Respuesta :
Respuesta:
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 orighoice 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 describes in any of the