Understanding Position Versus Time Graphs

Constructing a position versus time graph for a physical situation is straightforward. For instance, consider the animated figure [1] below which shows a block sliding across a flat surface. Note that the time that animation also shows the times when the block is at certain positions.

Figure 1. Animation of a block sliding across a flat surface.
(Reload to view animation again.)

The table of this position versus time data is shown in figure [2] below.

x	0	1	2	3	4	5
t	0	0.5	1.0	1.5	2.0	2.5

Figure 2. Table of data for sliding block.

In turn this data is plotted in figure [3] below.

Figure 3. Graph of position versus time for sliding block.

We are interested in the converse. Given a particular position versus time graph, we need to be able to describe the motion of the block. This handout describes how to do just that.

The Basics

The first thing to do, if examining a graph, is to make sure that it is in fact a position versus time graph. You do this by examining the vertical and horizontal axes. The vertical axis must represent position and may be noted by x or y or z or just the word position. The horizontal axis must show t or the word time. Consider the six diagrams in figure [4] below. Which are position versus time graphs and which aren’t?

(a)		(b)
(c)		(d)
(d)		(f)

Figure 4. Various graphs.

Check your answers here (popup link).

The second thing to do is to look at the graph and to note the starting position and the end position of the graph. In figure [5] below the start position, at t = 0, is x = 1. The end position, at t = 3, is x = 5. This information tells you whether the object has moved in the positive x direction (or right or upwards or up an inline) or in the negative direction (or left or downwards or down an incline) or ended up at the same place it started. Here, in figure [5], the object moved in the positive x direction.

Figure 5. Start and end position.

More complicated graphs like figure [6a] can be understood by considering different portions of the graph separately. Peaks and troughs on a graph usually separate easily identifiable portions of the graph as are flat or straight sections. In figure [6b] the graph has been divided into four regions. In the first region the object moves in the positive direction. In the second region it moves in the negative direction. The object moves in the positive direction in the third section. In the last section the start and end position are the same.

(a)
(b)

Figure 6. Analyzing a more complex graph.

Velocity

Information on how fast an object is travelling can also be found from a position versus time graph. To find the velocity (speed and direction) of an object, we draw a straight line tangent to the positive versus time curve at the time for which we want to know the velocity. The slope of this tangent line is the velocity. Note that the slope of the tangent line is given by change in position over change in time. In SI units this would be in m/s which is the unit of velocity as required. Mathematically we say that velocity is given by the derivative, or rate of change, of the position versus time graph and write .

In figure [7] tangent lines are drawn at various points on a position versus time graph. Note that the inclination of a tangent line can have positive, negative, or flat. A positive slope means that the object is moving in the positive direction, i.e. the object has positive velocity. A negative slope means that the object is moving in the negative direction, i.e. the object has negative velocity. A flat or zero slope, zero velocity, can occur in two different situations. The first is at a peak or a trough where the object is turning around. The second is when the portion of the position versus time curve is flat indicating that the object is not moving. Note that the sign of the slope of the tangent lines, in other words the direction of the velocity, already agrees with what we found when discussing figure [6b].

Figure 7. Finding velocity from a position versus time graph.

Acceleration

Acceleration is defined as the derivative, or rate of change, of the velocity with respect to time graph and written . Determining acceleration from a position versus time graph requires some work. First let us examine figure [8] which shows position versus time graphs that consist of straight lines. Small tangent lines have been added to show velocity at different times. Note that for each graph the slope of the tangent lines remains the same. That is the velocity is constant in each graph. If the velocity is constant, it is not changing, so the acceleration is zero.

Curved position versus time graphs as shown in figure [11] below, on the other hand, have non-zero acceleration. Determining whether the acceleration is positive or negative from a graph requires care. Both velocity and acceleration are vector or signed quantities. The sign indicates a direction. For velocity, interpreting the sign is simple. A positive velocity means the object is moving in the positive direction. A negative velocity means the object is moving in the negative direction. The sign of an acceleration, however, only makes sense if you also know the direction of the velocity. The details are summarized in the table shown in figure [9].

Figure 9. Sign of velocity and acceleration and resulting interpretation.

To find the sign of the acceleration we must look at an initial and final velocity. Next determine the direction of the motion and whether the object is speeding up or slowing down.  The sign of the acceleration can then be found using the table shown in figure [10].

Consider the curved position versus time graphs in figure [11] below. The two tangent lines drawn on each curve have different slopes, so the velocity has changed, and each graph represents an object that is accelerating.

Examing the graph in figure [11a], we see that the velocity is positive but that the object is slowing down. Hence the acceleration is opposite to the velocity, i.e. negative. Examine the other graphs in figure [11] and determine the sign of the acceleration. The answers are given here (popup link).

Fundamental Curves

In preparatory and first-year physics we deal only with constant acceleration. As a result the positive versus time graph that you will encounter will be built up of the only nine simple curves. The interpretation of each is given below.