1.6.2 Position-versus-Time Graphs

FIGURE 1.17 is a motion diagram, made at 1 frame per minute, of a student walking to school. You can see that she leaves home at a time we choose to call t=0 min and makes steady progress for a while. Beginning at t=3 min there is a period where the distance traveled during each time interval becomes less—perhaps she slowed down to speak with a friend. Then she picks up the pace, and the distances within each interval are longer.

FIGURE 1.17 The motion diagram of a student walking to school and a coordinate axis for making measurements.

The figure illustrates the motion diagram of a student walking to school using a coordinate axis.
FIGURE 1.17 Full Alternative Text

Figure 1.17 includes a coordinate axis, and you can see that every dot in a motion diagram occurs at a specific position. TABLE 1.1 shows the student’s positions at different times as measured along this axis. For example, she is at position x=120 m at t=2 min.

TABLE 1.1 Measured positions of a student walking to school

Time t (min) Position x (m) Time t (min) Position x (m)
0     0 5 220
1   60 6 240
2 120 7 340
3 180 8 440
4 200 9 540

The motion diagram is one way to represent the student’s motion. Another is to make a graph of the measurements in Table 1.1. FIGURE 1.18a is a graph of x versus t for the student. The motion diagram tells us only where the student is at a few discrete points of time, so this graph of the data shows only points, no lines.

FIGURE 1.18 Position graphs of the student’s motion.

The figure consists of two parts labeled ("a") and (b) that illustrate the position-versus-time graphs of a student’s motion.
FIGURE 1.18 Full Alternative Text

NOTE   A graph of “a versus b” means that a is graphed on the vertical axis and b on the horizontal axis. Saying “graph a versus b” is really a shorthand way of saying “graph a as a function of b.”

However, common sense tells us the following. First, the student was somewhere specific at all times. That is, there was never a time when she failed to have a well-defined position, nor could she occupy two positions at one time. Second, the student moved continuously through all intervening points of space. She could not go from x=100 m to x=200 m without passing through every point in between. It is thus quite reasonable to believe that her motion can be shown as a continuous line passing through the measured points, as shown in FIGURE 1.18b. A continuous line or curve showing an object’s position as a function of time is called a position-versus-time graph or, sometimes, just a position graph.

NOTE   A graph is not a “picture” of the motion. The student is walking along a straight line, but the graph itself is not a straight line. Further, we’ve graphed her position on the vertical axis even though her motion is horizontal. Graphs are abstract representations of motion. We will place significant emphasis on the process of interpreting graphs, and many of the exercises and problems will give you a chance to practice these skills.

EXAMPLE 1.7 Interpreting a position graph

The graph in FIGURE 1.19a represents the motion of a car along a straight road. Describe the motion of the car.

FIGURE 1.19 Position-versus-time graph of a car.

The figure consists of two parts labeled ("a") and (b) that illustrate the position-versus-time graph of a car.

MODEL We’ll model the car as a particle with a precise position at each instant.

VISUALIZE As FIGURE 1.19b shows, the graph represents a car that travels to the left for 30 minutes, stops for 10 minutes, then travels back to the right for 40 minutes.