In this explainer, we will learn how to use speed–time graphs to show the speed of an object in uniform motion.

We can recall that the speed of an object is the distance moved by that object per unit time.

Mathematically, if an object travels at a constant speed such that it moves a distance in a time , then speed is given by

We can use a speed–time graph to plot an object’s speed at different times.

We may recall that the horizontal axis of a graph is called the -axis and the vertical axis is the -axis. A speed–time graph measures time along the -axis, or the horizontal axis, and speed along the -axis, or the vertical axis.

Let’s look at a quick example problem in which we are asked to identify a speed–time graph.

### Example 1: Identifying Which of Two Graphs Is a Speed–Time Graph

Which of the following is a speed–time graph?

### Answer

The question asks us to identify which of the two graphs is a speed–time graph.

We can recall that a speed–time graph plots speed on the -axis against time on the -axis.

Looking at graph A, we see that it plots distance on the -axis and time on the -axis. Therefore, it cannot be a speed–time graph. It is, in fact, a distance–time graph.

Meanwhile, graph B does indeed plot speed on the -axis and time on the -axis. This means that this graph is a speed–time graph.

Our answer to the question is therefore that the graph in option B is a speed–time graph.

In the above example, we have seen what the axes of a speed–time graph look like.

Now let’s consider how we may plot data on such a graph.

We will imagine that we have an object that moves at a constant speed. We will further imagine that we have some means of measuring the speed of this object at any instant in time.

Suppose that we measure the object’s speed once every second, and obtain the following measurements.

Time (s) | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

Speed (m/s) | 3 | 3 | 3 | 3 | 3 | 3 |

Notice that since the speed of this object was constant, the same value of speed is measured each time such a measurement is made.

Now let’s plot these measurements on a speed–time graph.

We will begin with the measurement taken at a time of 0 seconds. The speed of the object at this time is 3 m/s. To plot this, we look on the time axis (or -axis) for a value of 0 s. At this horizontal position, we then go upward on the graph until we reach a height on the speed axis (or -axis) of 3 m/s. We place a cross here, where the vertical line through “time = 0 s” intersects the horizontal line through “speed = 3 m/s.”

Now we will consider the next measurement, taken at a time of 1 second. The speed of the object is, again, 3 m/s. We look on the time axis of our graph for a value of 1 s. We then follow a vertical line upward from this position until we get to a height of 3 m/s on the speed axis, where the horizontal line through “speed = 3 m/s” intersects the vertical line through “time = 1 s.”

Applying the same process to the remaining four measurements, we arrive at the following speed–time graph for the object.

We may draw a trend line through the points to help make the behavior of the object clear. In this case, all of the points lie at the same height, or same speed value, since the object moved at a constant speed, so the measured speed was the same at every value of time. This means that the trend line will be a horizontal line through the points.

This fact about horizontal trend lines holds more generally. Any object that moves at a constant speed for the entire time its speed is measured will have the same value of speed at all measured values of time. This means that all of the points on a speed–time graph of the object will have the same height on the speed axis, and so the trend line for the points will be a horizontal line.

Therefore, the motion of any object that moves at a constant speed can be represented on a speed–time graph by a horizontal line. We can also turn this statement on its head to say that any horizontal line on a speed–time graph must represent the motion of an object that moves at a constant speed.

From left to right along the horizontal axis, or time axis, the time is increasing. The further right a point is on the graph, the greater the value of time. Similarly, from bottom to top on the vertical axis, or speed axis, the speed is increasing. The higher up a point is on the graph, the greater the value of speed.

When we have a horizontal line, representing motion at a constant speed, the higher up that line is on the graph, the greater the speed of the object.

Suppose we have the following speed–time graph, which shows the motion of two objects that each move at a constant speed.

The graph shows two horizontal lines, each corresponding to a different object. Even without any numbers on the axes, we may simply look at this graph and see that the blue line corresponds to an object moving with a greater speed. This is because the blue line is higher up on the graph than the red line.

Let’s look at an example problem.

### Example 2: Working Out Which Line on a Speed–Time Graph Corresponds to the Greatest Constant Speed

Which color line shows the object that has the greatest speed?

- Orange
- Blue
- Red

### Answer

The graph shown in the question plots speed on the -axis against time on the -axis, which means that it is a speed–time graph.

The graph shows three horizontal lines, each representing the motion of a different object. Since the lines are all horizontal, this means that we know that each of the three objects moves at a constant speed.

We are asked to identify which line shows the object with the greatest speed.

We can recall that the higher up a line is on a speed–time graph, the further it is along the speed axis. This means that the higher up a line is, the greater the speed that it represents.

In this diagram, the orange line is lowest. Therefore, this line corresponds to the slowest moving object.

The blue line is higher than the orange line, and so the object represented by the blue line moves with a greater speed than the object represented by the orange line.

The red line is the highest line of the three. This means that the object represented by the red line moves with a greater speed than either of the other two objects.

Therefore, our answer to the question is that the object that has the greatest speed is shown by the red line. This is the answer given in option C.

The graph in this last example had no units or scale on either the time axis or the speed axis. This meant that we were not able to read off actual values of time or the speed of the objects. All that we could do was identify whether one object was faster than another.

We have already seen how we may take specific values of time and speed and plot them on a speed–time graph. In order to know where to plot the points, we needed a scale on both the time axis and the speed axis.

In the same way, if we have a speed–time graph with numbered scales on the axes, we may use these to read off values of time and speed for plotted points or a line on the graph.

Consider the following speed–time graph, showing just a single plotted point.

This point has two values associated with it: a time value, given by its position on the time axis, and a speed value, given by its position on the speed axis. The time value is the time at which that measurement was taken. The speed value is the value of speed that was measured at that time.

To read off the time value of the point, we trace directly vertically down from the point until we reach the time axis, as shown below.

The time value is the value on the time axis at the position where our vertical line meets it. In this case, that value is 4 seconds, which means that this particular measurement was taken at a time of 4 seconds.

To read off the speed value of the point, we trace directly horizontally across from the point until we reach the speed axis, as shown below.

The speed value is the value on the speed–time axis at the position where our horizontal line meets it. In this case, that value is 2 metres per second, which means that the speed of the object when this measurement was made was found to be 2 metres per second.

If we have a speed–time graph showing motion at a constant speed, we know that this will be represented by a horizontal line. For example, consider the graph below.

We may pick any point on this line and trace down vertically to the time axis to find the time value associated with a particular position along this line.

Similarly, we may choose any point on the line and trace across horizontally to the speed axis to find the speed of the object at that instant. Since the motion is at a constant speed, and hence the graph shows a horizontal line, any point on the line we choose will trace across to the same value on the speed axis. Physically, this simply means that the object has the same value of speed for all values of time.

If we are trying to find the speed of an object from a graph like this in which the plotted speed is a horizontal line, the act of tracing across from a particular point is unnecessary. All points on the line trace across to the same speed value—this is the value on the speed axis at the point where the horizontal line meets, or intersects, it.

Looking again at our particular graph, we can identify the value on the speed axis at the point where the horizontal line meets the axis.

The position at which the horizontal line meets the speed axis is indicated by a red arrow on the graph. We can see that this position is at a value of 4 m/s. Therefore, we now know that our object moves at a constant speed of 4 m/s.

Let’s have a look at another example question.

### Example 3: Reading the Value of a Constant Speed from a Speed–Time Graph

What is the speed shown by the speed–time graph?

### Answer

This question shows us a speed–time graph and asks us what speed is shown by the graph.

We can see that the graph shows a horizontal line. We may recall that this corresponds to motion at a constant speed.

To read off the value of the speed, we need to identify the height of this line on the speed axis, that is, the position on the speed axis at which the line meets, or intersects, this axis.

We can identify this position as shown in the diagram below.

The blue arrow points to the position at which the horizontal line meets the speed axis. We can see that this is at a value of 3 m/s, and we know that this value must be the speed shown by the graph.

Therefore, our answer to the question is that the speed shown by the graph is 3 m/s.

Sometimes, we might have information about an object’s motion given to us in the form of a distance–time graph. We can recall that a distance–time graph plots distance on the -axis against time on the -axis.

In this case, we can use the information given to us in the distance–time graph in order to work out how the speed–time graph for the object will look. We will see how to do this for a couple of simple cases.

We have seen that a horizontal line on a speed–time graph represents motion at a constant speed. In other words, the speed of the object is not changing as time goes on. So, what does a horizontal line on a distance–time graph mean?

By the same logic, a horizontal line on a distance time graph must represent an object with a distance value that is not changing in time.

We can recall that speed is defined as distance moved per unit time. This means that if we have an object with a distance value that does not change during the time that we measure it, then that object has a speed of 0 m/s; it is not moving.

Consider the following distance time graph.

All three of the lines on this graph are horizontal. Therefore, all three lines represent objects that are not moving during the period that the measurements used to plot this graph were taken.

We can notice that the lines are at different heights on the graph, corresponding to different values of distance. Perhaps the objects moved different distances before the start time of the measurements taken for the graph. For our purposes, it does not matter what the value of distance is; in all three cases, the distance does not change during the measurement period. This means that all three objects have a speed of 0 m/s during this time.

Therefore, all three objects would have identical speed–time graphs. Specifically, the speed–time graph for each of the objects would look like this:

This speed–time graph shows a value of 0 m/s for every value of time. Therefore, it corresponds to an object with a constant speed of 0 m/s, in other words, an object that is not moving.

Now let’s consider the following distance–time graph.

This graph shows an object whose distance increases at a constant rate. We can see this as follows.

In the diagram below, we have the same graph, but now we have highlighted two triangles. The hypotenuse of both triangles lies along the line plotted on the distance–time graph.

The horizontal and vertical sides of the red-dashed triangle each have a side length of 1. For the horizontal side, this is in units of seconds, while for the vertical side, it is in units of metres. What this triangle is telling us is that the object travels a distance of 1 m the first 1 s shown on the graph.

Now looking at the blue triangle, we see that the vertical and horizontal sides each have a side length of 2. This tells us that the object travels a total distance of 2 m in the first 2 s.

The sides of the blue triangle are in the same proportion as the sides of the red triangle. In fact, no matter where we draw a triangle, so long as its hypotenuse lies along the line on the graph, we will find that the horizontal and vertical sides of the triangle are in this same proportion. This shows that the object travels equal distances in equal times.

We may recall that if an object moves equal distances in equal times, then that object moves with a constant speed.

By drawing two triangles in the way we have shown, it is straightforward to verify that the horizontal and vertical sides of the triangles will be in the same proportion for any distance–time graph where the distance increases with time as a straight line.

In other words, all distance–time graphs that show a straight line represent motion at a constant speed. Therefore, the corresponding speed–time graph will always be a horizontal line.

The steeper a line is on a distance–time graph, the greater the distance moved by the object in each unit of time. We can also recall that the greater the distance moved per unit time, the greater the speed of an object. Therefore, we can see that the steeper the line on a distance–time graph is, the higher up the corresponding horizontal line will be on a speed–time graph.

For example, let’s consider the diagram below.

Each color line on the distance–time graph corresponds to the same color line on the speed–time graph. On the distance–time graph, the red line is the least steep. Therefore, this line shows the smallest speed. The smallest speed is represented by the lowest horizontal line on the speed–time graph. Similarly, the blue line on the distance–time graph is the steepest. Therefore, it corresponds to the highest horizontal line on the speed–time graph. Meanwhile, the green line lies between these two.

Let’s look at one more example problem.

### Example 4: Identifying Which Line on a Speed–Time Graph Corresponds to a Given Line on a Distance–Time Graph

Which color line on the speed–time graph shows the motion of the object on the distance–time graph?

### Answer

In this question, we are presented with a distance–time graph along with a speed–time graph. We are asked to identify which of the two lines on the speed–time graph shows the motion of the object on the distance–time graph.

Looking at the distance–time graph, we see that this graph shows a straight line. The distance increases in equal proportion to the time, which means that the object travels equal distances in equal times.

Therefore, we know that the distance–time graph shows the motion of an object that moves at a constant speed.

If we now look at the speed–time graph, we have two potential choices. We need to work out whether it is the green line or the red line that represents motion at a constant speed.

The red line shows speed increasing as time increases. Since the speed is increasing, it cannot be constant. So the red line cannot represent the motion at a constant speed shown in the distance–time graph.

The green line on the speed–time graph is a horizontal line. This shows that the speed of the object has the same value for all values of time. Therefore, the speed of that object is not changing; or, in other words, this green line shows motion at a constant speed.

So, our answer to the question is that it is the green line on the speed–time graph that shows the motion of the object on the distance–time graph.

Let us now finish by summarizing what has been learned in this explainer.

### Key Points

- We can show the motion of an object using a speed–time graph. This is a graph that plots speed on the -axis against time on the -axis.
- Motion at a constant speed is represented by a horizontal line on a speed–time graph. A line that is higher up on the graph represents a greater speed.
- If our speed–time graph has a numerical scale on the axes, we can read off the speed for a point plotted on the graph by tracing a horizontal line across from that point to the speed axis. The value on the speed axis where this line meets it is the speed at this point.
- Consider a horizontal line on a speed–time graph, for an object moving at a constant speed. The line will meet, or intersect, the speed axis at some height. Reading the value on the speed axis at this height gives us the speed of the object.