Module 1 -- Position, Displacement, Distance and Coordinate Systems

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Motion is a change in location. To develop a quantitative understanding of motion, we must begin by developing a mathematical description of location, which is called position.

Learning Goals

After working through this module, you should be able to:

  • Define a coordinate system.
  • Construct position vectors.
  • Explain the similarities and differences between position, displacement and distance.
  • Calculate the displacement between two locations.


Illustrative Example: Late for Physics

Consider the following example:

A student rushes from their dorm room to the physics building in 2 minutes. After spending 4 minutes turning in their homework, the student hurries to the cafeteria in 2 minutes. The student eats lunch for 12 minutes, then walks to the library in 6 minutes.

Position

Before we can determine the velocity of the student at each stage of the motion, we must assign positions to each of the various buildings visited by the student. For simplicity, imagine a school where all these buildings are on the same street. The street runs east to west. Suppose that the physics building is two blocks east of the dorm, the cafeteria is one block west of the dorm, and the library is three blocks east of the dorm. A simple way to convey this information is to construct a one-dimensional position axis such as the one shown below.

Campus 1a.png

This position axis conveys all the information given in the problem about the relative locations of the buildings. The process of constructing this axis is called choosing a coordinate system. In this coordinate system, the vector representing the position of the physics building is a right pointing arrow of size 2 blocks, whereas the vector representing the position of the cafeteria points to the left and has a size of 1 block. These two vectors only have an x - component given by:

 \mathcal{}r_{Physics,x} = 2 \mbox{ blocks}

\mathcal{}r_{Cafeteria,x} = -1  \mbox{ block}

Displacement

By using this position axis, we can find the displacement between any two locations. The displacement between two locations is the change in position and is calculated as the difference between the final and the initial position. For example, starting in the cafeteria the student walks to the physics building, the displacement vector between these two positions is the difference between the position of the physics building and the position of the cafeteria:

\Delta\vec{r} = \vec{r}_{Physics} - \vec{r}_{Cafeteria}

which gives a displacement of (+2 blocks) (1 block) = + 3 blocks. Note, however, that reversing this process (taking the position of the cafeteria and subtracting the position of the physics building) gives (1 block) (+2 blocks) = 3 blocks. The sign of these results has meaning, because displacement is a vector. The sign of the displacement indicates the direction. For the coordinate system we have defined the positive x direction points east and the negative x direction points west. Thus, the displacement required to go from the cafeteria to the physics building is + 3 blocks because it is an eastward movement. On the other hand, the displacement required to go from the physics building to the cafeteria is 3 blocks because the movement to go from the physics building to the cafeteria is westward.

(Note: The standard notation to represent a difference of two quantities is the use of the big Greek letter delta, (Δ), follow by the letter used to represent each quantity in the difference, in our case, is the position vector indicated with the letter r and an arrow on top).

We now summarize two important points about position and displacement:

  • Position itself is a vector and its mathematical representation will depend on the coordinate system chose to describe the motion.
  • The origin of the coordinate system and the positive direction are arbitrary choices. The displacement between any two locations on the axis will be independent of the choice of the origin and of the choice of the positive direction (provided that the direction assignment is clearly stated).

To illustrate the second point, consider the following two alternate coordinate systems for the example situation described above.

Campus 2a.png

Campus 3a.png

If you calculate the displacements between any two locations in any of the two coordinate systems, you will find the same answer provided that you are told that the negative x direction points east in the final coordinate system.

Distance versus Displacement

It is important to separate the concept of distance from that of displacement. Both quantities are measured in units of length. But it should be clear from common usage that distance is not a vector. Consider the first two legs of the trip described in the example at the top of this Module. In the first leg (dorm room to physics building) the student has a displacement of + 2 blocks in our initial coordinate system, or 2 blocks east. We would simply say that the student moved a distance of 2 blocks. No direction is needed when specifying distance. In the second leg of the trip (physics building to cafeteria) the displacement was 3 blocks, or 3 blocks west. Again, we would simply say the distance traveled was 3 blocks.

From this discussion, it may be tempting to say that the distance is equivalent to the magnitude of the displacement. There is an important problem with this definition, however. Consider the students movement from another perspective: put the first two legs together into one trip. For the student's trip from the dorm to the physics building and then to the cafeteria, the initial position was xi = 0 blocks (the dorm) and the final position was xf = − 1 block (the cafeteria). The total change in position for this trip was:

 \mathcal{}\Delta x = x_{f} - x_{i} = -1 \mbox{ block } - 0 \mbox{ blocks } = -1 \mbox{ block } = 1 \mbox{ block west}

In other words, the student ended the trip one block west of where they started (the cafeteria is one block west of the dorm). Tracing the complete trip on the map, however, makes it clear that the distance traveled by the student was actually 5 blocks (2 blocks to get to the physics building plus 3 blocks to get to the cafeteria). Clearly the size of the displacement is not equal to the distance. This discrepancy is not a problem, however. The distance measures the total length of the trip, while displacement measures the net effect of the trip. The student moved 5 blocks total, but because they turned around during the trip, two of the three blocks moved in the second leg (from the physics building to the cafeteria) were used to "undo" the first leg (by returning to the dorm). Thus, they ended up only 1 block from where they started.

Campus trip.png

Multi-Dimensional Coordinate Systems

Position and displacement can be generalized to more than one dimension by using the rules of vector addition and subtraction. Suppose that we re-imagine the campus of the example we have been using. Suppose that the distance from the dorm to the physics building is still 2 blocks and the distance from the physics building to the cafeteria is still 3 blocks, but suppose that the cafeteria is now 3 blocks south of the physics building rather than 3 blocks west of the physics building. If we choose a coordinate system with the physics building at the origin, we can represent this situation as shown in the left picture below. Note that we have been forced to define a y-axis as well as an x-axis, because the campus map is now two-dimensional.

The net displacement achieved by the student in this trip is now determined by using the Pythagorean theorem to be about 3.61 blocks. The direction of this displacement is usually reported as an angle. The angle shown in the figure (56.3°) was determined using trigonometry from the sides of the triangle formed by the green and red arrows. It is not, however, appropriate to simply state that the displacement of the student was 3.61 blocks at 56.3°. The angle that is reported must be referenced to the coordinate system chosen. The usual technique is to measure angles clockwise from the positive x-axis, so that in this case the appropriate angle to report is 303.7°. It is also fairly common to call this angle 56.3°.

Campus 4a.png

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