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9 November 2001

Snapshots of Reality

by George E. Hrabovsky, President of MAST

News from MAST

Hello from MAST! I have  gotten some good responses to my request for email! Thanks for the suggestions, keep them coming. I am always on the lookout for good material for future columns.

This week I will discuss a rather remarkable development in mathematics and physics. Without this development it is not really possible to discuss physics in any substantial way.

Frames of Reference

Last week we developed an intuition about time. We even managed to define time as a clock reading assigned to something that happens (in our case, an object moved). What good does that do us? Without some way to combine time and distance into a single entity, it is not very easy to use these concepts. When we do combine these ideas (he said, encapsulating something like two thousand years worth of fascinating research into a sentence) we come up with a truly remarkable concept.

When an object moves its path is called a trajectory. Each point on the trajectory represents the position of the object corresponding to a position on a time line. Recall that in a previous column, I wrote about functions. In this case we can view position as a function of time! This realization paves the way for rapid advance in the study of motion. First of all this realization allows us to represent the position of an object with a formula. Secondly, it allows us to represent this formula graphically as a graph (or plot) with the independent variable (in this case time) as the horizontal axis (or abscissa) and the dependent variable (in this case position) as the vertical axis (or ordinate). So long as the origin (or reference point) of both the position line and the time line are chosen to intersect we have developed what is called a frame of reference. In a future column I will discuss how we can view the world from different frames of reference at the same time (in fact we have no choice in this!)

Let's look at a specific example of this. Say that we measure the position of an object with respect to time .  Let us also say that we are observing the motion of a ball hanging from a perfect spring (that is we can neglect both friction and gravity). After analyzing the data we determine that a formula describes the motion,

We can then make a plot of this motion.

The ability to treat the position of an object as a function only of time is a truly awesome conept and is very subtle in its implications. The development of this concept in history is both fascinating and completely eclipsed by the apparent simplicity of the concept. We look at this statement and say, "Of course, how could it be otherwise?" But we see in the history of this concept that it is anything but obvious. It opens to us all of the tools of calculus to attack the problem of describing the motion of an object. That is what we will do next.

Theory Challenge Answer for Last Week's Column

The subject of this column is, in effect, the answer to last week's Theory Challenge. Here I will simply put it in a more formal language.

If we assign a set to describe the position line of a moving object we note that the domain is the entire set of real numbers. Perhaps I will explore the properties of the set of real numbers in another column (a thing I should do, since there is no reason for you to believe that this is a reasonable statement otherwise). For now, you will just have to trust me. The set of real numbers is denoted .

If we assign another set to describe the time line, then we find an interesting thing; the domain is also the set of real numbers .

Recall from the column on functions that a function relates the independent variable to the dependent variable (in this case time to position). This creates a set of ordered pairs where for every there exists a unique value for . This gives us a Cartesian Product,

Using functional notation we see that , where describes a rule assigning a unique value of for the position to every value of corresponding to time:

The result is the trajectory.

Theory Challenge

Can you justify why position and time can be considered to be described by the set of real numbers?

Books That I Like

There are two good sources on this subject:

Josef Honerkamp, Hartmann Römer (1993), Theoretical Physics A Classical Approach, Springer-Verlag. Section 2.1 covers the theory challenge in all of its gory detail, but at a level that assumes mastery of vector analysis and the linear algebra of vector spaces.

George Hrabovsky (1997), Kinematics, Lesson 1, Unpublished. Notes for the development of a MAST course in physics that is in development for publication on the MAST website.

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