## Motion

In order to understand motion, the first concept to be introduced is the concept of frame of reference.

### Frame of Reference

If a cast away, in the middle of the ocean, has been rowing for several hours, how does he know if he has been moving? The trouble in answering this question is associated with the ocean not having reference points. The most likely answer to this question is that, after rowing for several hours, the cast away does not know about a possible change of location. On the other side, for a person lost in a city the response will be different. In the city case, even when the person may not be able to conclude about progress toward reaching the final destination, the person will certainty know about being on the same or different place with respect to the starting point. The difference in result is because a city has different land marks that allow people to identified different locations. In mechanics, the city land marks are equivalent to frames of references.

 The figure on the left represents the most common one-dimension frame of reference. The one-dimensional frame of reference is characterized by a single straight line with equally spaced divisions proportional to the unit of length used. At the center, usually the so called origin of coordinates is located. In this case, the figure on the left is a two-dimensions coordinate system. Here, two axis are use to identify every point on the plane described by the coordinates. The identification of points is equivalent to the one used in elementary mathematics with the first coordinate representing the value of the point with reference to the x-axis and the second coordinate representing the y-axis; thus, any point in the plane will be represented by the ordinate pair (x, y).  The two dimensional coordinate system is represented by two orthogonal (perpendicular) axis that intercept at a point called the origin of coordinates, (0, 0). The segment of the axis to the right of the origin is the positive side of the x-axis and the segment of the axis to the left of the origin is the negative segment of the x-axis. Similarly, the top part of the y-axis is the positive side while the lower part of the y-axis is the negative side. The picture below shows a possible use of a two dimensional coordinate system to represent position of objects on the floor of the three dimensional room . For example, if the motion of an object such as a toy car is always on the floor of the room, the motion of the toy car can be completely study with a two dimensional coordinates system represented by the x and y axis.
 The figure at the left represents a three-dimensional coordinate system generated from the vertex of a room. The abstract representation of the same coordinate system is also presented on the figure . The negative portion of the axis are not represented in the picture. The negative z-axis is the projection down of the pictured positive z-axis. Tree-dimensional coordinates can represent points on the space by a set of three numbers, the x coordinate, y coordinate, and the z coordinate. The notation for representing a point in the plane is ( x, y, z). In these notes, most of the problems will be reduced to a single plane. Thus, the most use coordinate system will be the two dimensional system.
 The applet on the left can help to understand coordinate systems in two dimensions. Click on the graph, the coordinates of the point will be displayed. The number of points that can be display at a single time is limited. Press the Clear button to clean the points from the graph. What are the sign of the coordinate numbers for points on the left-top quadrant (II quadrant) of the graph? In which quadrant are the two coordinate numbers negative? and, positive?

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com