that time interval. Note that if we consider two equal time intervals, say At, and At2, at different times (early and later), the corresponding areas under the line are quite different. The area corresponding to the later time interval is larger, simply reflecting the fact that the speed is greater.

The corresponding distance versus time graph is shown in Fig. 3.8b. The distances traveled during the time intervals Atx and At2 are represented by the sep-

Figure 3.8. Comparison of speed versus time and distance versus time graphs for uniformly accelerated motion.

arations between the pairs of horizontal lines. The distance traveled per unit time increases with time. Note that the slope (i.e., the "steepness") of the distance versus time curve is just the speed. The distance versus time graph for uniformly accelerated motion is seen to be an upward curve rather than a straight line. The precise shape of the curve can be found by carefully finding the area under the speed versus time curve as time increases. For example, suppose the time intervals Aty and At2 are both taken to be 12 minutes, or \ hour. In the time interval Atx, the average speed is seen to be 10 mph, so that the car would travel about 10 mi/ h X 5 hour = 2 miles. In the time interval At2, the average speed is seen to be 30 mi/h, so that the car would travel about 30 mi/h X | = 6 miles. These respective distances are shown in Fig. 3.8b, corresponding to these two time intervals.

The curve shown in Fig. 3.8b is called a quadratic; that is, it turns out to be precisely described by the mathematical formula d^ t2

where oc is the mathematical symbol meaning is proportional to. Thus the formula (and the curve of Fig. 3.8b) says that for uniformly accelerated motion, if the time traveled is doubled, the total distance covered is quadrupled (increases four times), or if the time is tripled, the distance increases nine times, and so on. The mathematics of calculus confirms this result.

These various developments were extremely important because they made it possible, for the first time, to analyze motion precisely and in detail, whether it be falling bodies or projectiles, in terms of specific kinds of motion. The definitions of uniform and uniformly accelerated motions were critical to properly describe motion, so that the underlying nature of the motions could be discovered. The graphical representations are needed in order to properly identify these different types, as we have seen above. Thus the stage was finally set for someone to determine precisely the kinds of motion involved with falling bodies and projectiles.

2. Falling Bodies

Galileo Galilei (1564-1642) first determined exactly how objects fall, that is, with what kind of motion. Galileo was born and raised in northern Italy just after the height of the Italian Renaissance. He demonstrated unusual abilities in several areas as a young man (he probably could have been a notable painter), and studied medicine at the University of Pisa, one of the best schools in Europe. But Galileo early became more interested in mathematics and "natural philosophy" (i.e., science), and displayed so much ability that he was appointed professor of mathematics at Pisa while still in his mid-20s. Following some difficulties with the other faculty at Pisa (he was aggressive, self-assured, and arrogant), Galileo, at age 28, moved to the chair of mathematics at Padua, where he stayed for nearly 20 years. He quickly established himself as one of the most prominent scholars of his time and made important contributions to many areas of science and technology.

Galileo Galilei. (Courtesy of Editorial Photocolor Archives.)

Galileo became convinced fairly early in his career that many of the basic elements of Aristotle's physics were wrong. He apparently became a Copernican rather quickly, although he did not publish his beliefs regarding the structure of the heavens until he used the telescope to study objects in the night sky about 1610, when he was almost 46. More important for this chapter, Galileo also became convinced that Aristotle's descriptions of how objects fall and why projectiles keep moving were quite incorrect.

Apparently, Galileo became interested in how objects fall as a young man, while still a student at Pisa (although it is not true that he performed experiments with falling objects from the Leaning Tower of Pisa). He realized that it was necessary to "slow down" the falling process so that one could accurately measure the motion of a falling object. Galileo also realized that any technique devised to solve this problem must not change the basic nature of the "falling" process.

Aristotle's solution to the problem of how to slow down the falling process (so as to be able to study it) was to study falling objects in highly resistive media, such as water and other liquids. From these studies, Aristotle observed that heavy objects always fall faster than lighter ones and that objects fall with uniform speeds. He did observe that heavy and light objects fall at more nearly the same speeds in less resistive media, but felt they would have the same speeds only in a vacuum, which he rejected as being impossible to obtain. Therefore he felt that the medium was an essential part of the falling process. He concluded that a basic characteristic of the falling process was that heavy and light objects fall at different speeds.

Galileo, on the other hand, felt very strongly that any resistive effects would mask the basic nature of falling motion. He believed that one should first determine how objects fall with no resistive effects, then consider separately the resistive medium effects, and add the two together to get the net result. The important point here is that Galileo felt the basic nature of the falling process did not include these resistive medium contributions. Thus he wanted to find some other way to slow down falling without changing the basic nature of the process.

Galileo devised two methods that he felt satisfied his criteria. The first, which he began to study as a student, was a pendulum. The second method was that of a ball rolling down an inclined plane. In both of these phenomena, he believed that the underlying cause of the motion was the same as that responsible for falling (i.e., gravity). Galileo argued that the motions involved should have the same basic characteristics, but they should be somewhat diminished for pendulums and rolling balls on inclined planes. He obtained the same results from both methods.

Let us consider his second method, the inclined plane, in some detail. By choosing an inclined plane without too much slope, one can slow down the process in order to observe it. Galileo did more than just observe the motion of the rolling ball, however; he carefully studied how far the ball would travel in different intervals of time (i.e., he determined the distance versus time graph for the rolling ball). Galileo did this because he knew how to identify the kind of motion involved from such a graph. (Galileo claimed to be the first to define uniform and uniformly accelerated motions, although he was not.)

Such measurements were not easy for Galileo. For example, there were no accurate stopwatches at that time, so he had to devise ways to measure equal intervals of time accurately. He reportedly started measuring time intervals simply by counting musical beats maintained by a good musician. His final technique was to use a "water" clock, which measured time by how much water (determined by weighing) would accumulate when allowed to drain at a fixed rate.

The results of Galileo's inclined-plane experiments are represented (in a simplified form) in Table 3.1, and are plotted in Fig. 3.9. The distance traveled at the end of the first unit of time is taken to define one unit of distance. The succeeding distances are seen to follow a quadratic dependence on the time, that is, d oc t2

Table 3.1. Total Distance Traveled After Different Times for the Inclined Plane Experiment

Time Elapsed

Distance Traveled

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