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See "A Pure Point-of-View", MOT 12 October 2001

26 October 2001

How Far Have We Gone?

by George E. Hrabovsky, President of MAST

News from MAST

This week I have begun the microbiology experiment. The goal is to isolate some harnless bacteria from soil to do additional research. I will be making stock cultures of the bacteria I identify.

Continuing from last week I will continue to develop the principles of motion from first principles. I will also be answering last week's Theory Challenge.

What is the Distance an Object Travels?

Last week we discussed position. We determined that to discover the position of an object we need to begin by identifying a reference point and then measure the distance and direction to the object.

I will now expand that idea to cover the most basic quantity of motion: the distance an object travels. This quantity is called displacement. Recall our definition of motion, "The change of position in time." So, all motion starts with an initial position. We will call this position . The motion will end at the final position, we will call this . For now we will only think of motion in a straight line. This motion will give us the line segment . This segment is called the displacement. Look at the diagram below,

O is the origin of the coordinate system we are using to locate and . We can see that this forms a triangle. We can apply the Law of Cosines to find the length of :

Assuming we know and we can manipulate the equation to get

Thus, by knowing the position vectors for the initial and final points of motion and the angle between the two vectors we can find the displacement.

Theory Challenge Answer for Last Week's Column

This is going to be somewhat long, I am afraid. Get a pen and paper and follow along. First, I will introduce some notions from geometry:

1. We will understand that everything is happening in a place. We will call this place a space. We will make no attempt at this stage to characterize this space.
2. This space can have regions within it that we will call subspaces.
3. One such subspace has no size or shape. This is called a point.

How does the notion of a point correspond to that of a particle (see the last two columns)? How does this change the problem of locating the particle? Now that we have changed the problem of finding the location of a particle to finding the location of a point within a space, how do we do this? What is our frame of reference?

This last notion has the seeds for the idea of assigning a reference point . We now recall Euclid's First Postulate:

For every point and every point not equal to there exists a unique line l that passes through and . This line is denoted .

We now state the formal definition of a line segment. Definition 1: Any two points, and , and the collection of all points between them, that lie on the line combine to form a line segment. A segment is denoted .
    

We now state the formal definition of distance. Definition 2: The segment   is called the distance between and . We can measure this distance, which we denote .

We now state the formal definition of absolute value. Definition 3: Given a number x the absolute value of x is x itself so long as x is greater than or equal to 0; if x is less than 0 then its absolute value is -x. Symbolically we write it this way:

We can now state the so-called Ruler Axiom: For every line there exists a one-to-one correspondence between the points lying on the line and the set of real numbers (denoted ) such that the distance between any two points on is the absolute value of the difference between the numbers corresponding to the two points.

From this we can construct another derive an expression for distance by applying Definition 3 to the Ruler Axiom and then to Definition 2.

To measure this we have to establish a unit of length, some uniform segment that can be applied successively. Let us say that this segment is bounded by the points and (the segment is thus ).

We next have to find a point on such that ,

We apply the same technique a second time, finding a second point such that .

We continue this process until we have a set of equal segments .=20

And thus the distance for the segment is then,

Theory Challenge

How did I get the formula,

From the Law of Sines?

 

Books That I Like

The best book on this subject is not a physics book:

John Roe (1993), Elementary Geometry, Oxford University Press. This is particularly a good book for developing a geometric intuition about physics topics.

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Converted by Mathematica      October 24, 2001