1.3 Position, Time, and Displacement

To use a motion diagram, you would like to know where the object is (i.e., its position) and when the object was at that position (i.e., the time). Position measurements can be made by laying a coordinate-system grid over a motion diagram. You can then measure the (x, y) coordinates of each point in the motion diagram. Of course, the world does not come with a coordinate system attached. A coordinate system is an artificial grid that you place over a problem in order to analyze the motion. You place the origin of your coordinate system wherever you wish, and different observers of a moving object might all choose to use different origins.

Time, in a sense, is also a coordinate system, although you may never have thought of time this way. You can pick an arbitrary point in the motion and label it “t=0 seconds.” This is simply the instant you decide to start your clock or stopwatch, so it is the origin of your time coordinate. Different observers might choose to start their clocks at different moments. A video frame labeled “t=4 seconds” was taken 4 seconds after you started your clock.

We typically choose t=0 to represent the “beginning” of a problem, but the object may have been moving before then. Those earlier instants would be measured as negative times, just as objects on the x-axis to the left of the origin have negative values of position. Negative numbers are not to be avoided; they simply locate an event in space or time relative to an origin.

To illustrate, FIGURE 1.5a shows a sled sliding down a snow-covered hill. FIGURE 1.5b is a motion diagram for the sled, over which we’ve drawn an xy-coordinate system. You can see that the sled’s position is (x3, y3)=(15 m, 15 m) at time t3=3 s. Notice how we’ve used subscripts to indicate the time and the object’s position in a specific frame of the motion diagram.

FIGURE 1.5 Motion diagram of a sled with frames made every 1 s.

The figure consists of two parts labeled ("a") and (b) that illustrate a sled moving down an slope and its motion diagram on an x y coordinate system. Five frames made at every 1 second are shown simultaneously.
FIGURE 1.5 Full Alternative Text

NOTE   The frame at t=0 s is frame 0. That is why the fourth frame is labeled 3.

Another way to locate the sled is to draw its position vector: an arrow from the origin to the point representing the sled. The position vector is given the symbol r⃗. Figure 1.5b shows the position vector r⃗3=(21 m, 45°). The position vector r⃗ does not tell us anything different than the coordinates (x, y). It simply provides the information in an alternative form.