Coordinate System

A set of mathematical axes which serve as a quantitative map grid, allowing precise specification of positions of objects.  Cartesian coordinates are most common in introductory mechanics, but cylindrical coordinates are sometimes useful, especially for circular or orbital motion.

Definition by Example

Consider an example of motion. Suppose a student rushes from their dorm 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. In which portion of the trip was the student moving the fastest? Before we can answer, we need information about the [positions] of the various buildings visited by the student.

For simplicity, imagine a school where all these buildings are on the same street so that the motion we have described is effectively one-dimensional. Suppose the street runs east to west. 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 as shown below.

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. With the system that we have chosen, the position of the physics building is + 2 blocks, the position of the cafeteria is - 1 blocks, and so on. Based on this exercise, we define (one dimensional) position as:

This definition implies that it is really the differences between the points on our map that are important. For instance, on our map, taking the position of the physics building and subtracting the position of the dorm gives a difference of (+ 2 blocks - (0 blocks)) = + 2 blocks. This difference means that you have to move two blocks east (remember that the positive direction of our axis was assigned to point east) to go from the dorm to the physics building. Similarly, taking the position of the cafeteria and subtracting the position of the physics building gives ( - 1 blocks - (+ 2 blocks)) = - 3 blocks, indicating you must move 3 blocks west to get from the physics building to the cafeteria. It really doesn't convey any information to say that the cafeteria is at position - 1 block. What matters is the relationship between the positions of the buildings. Thus, it is possible to design many valid coordinate systems for the street in this example. Here are two possibilities:

You can see that the differences in the positions still convey all the information about how to get from one building to another, provided that the assignment of the positive direction (east or west) has been clearly specified.