Kinematics and Dynamics
KINEMATICS AND DYNAMICS
Webster's defines physics as "a science that deals with matter and energy and their interactions." Alternatively, physics can be described as the study of matter and motion, or of matter in motion. Whatever the particulars of the definition, physics is among the most fundamental of disciplines, and hence, the rudiments of physics are among the most basic building blocks for thinking about the world. Foundational to an understanding of physics are kinematics, the explanation of how objects move, and dynamics, the study of why they move. Both are part of a larger branch of physics called mechanics, the study of bodies in motion. These are subjects that may sound abstract, but in fact, are limitless in their applications to real life.
HOW IT WORKS
The Place of Physics in the Sciences
Physics may be regarded as the queen of the sciences, not because it is "better" than chemistry or astronomy, but because it is the foundation on which all others are built. The internal and interpersonal behaviors that are the subject of the social sciences (psychology, anthropology, sociology, and so forth) could not exist without the biological framework that houses the human consciousness. Yet the human body and other elements studied by the biological and medical sciences exist within a larger environment, the framework for earth sciences, such as geology.
Earth sciences belong to a larger grouping of physical sciences, each more fundamental in concerns and broader in scope. Earth, after all, is but one corner of the realm studied by astronomy; and before a universe can even exist, there must be interactions of elements, the subject of chemistry. Yet even before chemicals can react, they have to do so within a physical framework—the realm of the most basic science—physics.
The Birth of Physics in Greece
THE FIRST HYPOTHESIS.
Indeed, physics stands in relation to the sciences as philosophy does to thought itself: without philosophy to provide the concept of concepts, it would be impossible to develop a consistent worldview in which to test ideas. It is no accident, then, that the founder of the physical sciences was also the world's first philosopher, Thales (c. 625?-547? b.c.) of Miletus in Greek Asia Minor (now part of Turkey.) Prior to Thales's time, religious figures and mystics had made statements regarding ethics or the nature of deity, but none had attempted statements concerning the fundamental nature of reality.
For instance, the Bible offers a story of Earth's creation in the Book of Genesis which was well-suited to the understanding of people in the first millennium before Christ. But the writer of the biblical creation story made no attempt to explain how things came into being. He was concerned, rather, with showing that God had willed the existence of all physical reality by calling things into being—for example, by saying, "Let there be light."
Thales, on the other hand, made a genuine philosophical and scientific statement when he said that "Everything is water." This was the first hypothesis, a statement capable of being scientifically tested for accuracy. Thales's pronouncement did not mean he believed all things were necessarily made of water, literally. Rather, he appears to have been referring to a general tendency of movement: that the whole world is in a fluid state.
ATTEMPTING TO UNDERSTAND PHYSICAL REALITY.
While we can respect Thales's statement for its truly earth-shattering implications, we may be tempted to read too much into it. Nonetheless, it is striking that he compared physical reality to water. On the one hand, there is the fact that water is essential to all life, and pervades Earth—but that is a subject more properly addressed by the realms of chemistry and the biological sciences. Perhaps of more interest to the physicist is the allusion to a fluid nature underlying all physical reality.
The physical realm is made of matter, which appears in four states: solid, liquid, gas, and plasma. The last of these is not the same as blood plasma: containing many ionized atoms or molecules which exhibit collective behavior, plasma is the substance from which stars, for instance, are composed. Though not plentiful on Earth, within the universe it may be the most common of all four states. Plasma is akin to gas, but different in molecular structure; the other three states differ at the molecular level as well.
Nonetheless, it is possible for a substance such as water—genuine H2 O, not the figurative water of Thales—to exist in liquid, gas, or solid form, and the dividing line between these is not always fixed. In fact, physicists have identified a phenomenon known as the triple point: at a certain temperature and pressure, a substance can be solid, liquid, and gas all at once!
The above statement shows just how challenging the study of physical reality can be, and indeed, these concepts would be far beyond the scope of Thales's imagination, had he been presented with them. Though he almost certainly deserves to be called a "genius," he lived in a world that viewed physical processes as a product of the gods' sometimes capricious will. The behavior of the tides, for instance, was attributed to Poseidon. Though Thales's statement began the process of digging humanity out from under the burden of superstition that had impeded scientific progress for centuries, the road forward would be a long one.
MATHEMATICS, MEASUREMENT, AND MATTER.
In the two centuries after Thales's death, several other thinkers advanced understanding of physical reality in one way or another. Pythagoras (c. 580-c. 500
b.c.) taught that everything could be quantified, or related to numbers. Though he entangled this idea with mysticism and numerology, the concept itself influenced the idea that physical processes could be measured. Likewise, there were flaws at the heart of the paradoxes put forth by Zeno of Elea (c. 495-c. 430 b.c.), who set out to prove that motion was impossible—yet he was also the first thinker to analyze motion seriously.
In one of Zeno's paradoxes, he referred to an arrow being shot from a bow. At every moment of its flight, it could be said that the arrow was at rest within a space equal to its length. Though it would be some 2,500 years before slow-motion photography, in effect he was asking his listeners to imagine a snapshot of the arrow in flight. If it was at rest in that "snapshot," he asked, so to speak, and in every other possible "snapshot," when did the arrow actually move? These paradoxes were among the most perplexing questions of premodern times, and remain a subject of inquiry even today.
In fact, it seems that Zeno unwittingly (for there is no reason to believe that he deliberately deceived his listeners) inserted an error in his paradoxes by treating physical space as though it were composed of an infinite number of points. In the ideal world of geometric theory, a point takes up no space, and therefore it is correct to say that a line contains an infinite number of points; but this is not the case in the real world, where a "point" has some actual length. Hence, if the number of points on Earth were limitless, so too would be Earth itself.
Zeno's contemporary Leucippus (c. 480-c. 420 b.c.) and his student Democritus (c. 460-370 b.c.) proposed a new and highly advanced model for the tiniest point of physical space: the atom. It would be some 2,300 years, however, before physicists returned to the atomic model.
Aristotle's Flawed Physics
The study of matter and motion began to take shape with Aristotle (384-322 b.c.); yet, though his Physics helped establish a framework for the discipline, his errors are so profound that any praise must be qualified. Certainly, Aristotle was one of the world's greatest thinkers, who originated a set of formalized realms of study. However, in Physics he put forth an erroneous explanation of matter and motion that still prevailed in Europe twenty centuries later.
Actually, Aristotle's ideas disappeared in the late ancient period, as learning in general came to a virtual halt in Europe. That his writings—which on the whole did much more to advance the progress of science than to impede it—survived at all is a tribute to the brilliance of Arab, rather than European, civilization. Indeed, it was in the Arab world that the most important scientific work of the medieval period took place. Only after about 1200 did Aristotelian thinking once again enter Europe, where it replaced a crude jumble of superstitions that had been substituted for learning.
THE FOUR ELEMENTS.
According to Aristotelian physics, all objects consisted, in varying degrees, of one or more elements: air, fire, water, and earth. In a tradition that went back to Thales, these elements were not necessarily pure: water in the everyday world was composed primarily of the element water, but also contained smaller amounts of the other elements. The planets beyond Earth were said to be made up of a "fifth element," or quintessence, of which little could be known.
The differing weights and behaviors of the elements governed the behavior of physical objects. Thus, water was lighter than earth, for instance, but heavier than air or fire. It was due to this difference in weight, Aristotle reasoned, that certain objects fall faster than others: a stone, for instance, because it is composed primarily of earth, will fall much faster than a leaf, which has much less earth in it.
Aristotle further defined "natural" motion as that which moved an object toward the center of the Earth, and "violent" motion as anything that propelled an object toward anything other than its "natural" destination. Hence, all horizontal or upward motion was "violent," and must be the direct result of a force. When the force was removed, the movement would end.
ARISTOTLE'S MODEL OF THE UNIVERSE.
From the fact that Earth's center is the destination of all "natural" motion, it is easy to comprehend the Aristotelian cosmology, or model of the universe. Earth itself was in the center, with all other bodies (including the Sun) revolving around it. Though in constant movement, these heavenly bodies were always in their "natural" place, because they could only move on the firmly established—almost groove-like—paths of their orbits around Earth. This in turn meant that the physical properties of matter and motion on other planets were completely different from the laws that prevailed on Earth.
Of course, virtually every precept within the Aristotelian system is incorrect, and Aristotle compounded the influence of his errors by promoting a disdain for quantification. Specifically, he believed that mathematics had little value for describing physical processes in the real world, and relied instead on pure observation without attempts at measurement.
Moving Beyond Aristotle
Faulty as Aristotle's system was, however, it possessed great appeal because much of it seemed to fit with the evidence of the senses. It is not at all immediately apparent that Earth and the other planets revolve around the Sun, nor is it obvious that a stone and a leaf experience the same acceleration as they fall toward the ground. In fact, quite the opposite appears to be the case: as everyone knows, a stone falls faster than a leaf. Therefore, it would seem reasonable—on the surface of it, at least—to accept Aristotle's conclusion that this difference results purely from a difference in weight.
Today, of course, scientists—and indeed, even people without any specialized scientific knowledge—recognize the lack of merit in the Aristotelian system. The stone does fall faster than the leaf, but only because of air resistance, not weight. Hence, if they fell in a vacuum (a space otherwise entirely devoid of matter, including air), the two objects would fall at exactly the same rate.
As with a number of truths about matter and motion, this is not one that appears obvious, yet it has been demonstrated. To prove this highly nonintuitive hypothesis, however, required an approach quite different from Aristotle's—an approach that involved quantification and the separation of matter and motion into various components. This was the beginning of real progress in physics, and in a sense may be regarded as the true birth of the discipline. In the years that followed, understanding of physics would grow rapidly, thanks to advancements of many individuals; but their studies could not have been possible without the work of one extraordinary thinker who dared to question the Aristotelian model.
Kinematics: How Objects Move
By the sixteenth century, the Aristotelian world-view had become so deeply ingrained that few European thinkers would have considered the possibility that it could be challenged. Professors all over Europe taught Aristotle's precepts to their students, and in this regard the University of Pisa in Italy was no different. Yet from its classrooms would emerge a young man who not only questioned, but ultimately overturned the Aristotelian model: Galileo Galilei (1564-1642.)
Challenges to Aristotle had been slowly growing within the scientific communities of the Arab and later the European worlds during the preceding millennium. Yet the ideas that most influenced Galileo in his break with Aristotle came not from a physicist but from an astronomer, Nicolaus Copernicus (1473-1543.) It was Copernicus who made a case, based purely on astronomical observation, that the Sun and not Earth was at the center of the universe.
Galileo embraced this model of the cosmos, but was later forced to renounce it on orders from the pope in Rome. At that time, of course, the Catholic Church remained the single most powerful political entity in Europe, and its endorsement of Aristotelian views—which philosophers had long since reconciled with Christian ideas—is a measure of Aristotle's impact on thinking.
GALILEO'S REVOLUTION IN PHYSICS.
After his censure by the Church, Galileo was placed under house arrest and was forbidden to study astronomy. Instead he turned to physics—where, ironically, he struck the blow that would destroy the bankrupt scientific system endorsed by Rome. In 1638, he published Discourses and Mathematical Demonstrations Concerning Two New Sciences Pertaining to Mathematics and Local Motion, a work usually referred to as Two New Sciences. In it, he laid the groundwork for physics by emphasizing a new method that included experimentation, demonstration, and quantification of results.
In this book—highly readable for a work of physics written in the seventeenth century—Galileo used a dialogue, an established format among philosophers and scientists of the past.
The character of Salviati argued for Galileo's ideas and Simplicio for those of Aristotle, while the genial Sagredo sat by and made occasional comments. Through Salviati, Galileo chose to challenge Aristotle on an issue that to most people at the time seemed relatively settled: the claim that objects fall at differing speeds according to their weight.
In order to proceed with his aim, Galileo had to introduce a number of innovations, and indeed, he established the subdiscipline of kinematics, or how objects move. Aristotle had indicated that when objects fall, they fall at the same rate from the moment they begin to fall until they reach their "natural" position. Galileo, on the other hand, suggested an aspect of motion, unknown at the time, that became an integral part of studies in physics: acceleration.
Scalars and Vectors
Even today, many people remain confused as to what acceleration is. Most assume that acceleration means only an increase in speed, but in fact this represents only one of several examples of acceleration. Acceleration is directly related to velocity, often mistakenly identified with speed.
In fact, speed is what scientists today would call a scalar quantity, or one that possesses magnitude but no specific direction. Speed is the rate at which the position of an object changes over a given period of time; thus people say "miles (or kilometers) per hour." A story problem concerning speed might state that "A train leaves New York City at a rate of 60 miles (96.6 km/h). How far will it have traveled in 73 minutes?"
Note that there is no reference to direction, whereas if the story problem concerned velocity—a vector, that is, a quantity involving both magnitude and direction—it would include some crucial qualifying phrase after "New York City": "for Boston," perhaps, or "northward." In practice, the difference between speed and velocity is nearly as large as that between a math problem and real life: few people think in terms of driving 60 miles, for instance, without also considering the direction they are traveling.
One can apply the same formula with velocity, though the process is more complicated. To obtain change in distance, one must add vectors, and this is best done by means of a diagram. You can draw each vector as an arrow on a graph, with the tail of each vector at the head of the previous one. Then it is possible to draw a vector from the tail of the first to the head of the last. This is the sum of the vectors, known as a resultant, which measures the net change.
Suppose, for instance, that a car travels east 4 mi (6.44 km), then due north 3 mi (4.83 km). This may be drawn on a graph with four units along the x axis, then 3 units along the y axis, making two sides of a triangle. The number of sides to the resulting shape is always one more than the number of vectors being added; the final side is the resultant. From the tail of the first segment, a diagonal line drawn to the head of the last will yield a measurement of 5 units—the resultant, which in this case would be equal to 5 mi (8 km) in a northeasterly direction.
VELOCITY AND ACCELERATION.
The directional component of velocity makes it possible to consider forms of motion other than linear, or straight-line, movement. Principal among these is circular, or rotational motion, in which an object continually changes direction and thus, velocity. Also significant is projectile motion, in which an object is thrown, shot, or hurled, describing a path that is a combination of horizontal and vertical components.
Furthermore, velocity is a key component in acceleration, which is defined as a change in velocity. Hence, acceleration can mean one of five things: an increase in speed with no change in direction (the popular, but incorrect, definition of the overall concept); a decrease in speed with no change in direction; a decrease or increase of speed with a change in direction; or a change in direction with no change in speed. If a car speeds up or slows down while traveling in a straight line, it experiences acceleration. So too does an object moving in rotational motion, even if its speed does not change, because its direction will change continuously.
Dynamics: Why Objects Move
To return to Galileo, he was concerned primarily with a specific form of acceleration, that which occurs due to the force of gravity. Aristotle had provided an explanation of gravity—if a highly flawed one—with his claim that objects fall to their "natural" position; Galileo set out to develop the first truly scientific explanation concerning how objects fall to the ground.
According to Galileo's predictions, two metal balls of differing sizes would fall with the same rate of acceleration. To test his hypotheses, however, he could not simply drop two balls from a rooftop—or have someone else do so while he stood on the ground—and measure their rate of fall. Objects fall too fast, and lacking sophisticated equipment available to scientists today, he had to find another means of showing the rate at which they fell.
This he did by resorting to a method Aristotle had shunned: the use of mathematics as a means of modeling the behavior of objects. This is such a deeply ingrained aspect of science today that it is hard to imagine a time when anyone would have questioned it, and that very fact is a tribute to Galileo's achievement. Since he could not measure speed, he set out to find an equation relating total distance to total time. Through a detailed series of steps, Galileo discovered that in uniform or constant acceleration from rest—that is, the acceleration he believed an object experiences due to gravity—there is a proportional relationship between distance and time.
With this mathematical model, Galileo could demonstrate uniform acceleration. He did this by using an experimental model for which observation was easier than in the case of two falling bodies: an inclined plane, down which he rolled a perfectly round ball. This allowed him to extrapolate that in free fall, though velocity was greater, the same proportions still applied and therefore, acceleration was constant.
POINTING THE WAY TOWARD NEWTON.
The effects of Galileo's system were enormous: he demonstrated mathematically that acceleration is constant, and established a method of hypothesis and experiment that became the basis of subsequent scientific investigation. He did not, however, attempt to calculate a figure for the acceleration of bodies in free fall; nor did he attempt to explain the overall principle of gravity, or indeed why objects move as they do—the focus of a subdiscipline known as dynamics.
At the end of Two New Sciences, Sagredo offered a hopeful prediction: "I really believe that… the principles which are set forth in this little treatise will, when taken up by speculative minds, lead to another more remarkable result…." This prediction would come true with the work of a man who, because he lived in a somewhat more enlightened time—and because he lived in England, where the pope had no power—was free to explore the implications of his physical studies without fear of Rome's intervention. Born in the very year Galileo died, his name was Sir Isaac Newton (1642-1727.)
NEWTON'S THREE LAWS OF MOTION.
In discussing the movement of the planets, Galileo had coined the term inertia to describe the tendency of an object in motion to remain in motion, and an object at rest to remain at rest. This idea would be the starting point of Newton's three laws of motion, and Newton would greatly expand on the concept of inertia.
The three laws themselves are so significant to the understanding of physics that they are treated separately elsewhere in this volume; here they are considered primarily in terms of their implications regarding the larger topic of matter and motion.
Introduced by Newton in his Principia (1687), the three laws are:
- First law of motion: An object at rest will remain at rest, and an object in motion will remain in motion, at a constant velocity unless or until outside forces act upon it.
- Second law of motion: The net force acting upon an object is a product of its mass multiplied by its acceleration.
- Third law of motion: When one object exerts a force on another, the second object exerts on the first a force equal in magnitude but opposite in direction.
These laws made final the break with Aristotle's system. In place of "natural" motion, Newton presented the concept of motion at a uniform velocity—whether that velocity be a state of rest or of uniform motion. Indeed, the closest thing to "natural" motion (that is, true "natural" motion) is the behavior of objects in outer space. There, free from friction and away from the gravitational pull of Earth or other bodies, an object set in motion will remain in motion forever due to its own inertia. It follows from this observation, incidentally, that Newton's laws were and are universal, thus debunking the old myth that the physical properties of realms outside Earth are fundamentally different from those of Earth itself.
MASS AND GRAVITATIONAL ACCELERATION.
The first law establishes the principle of inertia, and the second law makes reference to the means by which inertia is measured: mass, or the resistance of an object to a change in its motion—including a change in velocity. Mass is one of the most fundamental notions in the world of physics, and it too is the subject of a popular misconception—one which confuses it with weight. In fact, weight is a force, equal to mass multiplied by the acceleration due to gravity.
It was Newton, through a complicated series of steps he explained in his Principia, who made possible the calculation of that acceleration—an act of quantification that had eluded Galileo. The figure most often used for gravitational acceleration at sea level is 32 ft (9.8 m) per second squared. This means that in the first second, an object falls at a velocity of 32 ft per second, but its velocity is also increasing at a rate of 32 ft per second per second. Hence, after 2 seconds, its velocity will be 64 ft (per second; after 3 seconds 96 ft per second, and so on.
Mass does not vary anywhere in the universe, whereas weight changes with any change in the gravitational field. When United States astronaut Neil Armstrong planted the American flag on the Moon in 1969, the flagpole (and indeed Armstrong himself) weighed much less than on Earth. Yet it would have required exactly the same amount of force to move the pole (or, again, Armstrong) from side to side as it would have on Earth, because their mass and therefore their inertia had not changed.
The implications of Newton's three laws go far beyond what has been described here; but again, these laws, as well as gravity itself, receive a much more thorough treatment elsewhere in this volume. What is important in this context is the gradually unfolding understanding of matter and motion that formed the basis for the study of physics today.
After Newton came the Swiss mathematician and physicist Daniel Bernoulli (1700-1782), who pioneered another subdiscipline, fluid dynamics, which encompasses the behavior of liquids and gases in contact with solid objects. Air itself is an example of a fluid, in the scientific sense of the term. Through studies in fluid dynamics, it became possible to explain the principles of air resistance that cause a leaf to fall more slowly than a stone—even though the two are subject to exactly the same gravitational acceleration, and would fall at the same speed in a vacuum.
EXTENDING THE REALM OF PHYSICAL STUDY.
The work of Galileo, Newton, and Bernoulli fit within one of five major divisions of classical physics: mechanics, or the study of matter, motion, and forces. The other principal divisions are acoustics, or studies in sound; optics, the study of light; thermodynamics, or investigations regarding the relationships between heat and other varieties of energy; and electricity and magnetism. (These subjects, and subdivisions within them, also receive extensive treatment elsewhere in this book.)
Newton identified one type of force, gravitation, but in the period leading up to the time of Scottish physicist James Clerk Maxwell (1831-1879), scientists gradually became aware of a new fundamental interaction in the universe. Building on studies of numerous scientists, Maxwell hypothesized that electricity and magnetism are in fact differing manifestations of a second variety of force, electromagnetism.
The term classical physics, used above, refers to the subjects of study from Galileo's time through the end of the nineteenth century. Classical physics deals primarily with subjects that can be discerned by the senses, and addressed processes that could be observed on a large scale. By contrast, modern physics, which had its beginnings with the work of Max Planck (1858-1947), Albert Einstein (1879-1955), Niels Bohr (1885-1962), and others at the beginning of the twentieth century, addresses quite a different set of topics.
Modern physics is concerned primarily with the behavior of matter at the molecular, atomic, or subatomic level, and thus its truths cannot be grasped with the aid of the senses. Nor is classical physics much help in understanding modern physics. The latter, in fact, recognizes two forces unknown to classical physicists: weak nuclear force, which causes the decay of some subatomic particles, and strong nuclear force, which binds the nuclei of atoms with a force 1 trillion (1012) times as great as that of the weak nuclear force.
Things happen in the realm of modern physics that would have been inconceivable to classical physicists. For instance, according to quantum mechanics—first developed by Planck—it is not possible to make a measurement without affecting the object (e.g., an electron) being measured. Yet even atomic, nuclear, and particle physics can be understood in terms of their effects on the world of experience: challenging as these subjects are, they still concern—though within a much more complex framework—the physical fundamentals of matter and motion.
WHERE TO LEARN MORE
Ballard, Carol. How Do We Move? Austin, TX: Raintree Steck-Vaughn, 1998.
Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.
Fleisher, Paul. Objects in Motion: Principles of Classical Mechanics. Minneapolis, MN: Lerner Publications, 2002.
Hewitt, Sally. Forces Around Us. New York: Children's Press, 1998.
Measure for Measure: Sites That Do the Work for You (Web site). <http://www.wolinskyweb.com/measure.html> (March 7, 2001).
Motion, Energy, and Simple Machines (Web site). <http://www.necc.mass.edu/MRVIS/MR3_13/start.ht ml> (March 7, 2001).
Rutherford, F. James; Gerald Holton; and Fletcher G. Watson. Project Physics. New York: Holt, Rinehart, and Winston, 1981.
Wilson, Jerry D. Physics: Concepts and Applications, second edition. Lexington, MA: D. C. Heath, 1981.
A change in velocity.
The study of why objects move as they do; compare with kinematics.
The product of mass multiplied by acceleration.
A statement capable of being scientifically tested for accuracy.
The tendency of an object in motion to remain in motion, and of an object at rest to remain at rest.
The study of how objects move; compare with dynamics.
A measure of inertia, indicating the resistance of an object to a change in its motion—including a change in velocity.
The material of physical reality. There are four basic states of matter : solid, liquid, gas, and plasma.
The study of bodies in motion.
The sum of two or more vectors, which measures the net change in distance and direction.
A quantity that possesses only magnitude, with no specific direction. Mass, time, and speed are all scalars. The opposite of a scalar is a vector.
The rate at which the position of an object changes over a given period of time.
Space entirely devoid of matter, including air.
A quantity that possesses both magnitude and direction. Velocity, acceleration, and weight (which involves the downward acceleration due to gravity)are examples of vectors. Its opposite is ascalar.
The speed of an object in a particular direction.
A measure of the gravitational force on an object; the product of mass multiplied by the acceleration due to gravity. (The latter is equal to 32 ft or 9.8 m per second per second, or 32 ft/9.8 m per second squared.)