P HYSICS NOTES
Introduction to Dynamics—Explaining Motion
In physics we make a useful distinction between the description of motion and the cause of motion. The modern
name for the mathematical description of motion, without regard to causes, is kinematics. The term comes
form the Greek word kinema, meaning “movement.” We know this word through its English variation cinema—
motion pictures!
Kinematics—The description of motion, without regard to what causes the motion.
What we have been doing so far is kinematics. By using the concepts of space and time, and by defining position,
velocity, and acceleration, we have the language and concepts to say, for instance, that an object’s position is
changing at a constant rate, while moving in one direction (uniform velocity); or that an object is speeding up
quite rapidly (a large acceleration in the direction of motion). We’ve also described motion in two dimensions.
For example, a cannon ball rushes through the air in a parabolic arc when undergoing projectile motion—its
velocity in the x-direction is perfectly constant, while in the y-direction the ball slows to a momentary stop
before speeding up toward the ground. Each of these is a description of a particular variety of motion.
But kinematics tells us nothing about the cause of motion, or why an object is at rest, not moving at all. For
instance, why does an object moving along at a constant speed in some direction not suddenly speed up and
move about in whirling loops? And what is it, exactly, that causes the cannon ball to return to the ground, and in
the way that it does so? For the more fundamental task of understanding the cause of motion, we turn our
attention to dynamics.
Dynamics is the study of what causes motion. Here we try to explain why things move the way they do.
Dynamics joins with kinematics to form mechanics—the general study of motion. We will first take up
dynamics in a qualitative way, and later develop it quantitatively in subsequent discussions.
Dynamics—The study of what causes motion. Mechanics—The general study of motion.
Efforts to explain motion date to antiquity. To get a sense of the richness and history of this question of what
causes motion, let’s go back some 2300 years, and join the wonderful thinker Aristotle in Athens, Greece.
Aristotle (384 – 322 BC) wrote extensively about the nature of moving objects. One question he asked was “what
is the natural state of an object if left to itself?” Aristotle observed that moving objects around him would
always come to rest—wheels may roll down a hill but they all come to rest at the bottom; stones skipped across
the water will eventually stop and sink; carts pushed across the floor always come to a stop. Having observed
many such examples, Aristotle concluded that the natural state of an earthly object is to be at rest.
There are four categories in nature—
earth, water, air, and fire.
All earthly objects, when left alone,
will naturally come to rest. It is their
natural state to stop and stay put.
Aristotle
For Aristotle, an object at rest requires no explanation—it is doing precisely what comes naturally to it.
A moving object, on the other hand, is not in its natural state and thus requires an explanation. Why is
this object moving? What keeps it going and prevents it from being in its natural state?
Why does an arrow fly through the air once it is
no longer being pushed by the bowstring?
This was an important question for Aristotle. He believed
the continual motion required an explanation, a cause.
Aristotle believed that the elements—earth, water, air, and fire—each had their own natural place in the
hierarchy of the universe. Earth, the heaviest, belonged to the lowest position. Water was next, then air and last,
fire. Aristotle reasoned that if any of these are out of its hierarchical position, its natural motion would
be to return. These natural motions occurred in straight lines, toward or away from the earth.
Consider this. Take some earth (dirt) and drop it into a jar of water.
The earth will naturally sink to the bottom, and bubbles of air will
naturally make their way to the top. The water will remain in the middle.
Furthermore, if there is a thin layer of kerosene atop the water, and we
light this on fire, the flames will naturally move upward, above the air,
toward their position in the hierarchy.
Now how about that arrow? To explain the motion of the arrow, Aristotle invoked an interaction between the
arrow and the air. As the arrow moves through the air it creates a partial vacuum (absence of air) behind it. The
air, returning to its natural position, rushes in behind the arrow to fill the void, pushing the arrow and causing
the continued motion. What’s most important here is that Aristotle believed that in order for the arrow to
continue moving forward, there must be something pushing it along.
But there is a problem here. This view predicts that continual motion, without any effort, is perpetual in air,
while being impossible in vacuum! But this is not what we observe. In fact, there are quite a few things
difficulties with Aristotle’s view; for instance, how would Aristotle explain the difference in the motions of a box
being pushed across a table, and that same box given the same push when it was on ice? Nevertheless, Aristotle
certainly laid the foundation for observing nature and using reason to try to explain the world around him.
Aristotle’s explanation of motion lasted for nearly 2000 years. The whole picture changed when Galileo
came on the scene. Galileo (1564 - 1642) reopened the question of the “natural state” of objects. But he asked a
different set of questions (an approach which often leads to breakthroughs in science). Here’s an example of the
way Galileo reasoned:
Consider a boy who sleds down a hill, out onto a horizontal patch of snow. Even if the snow is
quite smooth, we observe that the sled will eventually come to rest.
What if the boy slides down the hill onto some very slick ice? Now the sled could slide for quite a
distance before coming to a stop.
Galileo’s genius was in this next step. He imagined the case where all sources of resistance to the motion
were removed—perfectly smooth ice and no air resistance. This of course is an imaginary scenario, but
it does help us to see the essence of what’s happening. When the sled reaches this perfectly smooth
ice, it will continue in its motion forever, moving in a straight line with no loss of speed!
Moreover, it will continue to move without anything pushing it along!
The natural state of an object—its behavior if free from external influences—
is straight line motion with constant speed. We call this uniform motion.
In other words, when left alone, a moving object will continue to move in the
same direction with the same speed forever!
It’s not constant motion that needs to be explained. Rather, why objects
don’t continue in constant motion is what needs to be explained. Why does
the sled, in reality, come to rest?
Galileo’s new viewpoint was that since the sled does, in reality, eventually come to a stop, it must not be free of
external influences. Today we call such influences that lead to deviations from uniform motion forces.
Notice that “at rest” has no special significance in Galileo’s view of motion—it is simply uniform motion
that happens to have a velocity of zero. This means that an object at rest, in the absence of external
influences, will remain at rest forever!