In this explainer, we will learn how to use distance–time graphs to compare the speeds of objects.

A distance–time graph is a graph that shows time on the horizontal axis and distance moved on the vertical axis, as in the template graph below.

In this explainer, we will talk about how to plot data onto a distance–time graph and how to read a distance–time graph to understand the speed at which something is moving.

Let’s start by adding some data to the graph above. In this example, we will measure the motion of a person walking. Measuring from a starting point, we have markers at 2-metre intervals. We will start a stopwatch when the person is at the first marker. As they pass each marker, we record the amount of time that has elapsed, as shown in the diagram below.

We can summarize this data in a table, where we record each pair of distance and time measurements.

Time (s) | 0 | 3 | 6 | 9 |
---|---|---|---|---|

Distance Moved (m) | 0 | 2 | 4 | 6 |

A table is a useful way to summarize data, but it is difficult to see any pattern. To show this data visually, we can plot it on a graph. To do this, we find each time value on the horizontal axis, and then move upward to find the corresponding distance value on the vertical axis. For this example, we will start by adding a point at the start, with a time of 0 s and a distance of 0 m. We place this point at the origin of the graph, as shown below.

Next, we add the second point that has a time of 3 s and a distance of 2 m. Since these values are different, we have to make sure that we have correctly identified which axis is which. We do this by matching the descriptions, as shown below.

To plot this point, we first need to locate 3 s on the horizontal axis and 2 m on the vertical axis, as shown below.

We now draw two lines, one going up from 3 s on the horizontal axis and one going across from 2 m on the vertical axis. Where the two lines meet, we plot a point. This is demonstrated in the following graph.

We can add the rest of the points in the same way to produce this distance–time graph of the person’s motion.

These examples will give some practice in relating data in tables to points on a graph.

### Example 1: Relating Data in Tables to Points on a Graph

The table shows the distances three toy cars move each second. Which of the markers on the graph shows the motion of the car that is recorded in the table?

Time (s) | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

Distance Moved (m) | 0 | 2 | 4 | 6 | 8 |

### Answer

Here, we have a graph that features three sets of points, represented by three different symbols, and we need to determine which of the symbols shows the motion of the car recorded in the table.

If we start with (A), the blue crosses, we can see that the first point is at the origin, which corresponds to a time of 0 s and a distance of 0 m. This matches the table, but the other two markers also have points at the origin, so we cannot use this point to differentiate between the markers.

Moving to the next blue cross, this lines up with 1 s on the horizontal axis and 2 m on the vertical axis. This also matches the second entry in the table. Looking at the other markers that line up with 1 s on the horizontal axis, we have the red cross that aligns with 1 m on the vertical axis and the green dot at about 0.5 m. The blue cross is therefore the only one that is right at this point.

To check that the blue cross shows the correct values at all of the points, we can summarize the points’ values in a table.

Looking at the entries in the table, we can see that the blue crosses match the entries from the original table at every point; therefore, the answer is (A).

### Example 2: Identifying Graph Axes by Relating Points to Data Tables

The table shows five readings of the distance traveled by a toy car that moved for 10 seconds. The distance was measured at 2 s intervals. Which color axis of the graph must show the time that the car moves for?

Time (s) | 0 | 2 | 4 | 6 | 8 | 10 |
---|---|---|---|---|---|---|

Distance Moved (m) | 0 | 5 | 10 | 15 | 20 | 25 |

### Answer

In this example, we have a graph whose axes are not labeled, and we need to work out which axis is which based on the values in the table.

At first glance, it may appear as though each of the green dots is showing equal values on the horizontal and vertical axes. However, the axes are labeled with values. We can see that the horizontal axis extends to a maximum of 10 and the vertical axis extends to a maximum of 25. The simplest way to differentiate between the two axes is to look at the top-right point, where the values are labeled as 10 on the horizontal axis and 25 on the vertical axis.

Consulting the table, we can see that if one value is 10 and the other is 25, this must correspond to the final entry in the table. The 10 refers to a time of 10 s, and the 25 is a distance of 25 m. Therefore, the horizontal axis is time and the vertical axis is distance moved.

We can add the appropriate labels to the axes as follows.

The time that the car moves for is therefore shown by the red axis.

Returning to our graph of a person moving, we now have a distance–time graph with four points representing the four measurements we recorded of time and distance. However, the person clearly did not disappear from one position and materialize at the next; they were moving uniformly throughout their motion. We can show this by drawing a line through all of our points, as demonstrated below.

By looking at the line, we can now read off more information about the person’s motion. For example, although we did not take a reading of the distance at a time of 1 s, we can now estimate the distance the person had moved at a time of 5 s by reading the value from the graph. We start from 5 s on the horizontal axis and move upward until we reach the blue line, as in the graph below.

We can now trace horizontally to find the corresponding value on the vertical axis, as in the following graph.

Reading off the value on the vertical axis, we can say that after 5 s, the person had walked just over 3 m.

Looking at our graph above, we can see that joining together all of the points made a single straight line. If a line is straight, it changes by equal amounts in equal time intervals. For example, in the first 3 s, the person walked 2 m, and in the next 3 s (between the times of 3 s and 6 s), they walked a further 2 m, to reach a distance of 4 m. In the final 3 s (between the times of 6 s and 9 s), they again walked another 2 m, ending at a final distance of 6 m. These intervals are shown in the graph below.

A constant increase of distance per unit of time indicates constant speed. So, because this line is straight, we can say that the person walked with a constant speed for the entire time.

In the graph below, the blue line shows the same person walking at a constant speed for 9 seconds. The orange dashed line shows a second person whose distance is recorded at the same time intervals.

The first thing we can say about the orange dashed line is that it is not straight, so the second person is not walking with constant speed.

In the first 3 seconds highlighted below, the person represented by the orange dashed line does not change their distance at all. In other words, they were stationary, or their speed was zero.

In the next 3 seconds, between the times of 3 s and 6 s, the person’s distance moved increased to 2 s. This means that over that interval, the person represented by the orange dashed line increased their distance by the same amount as the person represented by the blue line, although they are further behind. This means that during this period, as shown in the graph below, from 3 s to 6 s, the two people were walking at the same speed.

In the final 3 seconds, between times of 6 s and 9 s, the person represented by the orange dashed line only moved from 2 m to 3 m, an increase of 1 m. This means that they moved a smaller distance over those 3 seconds than the first person, so they were walking at a slower speed. This is shown in the final segment below.

As this example suggests, the gradient, or slope, of a line on a distance–time graph tells us about the speed. A steeper line means greater speed.

In the graph below, we have two lines representing the motion of two people.

Here, we can see that the orange dashed line is steeper than the solid blue line. This means the person represented by the orange dashed line is moving with greater speed.

The fact that we can identify the speed from the slope of a line on a distance–time graph means that we can compare speeds between different objects on the same graph without knowing any of the precise values. For example, the graph below shows the motion of four objects represented by different lines.

There are no values on the axes, but we can tell immediately that the object represented by the orange dashed line is moving with the greatest speed, because it has the steepest slope.

Similarly, we can identify the object represented by the green dot-dashed line as the one having the lowest speed, because it has the shallowest slope.

These last two examples give some practice in interpreting distance–time graphs.

### Example 3: Understanding Distance–Time Graphs

The motion of a running man is shown by the blue line. The red line shows the motion of a woman who is also running. Which of the following is correct?

- The woman started to run before the man did.
- Both the man and the woman started to run at the same time.
- The woman started to run after the man did.

### Answer

In this example, we have two lines on a distance–time graph: the blue line represents a man and the red line a woman. There are no values on the axes, other than at the origin that corresponds to a time of 0 s and a distance of 0 m. We need to identify which person started running first.

Looking at the blue line, representing the man, we can see that he started at a distance of 0 m and then his distance immediately began to increase, suggesting he started running at a time of 0 s.

The woman, on the other hand, was still at a distance of 0 m some time later, and her distance did not begin to increase until the time corresponding to the first grid line on the graph.

We can therefore say that the woman started to run after the man did, so answer (C) is the correct option.

### Example 4: Identifying the Greatest Speed from Lines on a Distance–Time Graph

Which color line on the graph shows the greatest speed?

### Answer

In this example, we have three lines on a distance–time graph, showing the distance moved by three different objects over some time. We need to identify which line on the graph indicates the greatest speed.

There are no scales given on the axes, but recall that we can tell the speed from the gradient, or slope, of a line on a distance–time graph. The steepest line is the one that shows the greatest speed. The steepest line on this graph is the green line; therefore, the green line is the one that shows the greatest speed.

### Key Points

- A distance–time graph shows the distance moved by an object on the vertical axis and the time on the horizontal axis.
- A straight line on a distance–time graph indicates that an object is moving with constant speed.
- The steeper the line on a distance–time graph is, the faster the object is moving.