The acceleration of gravity constant g has been determined from experiment to be g = 9.8 m/sec2, or 32 ft/sec2, where m/sec2 means meters per second per second. Thus, if an object is dropped from a large vertical height, we can calculate the speed and total vertical distance fallen using these equations. These equations tell us that the velocity (speed) of a falling object will increase 9.8 m/sec every second, so that the object keeps falling faster and faster. Because of this increased speed, the distance fallen increases faster and faster with time, at a rate proportional to the square of the time elapsed. The result is given in Table 3.0.1.

These results are correct if one neglects the resistance of the air for the falling object and are strictly valid only in a vacuum. A small dense object (e.g., a rock) will follow the speeds and distances of Table 3.0.1 for a few seconds before the resistance will slow down the rate of increase and the object reaches its terminal (final) velocity.

As part of Galileo's studies of objects rolling down inclined planes, he was able to arrive at two very important generalizations regarding motion. Both of these new insights applied directly to Aristotle's old problem of how to explain projectile motion.

Galileo noted that a ball rolling down an inclined plane would continually experience uniform acceleration (see Table 3.1), even for only a very small angle of inclination. He then wondered what would happen to a ball rolling up an inclined plane. He discovered that a ball started along a flat surface with uniform motion (constant speed) would experience uniform deceleration (i.e., negative acceleration) if it rolled up an inclined plane. Because these two conclusions appeared always to be true, even for very small angles of inclination, he concluded that as the angle of the plane was changed from being slightly downward to slightly upward, the acceleration must change from being slightly positive to slightly negative, and an angle must exist for which there is no acceleration. This angle, he concluded must correspond to a horizontal plane, with no slope whatsoever.

Galileo's analysis was extremely important, because, as he realized, if a ball rolling on a flat plane experienced no acceleration (or deceleration), then it would naturally roll along forever. That is, Galileo realized that a ball rolling along a perfectly level surface would naturally keep rolling with uniform speed, unless something (e.g., friction or upward inclined slopes) acted on the ball to cause it to slow down. Thus Galileo recognized that Aristotle's question: "Why do projectiles keep moving?'' although addressing a very important issue, was actually the wrong question. The question should be: "Why do projectiles stop moving?" Galileo realized that the natural thing for a horizontally moving object to do is to keep moving with uniform motion.

He then seized on this result to explain how the Earth could behave like other planets without requiring an engine to drive it (Chapter 2, Section C4). Galileo reasoned that horizontal was parallel to the surface of the Earth, and therefore circular. Thus it was natural for objects to travel on a circular path, without requiring an engine, unless there were some resistance due to a medium. But if the Earth were traveling in "empty space" (i.e., in a vacuum), then there would be no resistance, and the Earth could travel in a circular orbit forever, just as Copernicus had postulated. The Earth's atmosphere would also travel along with it, and thus there would also be no problem of high winds as a consequence of the Earth's motion. Unfortunately, Galileo's reasoning was somewhat fallacious—he had drawn the wrong conclusion from this aspect of his rolling ball experiments. The ball rolling on the flat plane was rolling in a straight line, not on a level surface parallel to the Earth's surface.1

The property of a body by which it keeps moving we now call inertia. Mass is a measure of this inertia. Perhaps the most dramatic example of inertia is the motion of a spacecraft in "outer" space. Recall how the astronauts traveled to the Moon. First, their spacecraft' 'blasted-off' from Cape Canaveral and put them into orbit around the Earth. Next, after determining that everything was ' 'A-OK," they fired their rocket engines for several minutes in order to leave their Earth orbit and start toward the Moon. Once they reached the proper speed and direction, they shut off the engines. From there, they were able to simply "coast," for about 3 days and 240,000 miles, to a rendezvous with the Moon. Sometimes (but not always), they needed to fire their engines for a few minutes at about the halfway point, not to change their speed but only to apply a ' 'mid-course correction' ' to their direction. The point is that the trip used the inertia of the spacecraft to keep it moving at a large constant speed (about 3000 miles per hour) all the way from the Earth to the Moon. It worked so well because there was essentially no resistance in the form of friction, which acted upon the craft in space.2

Thus Galileo answered Aristotle's question by pointing out that it is natural for a moving object to keep moving. Of course, Galileo did not address why this

'It might be speculated that this misunderstanding was the reason that Galileo failed to recognize the significance of Kepler's elliptical orbits. Galileo believed in circular orbits and that his experiments had proven their validity. It would be difficult to explain a noncircular orbit. Not until the time of Isaac Newton was the proper explanation given of the physics of planetary orbits.

2Actually the spacecraft slowed down somewhat until the Moon's gravitational attraction exceeded that of the Earth, and then the spacecraft speeded up somewhat.

is the natural state. But it was extremely important to first determine what nature does before it was possible to address questions of why things happen the way they do. Thus Galileo (just as he did for falling objects) had for the first time properly recognized what happens in horizontal motion. In fact, Galileo went on to specifically consider projectile motion, the example that so perplexed Aristotle.

Galileo realized that the tricky thing about projectile motion (i.e., thrown or struck objects) was that both horizontal and vertical motions were involved. He had already determined that ' 'pure' ' horizontal motion was uniform motion in a straight line. But how should these be combined in order to describe a projectile? Galileo was also faced with this question in his use of inclined planes and pendulums to study falling. Both of these systems involve combined horizontal and vertical motions. Because he found that balls rolling down inclined planes always exhibited uniformly accelerated motion, no matter what the slope of the plane (although the magnitude of the acceleration increases as the slope increases), Galileo guessed that the effects of falling were independent of the horizontal motion. Because he found that all of his experiments with both inclined planes and pendulums gave results consistent with this hypothesis, he eventually concluded that it was always true. Thus Galileo was led to what is known as the superposition principle, which states that for objects with combined vertical and horizontal motions, the two motions can be analyzed separately and then combined to yield the net result.

According to the superposition principle, projectile motion involves both horizontal motion and vertical motion, which can be analyzed separately. Thus the vertical motion is just falling motion—the same falling motion that one would expect from an object with no horizontal motion. Similarly, the horizontal motion is uniform motion with whatever horizontal velocity the object has initially. The resulting combined motion for an object originally thrown horizontally from some height is represented in Fig. 3.11.

Distance (feet)

Figure 3.11. Graphical representation of height versus distance for object thrown horizontally from a height.

Distance (feet)

Figure 3.11. Graphical representation of height versus distance for object thrown horizontally from a height.

The object will strike the ground in the same amount of time as an object simply dropped from the same height. The thrown object will move horizontally at its initial horizontal speed until it strikes the ground. Thus for example, if the horizontal speed of the object is 60 mi/h (or 88 feet per second), and it takes 2 seconds to fall to the ground (which would be the case if it starts from a height of 64 feet), then the object will travel 176 feet horizontally (88 ft/sec X 2 sec). Note that the superposition principle says that a bullet shot horizontally from a high-powered rifle, and a bullet dropped from the same initial height will hit the ground at the same time! The bullet fired from the rifle simply has a very large horizontal velocity and is able to move a rather large horizontal distance in the short amount of time available. (This result clearly does not apply to an object that "flies" in the air or in any way derives some "lift" from its horizontal motion. It is strictly valid only in a vacuum.)

Galileo also applied the concept of projectile motion to an apple (for example) falling from a tree fixed to the moving surface of the Earth. The apple maintains its "forward motion" while falling and thus does not get left behind, as had been claimed by some of the opponents of the heliocentric theory.

The net result of all of Galileo's contributions to the development of the science of mechanics is great. He determined exactly how objects fall and introduced the idea of inertia for moving objects. He was able to correct Aristotle's conceptual analysis of falling bodies and was able to "answer" his question of why projectiles keep moving. However, Galileo did not give the why's of the motions. As discussed below, it was Isaac Newton who provided the "why" for falling motion.

C. Logic, Mathematics, and Science

Before discussing Newton's contributions to the science of mechanics, it is necessary to discuss the difference between inductive and deductive logic, to consider the value and use of mathematics in science, and to indicate what is basically involved in the branches of mathematics known as analytical geometry and calculus. This will provide a better understanding of Newton's contributions and how he made them. These discussions will be useful also for the material in subsequent chapters.

There are two principal kinds of logic used in science. When one performs, or collects the results from, a series of experiments and extracts a general rule, the logic is referred to as inductive. For example, if it is concluded from several carefully performed surveys that people prefer to watch mystery shows above all others, inductive logic is being used. The general rule was obtained directly from experiment and not through any line of reasoning starting from certain basic principles.

The famous English philosopher Francis Bacon (1561-1626) was a rather extreme proponent of the inductive method. He believed that if one were to carefully list and categorize all available facts on a certain subject, the natural laws governing that subject would appear obvious to any serious student. While it is certainly true that Bacon made several important contributions to science in this manner in some of his particular studies and experiments, it is also true that he did not accomplish one of his stated goals in life—namely, to rearrange the entire system of human knowledge by application of the power of inductive reasoning. He tried to rely too heavily on this one kind of logic.

The other principal type of logic is deductive reasoning. In this style of logic one proceeds from a few basic principles or theories and logically argues that certain other results must follow. Thus if one theorizes that all people love to read a good mystery and argues that therefore mysteries will be popular television shows, then one is using deductive logic. Perhaps the strongest proponent of the deductive method was the French philosopher René Descartes (1596-1650). He believed that one could not entirely trust observations (including experiments) to reveal the truths of the universe. Descartes believed that one should start from only a very few, irrefutable principles and argue logically from them to determine the nature of the universe. Although Descartes made many important contributions to philosophy and mathematics (he developed much of the subject of analytical geometry, which is discussed below), because of his insistence on starting with a few basic principles not obtained from observation, he was not able, as he wished, to obtain a consistent description of the physical universe.

The proper use of the inductive and deductive methods of logic is in combination. Newton was quite skilled at this combined technique. He started with certain specific observations and generalized from these observations to arrive at a theory indicating the general physical law that could explain the phenomenon. Once he arrived at a theory, he would deduce consequences from it to predict new phenomena that could be tested against observation. If the predictions were not upheld, he would try to modify the theory until it was able to provide successful descriptions of all appropriate phenomena. His theories always started from observations (the inductive method) and were tested by predicting new phenomena (the deductive method). Newton properly saw both methods as necessary in determining the physical laws of nature.

Before discussing some of the basic mathematical concepts that were developed about the time of Galileo and Newton (some by Newton himself) and that were necessary for Newton's synthesis of the science of mechanics, it is appropriate to discuss in a general way why mathematics is often used and even needed to describe physical laws.

As described succinctly by Schneer (see reference at the end of this chapter), mathematics may be said to be the science of order and relationship. Thus to the extent that the physical universe is orderly and has related parts, mathematics can be applied to its study. It is, perhaps, somewhat surprising that the universe is ordered and has related parts. The universe might not have been ordered and may have had no related parts. Such a universe would be hard to imagine. Nevertheless, it is still amazing that the more we learn about the universe, the more we know that it is ordered and appears to follow certain rules or laws. (As Einstein said, "The most incomprehensible thing about the universe is that it is comprehensible.") The more ordered and related the universe is found to be, the more mathematics will apply to its description.

Developments in physics often have followed developments in mathematics. Occasionally some areas of mathematics owe their origin to the need to solve a particular problem, or class of problems, in physics. We have already discussed how Kepler was able to deduce the correct shape of the orbits of the planets because of his knowledge of conic sections, including the ellipse. He also made extensive use of logarithms, which had just been invented, in his many calculations. Similarly, in order for Galileo to determine the true nature of falling, he needed proper definitions of motion and certain simple graphical techniques. We will see increased dependence on mathematics as we continue, and must now introduce two new mathematical developments in order to discuss Newton's works.

In order to use mathematics to describe physical motions and shapes, it was necessary to find ways to combine algebra and geometry. Describing geometrical figures with mathematical formulas (and vice versa) is known as analytical geometry. As indicated above, this field was largely developed just before the time of Newton. The French philosopher-mathematician René Descartes was one of the principal persons involved in this work.

Most simple and many rather complex figures can be described by mathematical formulas. Figure 3.12 provides several examples. The circle is described by a rather simple formula, as is the ellipse (see Optional Section 3.2, below). Some of the more complicated patterns involve the use of trigonometric functions (sine, cosine, etc.) in their algebraic descriptions. It is not necessary here for us to understand how these algebraic descriptions are obtained—or even how they work. What is important, though, is to be aware that such algebraic descriptions of figures often exist and are known.

A particularly important class of geometrical figures for the discussions of this chapter are those called conic sections. A conic section figure is obtained by slicing through a cone, as shown in Fig. 3.13. If one cuts through the cone parallel to the base, one obtains a circle. If one cuts through the cone parallel to the side of the cone, a parabola is obtained. A cut across the cone but not parallel to the base or the side yields an ellipse, and finally a cut vertically but not parallel to the side yields an hyperbola. All of these figures are found in nature, as water ringlets in a pool, planetary orbits, comet trajectories, and so on.

The Witch of Agrtesi

Figure 3.12. Various mathematical formulae and their graphical representations.

(By permission from W.A. Granville, P.F. Smith, and W.R. Longley, Elements of the Differential and Integral Calculus, John Wiley & Sons, New York, 1962.)

Figure 3.13. Conic sections: (a) circle, (b) ellipse, (c) parabola, and (d) hyperbola.

Optional Section 3.2 The Formulae for a Line, a Circle, and an Ellipse

Perhaps the simplest example of analytic geometry is the formula for a line. This formula is just y — mx + b

This equation describes a straight line with slope m and that intercepts the y axis at value b.

Another simple example is the formula for a circle. This formula is x2 + y2 = R2

This formula describes a circle of radius R centered at the origin, that is, at x = y = 0. Let us consider a simple case. Let R = 4. We can assume various values for x, and then using this equation, determine the values for y. These values are shown in Table 3.0.2. If we plot these values on a simple x-y grid, we will get a circle. (Note that neither x nor y can be greater than 4, i.e., x or y cannot be greater than R.) Note also because of the squares, ±x yields the same value of y (which can also be ±y). This circle is plotted in Fig. 3.0.1.

Table 3.O.2. |
A Circle |
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