One easy way to understand position-vs.-time graphs is to imagine a graph showing the position of a friend who’s walking away from you or toward you over time. Such a graph is shown in Figure 6.7, with time increasing to the right on the horizontal axis and your friend’s position increasing upward on the vertical axis. In this type of graph, you’re at position zero, so your friend’s position value is her distance from you.
Position
Time
tstart treverse tend
Walking away from you at constant speed (slope is positive and constant)
Walking toward you at constant speed (slope is negative and constant) Reversing direction (slope changes from positive to negative)
Figure 6.7 Position-vs.-time graph for friend walking away and back.
6.4 The expansion of the Universe 181
In the scenario shown in this figure, your friend starts out at your position (zero height along the vertical axis) at time tstar t = 0 and initially walks away from you at constant speed. Since her speed is constant, her position increases linearly (that is, along a straight line on the graph) over time. That’s because for constant speed, distance equals speed × time, and with distance (d) on the y-axis and time (t) on the x-axis, d = vt is the equation of a straight line. As you may recall from Section 6.4.1, the equation for a straight line is y = mx + b, where m is the slope of the line and b is the y-intercept. The y-intercept is zero in this case since your friend started out at your position at time t= 0, so the equation of this line can be written as y = mx. Comparing d = vt to y = mx, you can see another important relationship: your friend’s speed (v) is equal to the slope of the line (m) on her position-vs.-time graph.
That should make sense to you, since the slope of a line is defined as the rise (y) over the run (x), and in this case the rise is the change in position (distance) while the run is the time it takes for that position change. Hence on a position-vs.-time graph
slope= y
x = distance
time = speed, (6.12)
where positive slope means increasing distance.
In the scenario shown in Figure 6.7, after walking away for some time, your friend turns around and walks back toward your location at the same constant speed with which she walked away. This part of her journey is shown by the portion of the graph in which her position value gets smaller over time, since smaller position means less distance between you and your friend. For the return trip, the slope of your friend’s line is again constant, since she’s walk-ing at a constant speed, but now the slope of her line will be negative, since the change in her position (y) is negative when she’s moving toward you.
And since she walks back at the same speed and for the same amount of time as she walked away, your friend ends up back at your position at the end of her trip.
There’s one portion of the graph in Figure 6.7 in which the slope of the line is not constant. That’s the portion near the time (tr everse) at which she changes her direction from moving away to moving toward you. As she approaches the turn-around point, she slows down, making the slope of her line less pos-itive. At the instant she stops moving away, her slope momentarily equals zero, and as she begins moving toward you, her slope becomes negative – slightly at first, and then reaching a constant value as she gets up to her walking speed.
182 Black holes and cosmology
Position
Time Walking away
slowly (small positive slope)
Walking away quickly (large positive slope)
Speeding up (increasing positive slope)
Slowing down (decreasing positive slope)
Stopped (zero slope)
Figure 6.8 Position-vs.-time graph for varying speeds.
Example: Sketch a graph summarizing the position-vs.-time motion if your friend begins walking away from your position at a slow speed but then speeds up and walks away more quickly; after moving away at this higher speed for a short time, she slows down, comes to a halt, and remains stationary at her position for the remainder of the graph.
This example shows the effect of acceleration on position-vs.-time graphs. The details of this trip are annotated on the graph in Figure 6.8, and you should make sure you understand what’s happening in each portion of the graph. But you should also take a step back and look at the big picture. In the big-picture view, straight lines mean constant speed, with a positive slope indicating away, a negative slope toward, and a zero (flat) slope indicating no motion. Curved lines have a changing slope and thus show a changing speed, which indicates acceleration: speeding up if the slope is becoming steeper (more positive or more negative) or slowing down if the slope is becoming flatter (less positive or less negative).
The use of position-vs.-time graphs in cosmology is discussed in the next section, but before moving on, you should work through the following exercise to verify your understanding of this important type of graph.
Exercise 6.17. Describe the motion of an object for which the position-vs.-time graph looks like the graph in Figure 6.9.