CHAPTER 2

Physics progresses not by revolutions, which do away with all that went before, but rather by evolutions, which exploit the best about what is already understood. Newton's laws will continue to be as true today, no matter what we discover at the frontiers of science. --Lawrence M. Krauss The Physics of Startrek

We take motion for granted. Virtually every activity we do in a day involves motion. There are many different kinds of motions that occur in nature, and that are caused by humans and machines. Riding a bicycle to school, walking to class, the liquid flowing into your cup of coffee, and the motion of the text book you accidentally drop as you dash to class, all represent various forms that motion can take. Many sports activities are based on motions of balls which travel through the air from one player to another, or move after being hit, rolled, kicked or thrown. What determines the paths that these moving objects follow? What path does the space shuttle follow as it is launched initially, and then orbits the Earth once every 90 minutes? How have we succeeded in sending spacecraft all the way to the planet Neptune, 4.5 billion kilometers from Earth?

The apparent motions of the Sun and the moon each day through the sky have intrigued humans for millennia, as have the more subtle apparent motions of the stars we see cross the sky at night. Not until Isaac Newton's brilliant insights in the seventeenth century were the motions of the moon and the planets understood. With Newton’s three Laws of Motion and his Law of Gravitation, the scientific study of motion began in earnest, and constituted the beginnings of modern science.

Analysis of the motions of objects has formed the basis of physical science ever since the times of Galileo and Newton, in the seventeenth century. These two scientists were among the first and foremost to understand motion, and, more importantly, what causes changes in motion. Newton vastly expanded our understanding of motion with his three Laws of Motion that have stood the tests of time for more than three centuries, and remain one of the most monumental advances in science. Here you will explore some of the same phenomena that Galileo and Newton studied in the 17th century. Be observant and alert to all of the subtleties in your experiments, and you, too, will discover some of the most fundamental aspects of motion and its causes.

What is motion? In science we define motion as the change in position with time. How fast an object changes position or the rate that an object’s position changes with time is called the speed. If we know both the direction in which an object moves and its speed, then the combined direction and speed are called the velocity of an object. When either the speed or the velocity of an object is zero, then there is no motion.

Personal Lab 1: What is Motion?

Horizontal and Vertical Motions

Consider the motions of a tennis ball. If you hold a ball a meter or so above the ground and release the ball, what happens? When you give a push to the ball on a horizontal surface, what happens?

When moving objects are carefully observed, as in a photograph obtained with a stroboscopic light that can momentarily "freeze" the motion at regular time intervals, we can perceive differences in the motions of a rolling and a falling tennis ball. You have analyzed motions in directions that are horizontal and vertical to the surface of the Earth. What are the characteristics that distinguish horizontal from vertical motion of objects in free-fall?

Which ball is moving faster just before it stops? Explain your reasoning. What causes the motion of each ball to stop? What causes the motion of each ball to begin? Make a list of all of the different characteristics between horizontal and vertical motions you can think of. What happens when you toss a tennis ball across the room? What path does the thrown ball follow? On what variables does the path of the tossed ball depend? On what variables does the speed of the tossed ball depend? Would the motion of a tossed tennis ball be different on the Moon? If so, in what ways do you think the path and/or the speed would be different? Explain your reasoning.

A ball rolling on a smooth, horizontal surface (with no friction) has uniform motion or constant speed, as evidenced by the evenly spaced images of the ball in the stroboscopic photograph. What would a graph of position vs. time look like for this tennis ball?

What kind of motion does the falling tennis ball have? The rate the ball is falling as depicted in the stroboscopic image shows an increasing distance between images taken at equally spaced time intervals. What do you conclude about the speed of the falling ball? Is the ball moving faster as it falls or slower? If the ball moves faster with time, then its motion is accelerated. If the ball moves slower, then its motion is decelerating or we can also say that the ball has a negative acceleration.

Horizontal Motion

Modeling the Data

In science we begin an experiment with a working hypothesis or a proposed model for some phenomenon about which we have raised a question, but do not know the answer. A hypothesis is our assumed, initial explanation which we wish to test by experiment. A hypothesis is in the form of a statement which is our initial attempt to explain the phenomenon in question. A model is like a hypothesis in that we begin with an assumed or proposed model which must be tested against experimental data. However, a model is much more powerful than a hypothesis. A model can be graphical, mathematical, a 3-D representation, a diagram, or an explanatory statement. Thus a model can be much more detailed, and can make more specific, more accurate and useful predictions. In physical science we use models far more than hypotheses.

In science we try to perceive patterns in phenomena, and then generalize the patterns, so that predictions can be made. A model serves as a way to generalize the patterns we discover in natural phenomena.

We can "model" the motion of the rolling tennis ball in a variety of ways. Here we are justified in ignoring complicating factors such as friction and air resistance, since these effects are negligible for the purpose of this discussion. The stroboscopic picture above could be considered a pictorial model of the ball’s motion. Your sketches of graphs depicting the position and the speed of the ball as a function of time constitute graphical models. Note that the models not only summarize the motion of the rolling ball, but can also be used to predict the motion of the ball in the past and in the future. Predicting phenomena is basic to doing science. We construct models to explain phenomena, and use these models to predict phenomena both in the past and the future. A model begins as a hypothetical idea, and after comparison with experimental data, the model is improved to better match the experimental results. Once an acceptable match between a model and the experimental data has been found, then the model can be used to make predictions about the phenomenon being studied.

A conceptual model might be a generalized descriptive statement such as: A tennis ball given a push on a horizontal, frictionless surface will roll in a straight line path with a constant speed in the direction of the original push.

Another way to model the tennis ball's motion is by using a mathematical model. In the case of the rolling tennis ball, the position, s, as a function of time can be expressed by s = vt, where v is the ball’s rate of change of position (or velocity), and the time, t, is any instant in time. Note that this mathematical model is merely the equation of the straight line in the graph of position vs. time which you sketched above.

Mathematical expressions can be very powerful models in science, because the mathematical expressions which describe natural phenomena are sometimes quite simple and accurate, such as in the case of the rolling tennis ball. This mathematical model describes the motion of any object moving with uniform motion in a straight line. With the mathematical model for the motion of the ball, we could predict the position of the ball at any time, if we knew its velocity (i.e. its rate of change of position.). For the motion of any object the change in position, s, and the corresponding change in time, t, are related to the velocity of an object by

Examine the stroboscopic picture of the rolling tennis ball. What pattern do you notice in the time intervals (set by the flashes of the light) between positions of the tennis ball? Motion described by the model above and by your graph which indicated a linear relation between s and t, indicates a constant velocity, and we refer to this kind of motion as uniform motion. This is the simplest kind of motion, and the model above may be applied to any object which is moving with uniform motion, or constant velocity. Note that the direction of the ball did not change during the photograph. What would be required to change the direction? the speed? For velocity to be constant both the speed and the direction must be constant. If you were to swing a ball attached to a string around in a circle maintaining the ball's motion at a constant speed, could you describe the velocity of the ball as constant?

Below are several graphs of position vs. time. Think about the motions which are represented by each graph. With your fingers moving along a table top, trace the motions indicated in each graph.

Vertical Motion

Suppose a student dropped a tennis ball to the ground, while a photographer with a camera equipped with a strobe electronic flash recorded the event. About how fast do you think the ball would fall? How much distance would the ball cover in 0.1 sec? in 0.2 sec? These are interesting scientific questions that could be posed for the "falling tennis ball phenomenon". However, are these the most relevant questions one could ask about the fall of the tennis ball that would lead to an improvement in our understanding of this natural phenomenon. Wouldn’t it powerful to be able to predict exactly how long it would take the ball to fall, and better yet to be able to characterize its motion at each moment all the way down. Did the ball fall with a constant speed? a variable speed? Did its speed become slower as the ball approached the ground? Did the ball’s speed become faster as it neared the ground? A priori (that is, before we perform an experiment) we can do no better than guess at the answers to these questions. If we wanted to subject the vertical motion of a falling tennis ball to scientific study, we would need to think of an experiment to perform that would help us determine the nature of the Ball’s downward motion as a function of time. Initially, we might begin without any idea of a model to propose.

The first step, then would be to think of as many possible explanations (hypotheses) of the "falling ball phenomenon" as we could. To get things started, we might propose the hypothesis that: The tennis ball’s speed decreases as the ball falls to the ground. This will become our "working" hypothesis. Hypotheses represent our best considered opinions for the explanation or description of some natural phenomenon. To become accepted, all scientific hypotheses must be subjected to an experimental test. Doing science involves performing experiments which test hypotheses or models. Usually a hypotheses (or a model) must be revised several times after an experiments are performed. Not until the hypothesis (or model) provides an acceptable match with the experimental results does a scientist accept the validity of the hypothesis (or model).

The experiment we might propose to test our working hypothesis or "proposed model" is a photograph, obtained using the "stop action" effect of a strobe light in a darkened room. The stroboscopic photograph then becomes our experimental data.

The photographer’s image of the freely falling ball in the figure below was obtained by opening the camera shutter for one second while the ball was dropped in a darkened room. A stroboscopic light flashed for a split second (1/5000 sec) every 0.05 sec. How many flashes occurred while the camera shutter was open? If the full length of the picture represents roughly the distance to the ground, how long did the tennis ball take to fall roughly 0.3m to the ground? What are the time intervals between images of the ball in the picture?

What patterns do you notice in the image displayed? How would you describe the motion of the tennis ball during this fall to the ground? Do the experimental results confirm our working hypothesis that the ball’s speed decreases as the ball nears the ground? If not, how would you change the hypothesis to better match the experimental results?

Sketch a graph of distance vs. time for the tennis ball’s motion shown in the figure.

Is your current working hypothesis correct, or does it need to be revised? Recall our hypothesis: The tennis ball’s speed decreases as the ball falls to the ground. If not correct, how would you alter your working hypothesis to match the experimental data (the photograph)?

Is the ball moving faster or slower at the bottom of the drop? How do you know? What mathematical function best fits your distance vs. time graph? What do you think caused the differences in the distance vs. time graphs observed for the rolling and the falling balls? Is the falling ball moving with uniform motion? Explain why or why not.

Sketch a graph of velocity vs. time for the tennis ball’s motion shown in the photograph.


Below are some graphs of distance vs. time for various objects. Move your finger over the table top to demonstrate the speed of the object represented by each graph.


The motion of the falling tennis ball is not uniform. Instead as inspection of the image above indicates, the ball’s position changes more each second near the ground than it does near the top of the fall. Thus defining an average velocity as we did in the case of the uniform motion would not be meaningful.

Instead it is better to talk about the velocity at different times or positions during the fall of the ball to the ground, because the velocity of the ball is continuously changing. For a specific clock time or instant in time, we can then define an instantaneous velocity as the rate of change of the distance with time at a given instant. If d represents the change in distance over the time interval, t, then the instantaneous velocity of the falling tennis ball can be found from

Since the velocity is continuously changing, we can also define a rate at which the velocity is changing,

The above expression defines acceleration as the rate of change in velocity as a function of time. Thus the falling ball is dropping with an accelerated motion because the instantaneous velocity is continuously increasing.

How can you find the instantaneous velocities for several times during the fall of the ball from the photograph above? Determine a few values of v and t for the falling ball. Sketch a plot the accelerations vs. time for the falling tennis ball. What does your graph look like? What does the graph tell you about the acceleration of the ball as it falls? What is the average value of the acceleration of the falling tennis ball? What do you think causes the acceleration of the tennis ball? How is the motion of the falling ball different from that of the rolling ball?

Combined Horizontal and Vertical Motion

Another strobe light image is shown below. The light flashes occurred every 0.1 second, and the ball was tossed upward beginning its motion in the lower left corner of the picture. What kind of motion can you identify in this picture? What causes the path you see for the ball? Remembering that the flashes which exposed the film occurred in even time intervals, what can you say about the speed of the balls at different parts of its path?

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Last modified 9 Aug 1997
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