In this explainer, we will learn how to determine speeds from distance–time graphs.
We recall that, on a distance–time graph, a line through a set of points plotted on the graph represents a speed.
The following figure shows a distance–time graph. On the graph, two lines are shown.
We see that the dashed line represents a greater speed than the dotted line.
For the dashed line, there was a greater change of distance than for the blue line. Both lines start at the same time value and end at the same time value.
No numbers are shown on the distance or time axes of the graph. This means that there is no way to know what value of speed is for either line. We can only say that the speed of the dashed line must be greater than that of the blue line.
Let us now consider a distance–time graph that has numbers on its axes. We call this a scaled graph. An example is shown in the following figure.
As the graph is scaled, it is now possible to count distances traveled and times taken to travel these distances.
Let us first consider the dashed line.
The dashed line has a start point and an endpoint. These are shown in the following figure.
First, let us consider what the start point and endpoint show us about the distance traveled.
The start point and endpoint both correspond to numbers on the distance axis of the graph. This is shown in the following figure.
We can see that the start point corresponds to a distance of 0 metres and the endpoint corresponds to a distance of 5 metres.
We call 0 metres the initial distance and we call 5 metres the final distance.
We now compare the final distance and the initial distance to see how much the distance changes. This is shown in the following figure.
We can see that the final distance is 5 metres greater than the initial distance. The change in distance is 5 metres.
We can write this as where is the change in distance.
To know the speed, we must also know the amount of time in which the distance changed by 5 metres.
Let us again look at our start point and endpoint of the line, shown in the following figure.
What we just did for change in distance we can also do for change in time. This is show in the following figure.
From the initial time and final time, we can find the change in time. This is shown in the following figure.
We can see that the final time is 5 seconds greater than the initial time. The change in time is 5 seconds.
We can write this as where is the change in time.
We now have all the information necessary to determine what value of speed is represented by the dashed line.
We can recall that the average speed of an object, , is given by the formula
We have just found the values of and for the dashed line. Using these values, we find that
We can see that
The value of the speed is 1.
The unit of the speed must also be determined. The unit of the speed is given by
The unit is m/s. In words, this is written as “metres per second.”
The speed is 1 metre per second, or 1 m/s.
This speed is shown in the following figure.
Let us now repeat all these steps for the dotted line. This is shown in the following figure.
We can see that and that
We can use the formula with these values. This gives us
This speed is shown in the following figure.
We see that the value of the speed of the dotted line is less than that of the dashed line. This is shown in the following figure.
The change in distance divided by the change in time for a distance–time graph is the gradient of the graph.
This means that the gradient of a distance–time graph equals the speed of the object that has the motion represented by the line showing the change in distance with time.
Let us now practice some examples of finding speed using distance–time graphs.
Example 1: Determining Distance Moved at a Constant Speed Using a Distance–Time Graph
The distance–time graph shows an object moving at a constant speed.
- What distance does the object move between 0 seconds and 1 second?
- What distance does the object move between 4 seconds and 5 seconds?
Answer
Part 1
At a time of 0 seconds, the distance moved is 0 metres.
At a time of 1 second, the distance moved is 1 metre.
The distance moved between 0 seconds and 1 second is given by
The distance changes by 1 metre.
Part 2
At a time of 4 seconds, the distance moved is 4 metres.
At a time of 5 seconds, the distance moved is 5 metres.
The distance moved between 4 seconds and 5 seconds is given by
The distance changes by 1 metre.
This is the same change of distance as between 0 seconds and 1 second. The equal changes in time are to be expected. The change in time between 4 seconds and 5 seconds is equal to the change in time between 0 seconds and 1 second. In each case, time increases by 1 second.
The object has a constant speed and so must have equal changes in distance for equal changes in time throughout its motion.
In the explanations shown in the explainer, the graph is divided into squares.
The vertical length of a square represents a change in distance of 1 metre.
The horizontal length of a square represents a change in time of 1 second.
These are not necessarily the values used for the scaling of the axes of a distance–time graph.
Let us now look at an example where the scale on the graph axes is not 1 metre per square on the distance axis nor 1 second per square on the time axis.
Example 2: Determining Speed Using a Distance–Time Graph
The distance–time graph shows an object moving at a constant speed. What is the speed of the object?
Answer
The speed of the object is found by dividing the distance moved by the object by the time taken to move that distance.
We can see from the graph that the line representing speed goes through the corners of a set of squares on the graph that are shown in the following figure.
Looking at the distance axis, we see that the first square starts at 0 metres, the second square starts at 10 metres, the third square starts at 20 metres, and so on.
The distance increases by 10 metres for each square.
Looking at the time axis, we see that the first square starts at 0 seconds, the second square starts at 10 seconds, the third square starts at 20 seconds, and so on.
Time increases by 10 seconds for each square.
This shows us that each 10-metre increase in distance corresponds to a 10-second increase in time.
The speed of the object is given by the change in distance divided by the change in time. This is given by
The speed has a value and a unit. The value of the speed is given by
The unit of the speed is given by
The speed is 1 m/s.
Let us now look at an example where the changes in distance traveled and time traveled for are not equally scaled on the axes of a graph.
Example 3: Determining Speed Using a Distance–Time Graph
The distance–time graph shows an object moving at a constant speed. What is the speed of the object?
Answer
The speed of the object is found by dividing the distance moved by the object by the time taken to move that distance.
We can see from the graph that the line representing the speed goes through the corners of a set of squares on the graph that are shown in the following figure.
Looking at the distance axis, we see that the first square starts at 0 metres, the second square starts at 2 metres, the third square starts at 4 metres, and so on.
The distance increases by 2 metres for each square.
Looking at the time axis, we see that the first square starts at 0 seconds, the second square starts at 1 second, the third square starts at 2 seconds, and so on.
Time increases by 1 second for each square.
It is very important to understand that this means that although the vertical sides of the squares are the same length as the horizontal sides of the square, the vertical and horizontal sides do not represent changes of the same magnitude.
Each 2-metre increase in distance on this graph only corresponds to a 1-second increase in time.
The speed of the object is given by the change in distance divided by the change in time. This is given by
The speed has a value and a unit. The value of the speed is given by
The unit of the speed is given by
The speed is 2 m/s.
Let us now look at an example where the changes in distance traveled and time traveled for by an object have different values.
Example 4: Determining Speed Using a Distance–Time Graph
The distance–time graph shows an object moving at a constant speed. What is the speed of the object?
Answer
The speed of the object is found by dividing the distance moved by the object by the time taken to move that distance.
We can see from the graph that the line representing the speed goes through the corners of a set of rectangles on the graph that are shown in the following figure.
Looking at the distance axis, we see that the first rectangle starts at 0 metres, the second rectangle starts at 2 metres, the third rectangle starts at 4 metres, and so on.
The distance increases by 2 metres for each rectangle.
Looking at the time axis, we see that the first rectangle starts at 0 seconds, the second rectangle starts at 1 second, and so on.
Time increases by 1 second for each rectangle.
This shows us that each 2-metre increase in distance corresponds to a 1-second increase in time.
The speed of the object is given by the change in distance divided by the change in time. This is given by
The speed has a value and a unit. The value of the speed is given by
The unit of the speed is given by
The speed is 2 m/s.
Let us look at another such example.
Example 5: Determining Speed Using a Distance–Time Graph
The distance–time graph shows an object moving at a constant speed. What is the speed of the object?
Answer
The speed of the object is found by dividing the distance moved by the object by the time taken to move that distance.
We can see from the graph that the line representing the speed goes through the corners of a set of rectangles on the graph that are shown in the following figure.
Looking at the distance axis, we see that the first rectangle starts at 0 metres, the second rectangle starts at 1 metre, the third rectangle starts at 2 metres, and so on.
The distance increases by 1 metre for each rectangle.
Looking at the time axis, we see that the first rectangle starts at 0 seconds, the second rectangle starts at 2 seconds, and so on.
The time increases by 2 seconds for each rectangle.
This shows us that each 1-metre increase in distance corresponds to a 2-second increase in time.
The speed of the object is given by the change in distance divided by the change in time. This is given by
The speed has a value and a unit. The value of the speed is given by
The unit of the speed is given by
The speed is 0.5 m/s.
Let us now summarize what has been learned in this explainer.
Key Points
- A scaled distance–time graph can be used to determine a constant speed.
- The speed represented by a straight line on a distance–time graph equals the gradient of the graph.
- The gradient of a distance–time graph is the change in distance divided by the change in time. This can be written as where is the speed, is the change in distance, and is the change in time.
- The scaling of distances and times on the axes of a distance–time graph can have any values.
