Lesson Video: Calculating Speed from Distance-Time Graphs Science

Video Transcript

In this video, we will learn how to determine speeds from distance–time graphs. Let’s begin by recalling what a distance–time graph shows us. A distance–time graph is a graph that measures distance on the 𝑦-axis against time on the 𝑥-axis. If the graph is labeled with units and a numerical scale for both of the two axes, then we can read off the distance and the time for any given point plotted on the graph.

For example, let’s consider this graph here. We’ll suppose that we want to read the distance and time values for this point. To find the time value, we draw a vertical line down from the point until we get to the time axis. So that’s the horizontal or 𝑥-axis. We then read the value on this time axis at the point where the vertical line meets it. In this case, that value is four. And we can see that the time axis is in units of seconds. So the time at this point is equal to four seconds.

To find the distance value, we go back to this point and draw a horizontal line from it until we get to the distance axis. So that’s the vertical or 𝑦-axis. Reading the value on the distance axis at the point where this line meets it, we find that this value is two. We can see that the distance axis has units of meters. So we have read from the graph that the distance at this point is two meters.

Okay, so now that we’ve reminded ourselves what a distance–time graph is, let’s think about the physical quantity speed. We can recall that the speed of an object is defined as the distance traveled by that object per unit of time. Let’s suppose that we have an object that is moving at a constant speed that we’ll label as 𝑣. If we know that the object moves a distance 𝑑 in a time 𝑡, then the speed 𝑣 of that object is given by 𝑑 divided by 𝑡.

Since a distance–time graph gives us information about both distance and time, then it also tells us about speed. We may recall that a line drawn through points on a distance–time graph represents a speed. In particular, a straight line means a constant speed. The object travels equal amounts of distance in equal amounts of time. Let’s add some units and numerical values to the axes of this graph to see an example of this. First off, let’s look at what happens between a time of zero seconds and a time of one second. We can see that at zero seconds, the object has moved a distance of zero meters. Let’s label this pair of time and distance values with a subscript one. Then, at a time of 𝑡 one equal to zero seconds, the object has moved a distance of 𝑑 one equal to zero meters.

Next, we’ll look at the point on the graph with a time value of one second. Let’s label this with a subscript two so that we have 𝑡 two is equal to one second. We want to find the distance that the object has moved at this time 𝑡 two. We’ll label this distance as 𝑑 two. To find the value of 𝑑 two, we draw a vertical line up from the time of 𝑡 two, which is one second, until this vertical line meets the line plotted on the graph. This point on the graph then corresponds to this time of 𝑡 two equal to one second.

Then, to find the distance 𝑑 two, we draw a horizontal line from this point across to the distance axis. The line hits the axis at a value of one, and the distance axis has units of meters. So the value of 𝑑 two is equal to one meter. In other words, between a time 𝑡 one equal to zero seconds and a time 𝑡 two equal to one second, the object moves from a distance 𝑑 one equal to zero meters to a distance 𝑑 two equal to one meter.

The time taken between this point and this point on the graph is equal to the final time, 𝑡 two, minus the initial time, 𝑡 one. We’ve labeled this time taken as Δ𝑡. Δ is a Greek letter that we typically use to represent a change in a quantity. In this case, that quantity is time. Δ𝑡 is the change in time between the time 𝑡 one and the time 𝑡 two. Subbing in our values for 𝑡 one and 𝑡 two, we have that Δ𝑡 is equal to one second minus zero seconds, which is one second. In the same way, we can define the change in distance moved between this point and this point as Δ𝑑 is equal to 𝑑 two minus 𝑑 one. Subbing in the values for 𝑑 one and 𝑑 two gives us that Δ𝑑 is equal to one meter minus zero meters, which is one meter.

Now, we saw earlier that if we divide the distance moved by an object by the time taken to move that distance, this gives us the object’s speed. In our case, Δ𝑑 is the distance moved between this point and this point on the graph. And Δ𝑡 is the time between these same two points. So between these two points on the graph, the object has moved a distance of Δ𝑑 in a time of Δ𝑡. This means that we can write the speed of the object as the distance moved, Δ𝑑, divided by the time taken, Δ𝑡. For the section of the graph that we’ve looked at, we have Δ𝑑 is equal to one meter and Δ𝑡 is equal to one second. Subbing these values into this expression for the speed, we find that 𝑣 is equal to one meter divided by one second. This works out as a speed of one meter per second.

Now, we said that a straight line on a distance–time graph represents a constant speed. This graph here is clearly a straight line. So let’s check that it does in fact show a constant speed. We’ve already found the speed between zero seconds and one second. This speed was one meter per second. Let’s now check that we get the same value using the part of the graph between two seconds and three seconds. First, we’re going to need to clear some space to do this.

Okay, so to calculate the speed between two seconds and three seconds, we’ll use the exact same approach as we did to find this speed between zero seconds and one second. The first step is to find the distance moved at each of the two time values. We draw a vertical line up from the time of two seconds until we get to the plotted line. Then, we draw a horizontal line across to the distance axis. We can then read from the graph that at a time of two seconds, the object has moved a distance of two meters. We’ll label this time as 𝑡 one and this distance as 𝑑 one.

Now, we need to do the same thing for the time of three seconds. We find that at three seconds, the object has moved a distance of three meters. We’ll label this second pair of values with a subscript two. Just as we did before, we can calculate the distance moved between this point on the graph and this point. This distance moved, Δ𝑑, is equal to 𝑑 two minus 𝑑 one. Subbing in our values, we have that Δ𝑑 is equal to three meters minus two meters, which is one meter.

Similarly, the time interval between the two points, Δ𝑡, is equal to 𝑡 two minus 𝑡 one. Again, subbing in our values, we have that Δ𝑡 is equal to three seconds minus two seconds, which is one second. Let’s now use this equation to calculate the speed of the object between the two points. When we sub in our values for Δ𝑑 and Δ𝑡, we have that the speed 𝑣 is equal to one meter divided by one second. This works out as a speed of one meter per second.

So we have looked at two different parts of the graph and calculated the same speed of one meter per second in each case. In fact, no matter what part of the graph we used, we would get the same result. This graph represents an object moving with a constant speed of one meter per second. To work out this speed, what we did was to calculate the slope of the line on the graph.

The slope of a straight line, which is also known as the gradient of the line, is defined as the change in the vertical coordinate between two points on that line divided by the change in the horizontal coordinate between the same two points. This is exactly what we calculated to find the speed 𝑣. Δ𝑑 is the change in distance or vertical coordinate. And we divided this by Δ𝑡, which is the change in time or horizontal coordinate. So the slope of a line on a distance–time graph tells us the speed of the object. We calculated that this orange line has a slope or speed of one meter per second.

Now, let’s consider this pink line that we have added. Since it’s a straight line, then we know that it represents a constant speed. This means we can use any part of the line to work out what the speed is. Let’s consider the part between zero seconds and one second. We’ll label these times as 𝑡 one and 𝑡 two, respectively. We can see from the graph that at zero seconds, the object has moved a distance of zero meters. We’ll call this distance 𝑑 one. At one second, the object has traveled a distance of two meters. We’ll call this distance 𝑑 two.

We want to find the slope of the line or the speed of the object, which is equal to the change in distance, Δ𝑑, divided by the change in time, Δ𝑡. Δ𝑑 is equal to 𝑑 two minus 𝑑 one, while Δ𝑡 is 𝑡 two minus 𝑡 one. Subbing in our values and evaluating the expressions, we find that Δ𝑑 is two meters and Δ𝑡 is one second. Then, 𝑣 is equal to the slope Δ𝑑 divided by Δ𝑡. For our values, this is two meters divided by one second. This works out as two meters per second.

So on this distance–time graph, we found that the orange line represents a speed of one meter per second, while the pink line represents a speed of two meters per second. The pink line has a greater slope, and so it represents a greater speed. Without any calculation, we can see that the pink line looks steeper than the orange line. For any two straight lines drawn on the same distance–time graph, the steeper line has the greater slope and so represents the greater speed.

The same logic applies when we have a distance–time graph like this, where the line has different sections with different steepnesses. With no calculations, we can look at this graph and see that the line is steepest between zero seconds and one second. This segment of the graph therefore has the greatest slope. And so we can say that the object has the greatest speed between zero seconds and one second. However, for two lines drawn on different distance–time graphs, then the slope of the lines is not necessarily the same as how steep the lines look.

For example, let’s consider these two distance–time graphs. The left-hand one is the same as the first graph that we looked at. We already worked out that this has a slope of one meter per second. The line on the right-hand graph looks steeper than the line on the left-hand graph. Let’s now calculate the slope of this right-hand line. We can recall that the slope is the change in vertical coordinate, distance, divided by the change in horizontal coordinate, time. Let’s consider the part of the graph between these two points on the time axis. Reading off the scale on the time axis, we see that the first point is at a time of zero seconds and the second point is at a time of 10 seconds. The time interval Δ𝑡 between the two points is equal to the second time, 10 seconds, minus the first time, zero seconds. This works out as 10 seconds.

We can see from the graph that at zero seconds the distance moved is zero meters and that at 10 seconds the distance moved is three meters. This means that the distance Δ𝑑 moved in the time Δ𝑡 is equal to three meters minus zero meters. This works out as three meters. So the slope of this graph, Δ𝑑 divided by Δ𝑡, is equal to three meters divided by 10 seconds, which is 0.3 meters per second.

Despite the fact that the right-hand distance–time graph looks steeper than the left-hand graph, the slope of the right-hand graph has a smaller value than the slope of the left-hand one. So the right-hand graph represents a smaller speed. The reason that this was possible is that the two graphs have different scales on the time axis. When we use this equation to calculate the slope, we’re using the numerical values from the axes. So this slope takes the axis scales into account. However, this isn’t the case when we just look at how steep the line looks. So we need to be extra careful when comparing lines drawn on different sets of axes. And, in general, whenever we’re calculating a slope, it’s important to take care to check the scales on each of the two axes.

Okay, let’s finish up by summarizing what we have learnt. First off, we reminded ourselves that a distance–time graph plots distance on the vertical axis against time on the horizontal axis and that a straight line on a distance–time graph represents an object that is moving with a constant speed. We then saw that the slope of a straight line on a graph is defined as the change in the vertical coordinate between two points divided by the change in horizontal coordinate between the same two points.

In this case, the vertical coordinate is distance and the horizontal coordinate is time. So the slope of a line on a distance–time graph is the change in distance, Δ𝑑, divided by the change in time, Δ𝑡. We learnt that the slope is equal to the speed of the object. We then saw that for two lines drawn on the same distance–time graph, the steeper line has the greater slope and so represents the greater speed. However, the same is not necessarily true for lines drawn on different distance–time graphs because the two graphs could have different axis scales to each other.