Lesson Video: Interpreting Graphs of Motion Science • Third Year of Preparatory School

Video Transcript

In this video, we’ll learn how to interpret graphs of motion. Graphs like this one can be a really useful way of representing motion. They can tell us a lot about how an object is moving without needing us to do any calculations. But we first need to learn how to interpret what they’re saying. In this video, we’re going to learn how to interpret one specific type of graph, distance–time graphs.

Distance–time graphs show distance on the vertical axis and time on the horizontal axis. So a line drawn on a distance–time graph shows us how the distance traveled by an object changes over time. Now, before we look at how to interpret some specific examples of distance–time graphs, we first need to clarify exactly what we mean when we say distance.

Now, distance seems like a pretty straightforward concept. However, the word distance can actually be used to refer to two slightly different things. One way that we use the word distance is to refer to the total distance that something has traveled. For example, let’s imagine that we have a runner on a race track, where a full lap of the track is 200 meters long. Let’s say that our runner runs one lap, coming to a stop just one meter over the finish line. The runner could then say that their distance is 201 meters because that is the total distance that they’ve traveled.

However, another way that we use the word distance is to refer to the distance between two points. So an observer might say that our runner’s distance is actually just one meter from the start line, referring to the current distance between the runner and the start line. We can see how different definitions of the word distance give us different measurements. The total distance that the runner has traveled is 201 meters. But the distance between the runner and the start line is only one meter. In this video, whenever we talk about distance, we’ll always be using this first meaning. So distance will always refer to the total distance traveled by an object, not the distance between two points. This means that the distance–time graphs that we’re going to look at will show us how the total distance traveled by an object varies over time.

As a result of this, we’ll never see a graph where the distance decreases over time like this because the total distance traveled by an object can never decrease. So now that we’ve established exactly what we mean by distance, let’s take a look at our first example of a distance–time graph. Here we can see a line plotted on the axes. We can see that the line is completely straight and horizontal. Like all distance–time graphs, this tells us about the motion of some object. But how can we interpret this graph? Let’s start by looking on the left-hand side of the graph. The reason we start on the left is because the horizontal axis represents time. This means that the left of our graph represents the earliest time at which distance was measured. In other words, it represents the start of the object’s journey. Let’s mark this time with an arrow.

Now, because we have distance on the vertical axis, we know the height of the line on the graph represents distance. Now, at this time we’ve indicated, we can see that the height of the line plotted on the graph is zero. That is, it touches the horizontal axis. This means that the distance traveled by the object at this point in time is zero. Of course, it makes sense that an object’s distance would be zero at the very start of its journey. And as we’ll see, all distance–time graphs actually start with a distance of zero.

Now, on this graph, the plotted line is horizontal. This means that no matter what point in time we look at, the height of the graph will always be the same, in this case zero. This means that the distance doesn’t change over time. So our interpretation of this graph is that it shows us an object which isn’t moving, in other words, a stationary object. This gives us a general rule. On any distance–time graph, a horizontal line means the object is stationary.

Okay, now let’s look at another example of a distance–time graph. So here we once again have a straight line plotted on the graph, but this time it’s sloped instead of horizontal. Once again, we can interpret this graph by starting at the earliest possible time, that is, the start of the object’s journey, and thinking about how the distance changes as we move to the right. We can see that just like before, at the earliest time indicated on the graph, the line is touching the horizontal axis. This means that the distance of the object at the start of its journey is zero. If we look at a later point in time, for example, here, we can see that the height of the line has changed, which means that the distance of the object has changed.

Starting at the time we’ve indicated, we can draw a dashed line vertically upward from the axis until it meets the line plotted on the graph. We can then draw a line horizontally from this point until it meets the distance axis. The point where this dashed line meets the distance axis represents the distance of the object at the time we’ve chosen. So this graph shows us that the distance traveled by the object is increasing over time. In other words, the object is moving. Now, because the line plotted on the graph is straight, in other words, it has a constant slope, we know that the distance of the object is increasing at a constant rate. So this graph shows us an object which is moving at a constant speed.

This gives us another general rule. A straight sloped line on a distance–time graph means the object is moving at a constant speed. Now, as well as telling us that an object is moving at a constant speed, a graph like this can also tell us how fast an object is moving. To see how this works, let’s compare two different distance–time graphs. Now, both of these graphs consist of a straight sloped line, which means they both show us an object that’s moving at a constant speed. And if we look at the earliest time on each graph, we can see that the initial distance is zero as we would expect. But what did these graphs show us about the change in distance over time?

Now it’s important to remember here that if we want to compare two distance–time graphs accurately, they should both have identical axes. That means they should both use the same units as each other for distance and time and the axes on both graphs should have the same scales. Assuming that these graphs have identical axes, we can see that the steeper graph shows distance changing more rapidly than the less steep graph. This becomes clear if we pick a certain moment in time and compare the distance shown by each graph at that time. Once again, we do this by drawing a dashed line vertically upward from our chosen time until it reaches the line plotted on the graph. We then extend the line horizontally to the left until it reaches the distance axis.

We can now clearly see that the distance represented by the top graph at our chosen moment in time is much less than the distance represented by the bottom graph at the same moment in time. This means that the object represented by the top graph must be moving more slowly than the object represented by the bottom graph. We can conclude that on a distance–time graph, a steeper line corresponds to faster motion. In fact, the general rule is that the slope of the line is equal to the speed of the object. This is the most important rule that we’re gonna learn about distance–time graphs. Not only does it tell us that steeper lines on distance–time graphs correspond to faster motion, but it also sums up the previous two rules that we’ve learned.

For example, the first rule tells us that a horizontal line means a stationary object. However, we can get this from the third rule as well. A horizontal line is a line whose slope is zero. And the third rule tells us that this means the speed of the object is zero. We can see that the fact a horizontal line represents a stationary object is actually a result of the fact that the slope of the line is equal to the speed of the object. Similarly, the second rule tells us that a straight sloped line represents an object moving at a constant speed. But we can also get this from the third rule. After all, a straight line is a line whose slope is constant. It doesn’t change over time. And according to the third rule, if the slope is constant, then the speed of the object must be constant. So we can see that this second rule too is just a result of the third rule.

So now that we’ve looked at some simple examples of distance–time graphs and learned some of the rules that we can use to interpret them, let’s look at an example question that asks us to interpret a slightly more complicated graph.

Which of the following distance–time graphs shows an object initially moving with constant speed that then stops moving and then starts moving again with a greater constant speed?

Let’s start by recalling that a distance–time graph shows us how the total distance traveled by an object varies over time. As we can see from the answer options provided, distance–time graphs show distance on the vertical axis and time on the horizontal axis. We need to figure out which of these graphs shows an object which moves in the manner described in the question. So we want to choose the graph that shows us an object which moves at a constant speed, then stops, then moves again at a greater constant speed.

There are a couple of rules we can use to help us do this. Firstly, a horizontal line on a distance–time graph represents a stationary object. And secondly, a straight sloped line represents an object moving at a constant speed. Looking at the two graphs provided, we can see that they both share some characteristics. Firstly, they both consist of three straight-line segments. And these straight-line segments follow a similar pattern on each graph. Let’s look at these segments in time order, that is, moving from left to right on each graph.

We can see that the first line segment on each graph is a straight sloped line. We know that this represents an object moving at a constant speed. The next segment on each graph is horizontal, which we know represents a stationary object. And the third segment on each graph is again a straight sloped line representing constant speed. So we can see that both of these graphs show us an object which moves at a constant speed, then stops moving, and then moves at a constant speed again. The difference between these graphs relates to how steep these sloped sections are. In order to interpret these differences, we need to remember another rule. The slope of the line is equal to the speed of the object. This means that steeper sections of the graph, like this and this, correspond to faster motion than less steep sections of the graph, like this and this.

Once we’ve spotted this, we can see that the graph given in option (A) shows an object which moves relatively quickly, which then stops moving for a bit and then moves relatively slowly, whereas the graph given in option (B) shows us an object which moves relatively slowly, then stops moving, and then moves again relatively quickly. So we can see that the correct answer is option (B). This graph shows us an object initially moving with a constant speed that then stops moving and then starts moving again with a greater constant speed.

Now, so far in this video, we’ve only looked at distance–time graphs which are made up of straight lines. Before we finish, let’s quickly take a look at a distance–time graph where this isn’t the case. Here, we can see a distance–time graph where the plotted line is curved. But what exactly does this mean? Well, the key to interpreting graphs like this is to remember our rule from before. The slope of the line on a distance–time graph is equal to the speed of the object. Now, on a curved graph, the slope of the line is different in different places. If we choose this point on the time axis, for example, we can see that at this point the slope of the graph is something like this, whereas at this later point in time, the slope of the graph is more like this. We can clearly see that this slope is much steeper than this slope.

Since we know that the slope of the line is equal to the speed of the object, the way that we interpret this change in slope is that the speed of the object must be changing. And the slope of the graph at these specific times represents the speed of the object at those times. So at the moment in time where the graph is very steep, we know that the object is moving very quickly. And at the time where the graph is not very steep, we know that the object is not moving very quickly. In this case, the graph is curved in such a way that it gets steeper and steeper as we move forwards in time. This means that the object must be getting faster and faster as time goes on. This means we can say that the object shown by this graph must be accelerating.

However, distance–time graphs can curve in different ways like this, for example. But we can always interpret these graphs just by remembering this rule. This time, we can see that the graph starts off fairly steep, but the slope decreases as we go forward in time. This tells us that the object starts off fast but gets slower and slower as time goes on. So our interpretation of this graph is that it shows us an object which is decelerating.

With all of this in mind, let’s take a look at one more practice question. This time, we’ll be interpreting a distance–time graph where the line is curved.

Which of the following distance–time graphs shows an object that initially accelerates and then decelerates?

We’ve been given three options to choose from: (A), (B), and (C). In order to figure out which one of these distance–time graphs shows an object that accelerates and then decelerates, we just need to remember one rule. The slope of the line on a distance–time graph is equal to the speed of the object. With this in mind, let’s look at the variation of the slope of the line on each of these graphs. In each case, we’ll start by looking on the left of the graph, which corresponds to the start of the object’s journey. And we’ll see how the slope of the line changes as we move forwards in time.

Let’s start with option (A). We can see that this graph is made up of two straight-line segments. The first of these is both straight and sloped. In other words, it has a constant slope. Since this rule tells us that the slope of the line is equal to the speed of the object, this line segment with a constant slope indicates that the object has a constant speed. So for the first part of the journey shown in option (A), the object moves with a constant speed.

For the next part of the object’s journey, the graph is just a straight horizontal line. Since the graph is horizontal, this means that slope is zero. And because the slope of the line is equal to the speed of the object, that means that this section of the graph shows the object is stationary. So overall, graph (A) shows us an object which initially travels at a constant speed and then immediately comes to a stop. It doesn’t show us an object that accelerates and then decelerates. So we know that option (A) is not the correct answer.

Let’s take a look at graph (B). We can see that the first part of this graph is curved upward. This means that the slope of the line is increasing as we move forward in time. And we know from this rule that if the slope of the line is increasing, then the speed of the object must be increasing. In other words, the first part of this graph shows us an object which is accelerating. Now, although this graph initially curves upward, we can see that this changes and later on it curves the other way. We can say that at this part of the graph, the slope is decreasing. And if the slope is decreasing, then the object’s speed must be decreasing. We can say that at this point, the object is decelerating. So it looks like option (B) is the correct answer. But let’s take a quick look at option (C) just to be sure.

Now, the first part of this graph is actually a straight line, which means that the object represented by this graph is initially traveling at a constant speed. Next, the graph curves in such a way that its slope decreases. So this portion of the graph shows us that the object’s speed is decreasing, in other words, is decelerating. The last section of this graph is also a straight line, showing us that the object again travels at a constant speed, although we know that it’s traveling slower than it was previously because the slope here is less steep than the slope here.

But overall we can see that graph (C) does not match the description of motion given in the question. So we can now be sure that option (B) is the right answer. The fact that the slope of this graph initially increases and then decreases means that it shows us an object which initially accelerates and then decelerates.

Let’s now finish by recapping the key points that we’ve learned in this video. Firstly, we’ve seen that a distance–time graph shows the total distance traveled by an object on the vertical axis and time on the horizontal axis. The most important rule to remember when interpreting distance–time graphs is that the slope of the graph at a given time is equal to the object’s speed at that time. This means that a horizontal line which has zero slope means the object is stationary. A straight sloped line means the object is moving at a constant speed. And a curved line means the object is accelerating or decelerating.