# Lesson Explainer: Interpreting Graphs of Motion Science

In this explainer, we will learn how to compare a distance–time graph to a description of the motion of an object.

Graphs are a really useful way of representing motion. There are many different kinds of graph that we can use to do this, but in this explainer, we will be looking at one specific type of graph: the distance–time graph.

A distance–time graph shows distance on the vertical axis and time on the horizontal axis, like this:

There are actually two different meanings of “distance”:

• the total distance traveled by an object,
• the distance of an object from some point.

These are similar but sometimes give us different measurements. In this explainer, we will be using the first definition of distance. This means that all of our distance–time graphs will show a plot of how the total distance traveled by an object changes over time. Furthermore, it means that we will never see a distance–time graph where the distance decreases over time, because the total distance traveled by an object can never decrease.

Let’s look at some examples of distance–time graphs and see how we can interpret them.

Here is our first example:

Here, we can see a red line plotted on the axes of a distance–time graph. We can see that this line is completely straight and horizontal.

When interpreting graphs of motion, it is a good idea to always start on the left. This is because the horizontal axis is for time, so the left of our graph represents the earliest time at which the distance was measured. Let’s mark this time on our graph with a orange arrow.

Since we have distance on the vertical axis, the “height” of the line above the horizontal axis represents the total distance traveled by an object at a certain time. Let’s think about the height of the red line at the moment in time labeled by the orange arrow. In this case, the plotted line is actually touching the horizontal axis, so its height above this axis is zero. In other words, the distance traveled by the object is zero at this point in time.

Just like we have used a orange arrow to represent a specific time on the time axis, let’s use a blue arrow to represent the distance traveled by the object at that time, as measured on the distance axis:

Because our definition of “distance” is “the distance an object has traveled”, we know that the initial distance of an object on a distance–time graph will always be zero.

Now, let’s look to the right and see how the distance traveled by the object changes as time goes on. In this diagram, we have marked a later moment in time with a orange arrow, and the distance of the object at this time (that is, the height of the graph) is marked with a blue arrow:

In this case, we can see that as we look further to the right, the height of the graph remains unchanged. This means that the distance traveled by the object remains unchanged! So, our interpretation of this graph is that it just shows a stationary object.

### Rule: Horizontal Distance–Time Graphs

A straight horizontal line on a distance–time graph represents a stationary object.

Let’s look at another example.

Here, we can see that the graph is once again a straight line, but this time, it has a slope. Again, we will interpret this graph by considering how the height of the graph changes as it goes from left to right. Let’s mark the earliest time on our graph with a orange arrow and the distance of the object at this time with a blue arrow:

Once again, the graph touches the horizontal axis on the left. The distance traveled by the object when we first start measuring will always be zero, so distance–time graphs should always start from the origin point (where the axes meet).

Let’s now look forward in time. In this diagram, we will mark a point further to the right along the time axis with a orange arrow.

The distance traveled by the object at this time is given by the height of the graph at this point. Let’s draw a vertical dashed line from the time we have marked on the horizontal axis up to where it meets the line on the graph. Then, we can draw a dashed line horizontally from this point, going left to where it meets the vertical axis:

The point where the dashed line meets the distance axis is marked on the diagram with a blue arrow. This point on the distance axis represents the total distance traveled by the object by this point in time.

So, on this graph, the object’s distance has changed over time. In other words, the object is moving! Because the line is straight (that is, it has a constant slope), we know that the object’s distance is changing at a constant rate. In other words, this graph shows us that the object is moving at a constant speed.

### Rule: Distance–Time Graphs with a Constant Slope

A straight, sloped distance–time graph represents an object moving at a constant speed.

A distance–time graph like this can also give us information about how fast an object is traveling. Let’s compare the previous graph to a graph with a steeper slope. The following diagram shows the previous graph on the left and a new graph with a steeper slope on the right.

On each of these graphs, we have marked the same time on the time axis with a orange arrow. The blue arrows indicate the distance that each object has traveled by that moment in time. On the steeper graph, we can see that the object has traveled a greater distance within that time. This means that the object represented by the graph on the right must be traveling faster than the object represented by the graph on the left.

In fact, the slope of the line on a distance–time graph is equal to the speed of the object. This means that steeper distance–time graphs correspond to faster motion.

### Rule: Slope of a Line on a Distance–Time Graph

The slope of a line on a distance–time graph is equal to the speed of an object. The steeper the line, the faster the object is traveling. Lines with zero slope (that is, horizontal lines) represent stationary objects.

All the graphs we have looked at so far just have a single straight line on them. But this is not always the case! In the next example, the line on the distance–time graph consists of several “line segments” with different slopes:

To interpret the motion represented by this graph, we can look at each straight line segment separately. First, let’s pay attention to the line segment on the left as shown in the following diagram. This represents the first part of the object’s motion.

We can see that this line segment is straight, which tells us that it represents motion at a constant speed. The object moves at this constant speed between the two times indicated on the time axis below:

Now, let’s look at the horizontal line segment as shown in the following diagram. This represents the second part of the object’s motion.

We can see that this line segment is also straight; however, this time it is horizontal (i.e., it has a slope of zero). This tells us that the object is stationary (its distance does not change) between the two times indicated on the time axis below.

Now, let’s look at the final part of the graph, as shown in the diagram below.

This line segment is also straight, and we can see that it is sloped. We can compare this to the other sloped section of the graph on the left. The sloped section on the right has a steeper slope, which tells us that the object is traveling faster between the two times indicated in this diagram than it was at the beginning of its journey.

We can put all of this information together to describe the total journey represented by the distance–time graph. First, the object moves at a constant speed. Next, the object remains stationary at some distance away from its starting point. Finally, the object travels at a constant speed again but faster than it did previously.

Let’s look at an example question where we interpret a distance–time graph that shows several stages of motion.

### Example 1: Interpreting a Distance–Time Graph Showing Motion at Different Speeds

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

### Answer

Let’s start by remembering that a distance–time graph shows how the distance of an object varies over time. Distance is measured on the vertical axis, and time is measured on the horizontal axis.

Remember that a straight, sloped line on a distance–time graph represents motion at a constant speed. In addition, a straight horizontal line on a distance–time graph represents the fact that an object is stationary.

Looking at the two graphs provided, we can see that they both share some characteristics. First, they both consist of three straight line segments. In addition to this, the three straight line segments follow a similar pattern on each graph. Reading from left to right, the first line segment (on the left of each graph) is sloped, the second line segment (in the middle of each graph) is horizontal, and the third segment (on the right of each graph) is sloped too.

This means that each graph shows an object that moves at a constant speed, then remains stationary, then moves again at a constant speed.

The difference between the two graphs relates to the slopes of the first and third line segments. Here, we need to remember that the slope of a straight line on a distance–time graph is equal to the speed of an object. So, a steeper line corresponds to faster movement.

On graph A, the first line segment is less steep than the third line segment. This means that, for the object shown by graph A, the first part of the object’s motion is slower than the last part of the object’s motion. In other words, graph A shows an object moving slowly, then stopping for a while, then moving quickly.

On graph B, the opposite is true: the first line segment is steeper than the third line segment, meaning the first part of the object’s motion is faster than the last part of its motion. In other words, graph B shows an object moving quickly, then stopping for a while, then moving slowly.

Now we can answer our question: the graph that shows an object initially moving with constant speed that stops moving and then starts moving again with a greater constant speed is graph A.

So far, we have only looked at distance–time graphs consisting of straight lines. However, it is common for the line on a distance–time graph to be curved. For our next example, let’s consider this distance–time graph:

The key to interpreting curved lines on distance–time graphs is to remember that slope on a distance–time graph corresponds to speed. The slope of a curve is always changing over time. This means that a curve represents motion where the speed is always changing over time.

Let’s look at a specific point in time on our graph, marked by the orange arrow in this diagram:

What is the slope of the graph at this point? We can draw a dashed pink line vertically up from the time we have chosen on the horizontal axis until the point where it meets the graph. Then, let’s draw a green line to show the slope of the graph at this point:

The slope of the graph at this point represents the speed of the object at the moment in time we have chosen to look at. Let’s compare this to the slope of the graph at a later moment in time.

In the above diagram, we can clearly see that the slope of the line is much greater at this later moment in time than it was at the earlier time we chose. In fact, the slope of the line increases more and more as we get further along the time axis. This tells us that the object is speeding up. In other words, it is accelerating.

### Rule: Curved Lines on Distance–Time Graphs

A curve on a distance–time graphs shows acceleration (change of speed). If the slope of the curve is increasing, the speed of the object is increasing. If the slope of the curve is decreasing, the speed of the object is decreasing.

It is important to remember that, technically, an object is “accelerating” if it is speeding up or slowing down. However, in everyday speech, we often only say an object is “accelerating” when it is speeding up. When an object is slowing down, we often say it is “decelerating.”

### Example 2: Identifying a Distance–Time Graph Showing Uniform Deceleration

Which of the following distance–time graphs shows an object that has uniform deceleration?

### Answer

We need to identify the graph that shows an object experiencing uniform deceleration. In this context, “uniform” just means the object is decelerating (that is, slowing down) at a constant rate.

In order to answer this question, we can start by remembering that the slope of a distance–time graph at any point represents the speed of an object at that moment in time. Let’s keep this in mind as we look at the variation in the slope of the line on each graph.

In graph A, the slope of the line smoothly increases as time goes on (going from left to right). This tells us that the object is going from a low speed to a high speed—it is speeding up! This means that option A cannot be the correct answer.

In graph B, the slope of the line again smoothly increases over time, so option B cannot be correct either. We should also take note that, unlike graph A, graph B shows distance decreasing over time. Remember that a distance–time graph shows the total distance traveled by an object. This means that we can never have a distance–time graph that shows distance decreasing, like this graph does. This graph is not a valid distance–time graph.

In graph C, we can see that the line that starts off relatively steep and smoothly becomes less steep as time goes on. This means that the object must be smoothly decreasing its speed. In other words, this graph shows an object decelerating! It is actually difficult to know whether this is truly uniform deceleration without having more information, but, for now, C looks like a good answer. Let’s take a look at the fourth option to make sure.

In graph D, the line initially has a steep slope. As time goes on, the slope of the line smoothly becomes less steep. This suggests that the graph shows an object smoothly decreasing its speed. In other words, this graph looks like it shows an object that could be experiencing uniform deceleration too. However, we need to be careful here! This graph, like graph B, actually shows distance decreasing. This means that graph D is actually not a valid distance–time graph either.

So, since options A, B, and D cannot be correct, option C must be the correct answer.

### Example 3: Identifying the Distance–Time Graph of an Object That Accelerates and Decelerates

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

### Answer

To answer this question, we should start by recalling that the slope of a line on an object’s distance–time graph corresponds to the speed of the object.

With this in mind, let’s look at the variation in the slope of the line on each graph.

In option A, the line on the graph is initially straight. This graph therefore shows us an object that is initially moving at a constant speed. After some time, the line bends smoothly and its slope decreases. This means that the the speed of the object shown by this graph is steadily decreasing. In other words, the object is decelerating. The line then straightens out again but is less steep than before. This tells us that the object shown by this graph continues moving at a constant speed but more slowly than before.

In option B, the line on the graph is initially straight, so the object shown by this graph must be initially moving at a constant speed. The line then suddenly becomes horizontal. This means that the object suddenly stops moving.

In option C, the line on the graph is initially curving upward: its slope is increasing. This means the object’s speed is increasing, or, in other words, the object is accelerating. At a point about halfway along the time axis, the line on the graph starts to curve the other way such that its slope is decreasing. This means that the object is slowing down, or decelerating.

Out of the available options, only option C shows us an object accelerating and then decelerating. This means C is the correct answer to the question.

### Key Points

• A distance–time graph shows the total distance traveled by an object on the vertical axis and time on the horizontal axis.
• The slope of a distance–time graph at a given time is equal to the speed of the object at that time. A steeper slope corresponds to a higher speed. This provides the following conclusions
• A straight horizontal line means the object is stationary.
• A straight sloped line means the object is moving at a constant speed.
• A curved line means the object is accelerating.

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