# Lesson Explainer: Graphing Speed Physics

In this explainer, we will learn how to interpret graphs of distance and time and graphs of speed and time that represent the motion of objects.

A graph is a convenient way to show how two quantities change together. One of the quantities in a graph can be time that passes, so that the graph shows how the other quantity on the graph, whatever it is, changes as time passes. In this explainer, we will look at graphs showing either of the following:

• how the distance traveled by an object changes with time,
• how the speed of an object changes with time.

A graph of how the distance traveled by an object changes with time is called a distance–time graph, or a graph.

A graph of how the speed of an object changes with time is called a speed–time graph, or a graph.

Let us begin by looking at the distance–time graph and speed–time graph for an object that is at rest when time equals zero and does not change speed. This is shown in the following figure.

For both graphs, all the points plotted are on the time axis. This is because the time axis corresponds to zero distance on the distance–time graph and to zero speed on the speed–time graph.

Let us now look at the distance–time graph and speed–time graph for an object that is moving at a constant speed when time equals zero and does not change speed. This is shown in the following figure.

We can draw lines of best fit for both these graphs, as shown in the following figure.

We see that the line for the speed–time graph is horizontal and that the line for the distance–time graph is straight and slopes upward.

The value of on the speed axis of the speed–time graph equals the constant speed of the object.

The speed of an object is related to the distance that the object travels and the time that it travels for by the formula where is the speed of the object, is the distance the object travels, and is the time for which the object travels.

As the speed of the object for these graphs is constant, for equal changes in time there are equal changes in distance, as shown in the following figure.

The gradient of the line of the distance–time graph is equal to the speed of the object.

The following figure shows distance–time and speed–time graphs for two objects, where the speed of the faster object (shown by the green line) is twice the speed of the slower object (shown by the blue line).

We can see that on the distance–time graph the gradient of the green line is twice that of the blue line.

Let us now look at an example of interpreting a distance–time graph.

### Example 1: Identifying the Time Interval in Which an Object Is at Rest

The changes in the distance covered by an object during a time interval are shown in the graph. The graph is split into three sections: I, II, and III. In which section of the graph does the object have a speed of zero?

### Answer

An object has a speed of zero when the change in the distance it travels, , is zero in a time interval . This happens on the graph in a section where the distance does not change throughout the section.

In sections I and III, the distances traveled by the object are greater at the end of the section than at the start of the section. Only in section II are the distances at the start and the end of the section equal. The line of best fit in section II is horizontal, so there is no change in distance in this section. This must mean that the speed in section II is zero.

Let us now look at another similar example.

### Example 2: Identifying the Time Interval in Which the Speed of an Object Is Greatest

The changes in the distance moved by an object during a time interval are shown in the graph. The graph is split into three sections: I, II, and III. In which section of the graph is the speed of the object greatest?

### Answer

An object has its greatest speed when the distance it travels, , in a time interval is greatest.

The following figure shows the line in each section of the graph in a different color. Below this, the figure shows, in the same colors, the distance traveled in the same time interval for each section, directly compared.

We see that is the greatest for the blue line, and so in section I the speed is greatest.

Let us now look at another example.

### Example 3: Analyzing the Distance–Time Graph for an Object That Changes Speed

The graph shows the changes in the distance walked by a dog in a time interval of 8 seconds.

1. At what time did the dog change its speed?
2. Was the dog’s speed higher or lower before the point when its speed changed?
3. What is the difference between the speed of the dog before and after it changed speed?

### Answer

Part 1

The gradient of the line of best fit of the graph changes at a time value of 4 seconds. This is when the speed changes.

Part 2

The gradient of the line is less steep after 4 seconds. The speed of the dog must have decreased.

Part 3

In the first 4 seconds that the dog walks, it walks a distance of 12 metres. The speed of the dog in this time interval is given by where is the speed of the dog, is the distance the dog travels, and is the time for which the dog travels.

We see that

If the motion of the dog over less than full 12 metres is used, we would get the same answer.

For example, we could take the motion of the dog between 6 metres and 12 metres.

We see for the motion of the dog over this distance that

After 4 seconds, the dog walks for a further 4 seconds. In this time, the dog walks a distance of 8 metres. In this time interval, the speed of the dog is given by

We see then that the speed of the dog changes by

Consider the speed–time graph for an object that is shown in the following figure.

We can see that the speed of the object increases as time passes. We can compare this to the speed–time graph for an object that is shown in the following figure.

We can see that the speed of the object decreases as time passes.

Let us look at an example involving speed–time graphs for objects with different speeds.

### Example 4: Comparing the Speeds of Multiple Objects Using a Speed–Time Graph

The changes in the speeds of three objects during the same time interval are shown in the graph.

1. Which object has the greatest initial speed?
2. Which object has the greatest final speed?
3. Which object has the greatest average speed?
4. Which object was not moving?

### Answer

Part 1

The initial speed of an object shown by the graph is the speed of the object at the origin of the time axis. We see that at this time, the yellow line has the greatest speed, so object III has the greatest initial speed.

Part 2

The final speed of an object shown by the graph is the speed of the object at the furthest point along the time axis from its origin. We see that at this time, the blue line has the greatest speed, so object II has the greatest final speed.

Part 3

The average speed of an object with constant speed is the same as the constant speed of the object. Only object I has a constant speed, as the other objects change speeds.

The speeds of object II and object III change uniformly, so the average speed for each object is given by

No numbers are given in the question, but the average speeds for objects II and III can be estimated from the graph by inspection, as shown in the following figure.

We see from this that object I has the greatest average speed.

If the initial and final speed values were given, the average speeds could have been calculated using the equation for average speed.

In this question, no values are stated for the speeds, as the question is intended to require visually interpreting lines on a graph in terms of how their gradients and intercepts compare.

It is important to understand that the scale drawing method for obtaining the answer shown here is included to give a very clear explanation of how to identify the line corresponding to the greatest average speed.

A much simpler explanation might have simply stated that by inspection it can be seen that the red line corresponds to the greatest average speed. This would be correct but would not clarify how to use inspection to compare the speeds.

Part 4

The line for object I is horizontal. A horizontal line on a distance–time graph indicates zero speed, but this graph is a speed–time graph, and the speed of object I is constantly greater than zero. Object I, and the other objects, move. None of the objects do not move.

We have seen that the speed of an object can remain constant, increase, or decrease. The distance moved by an object can only ever remain constant or increase, however.

The following figure shows speed–time and distance–time graphs for an object that increases in speed and an object that decreases in speed. It is important to note that the distance moved by the object increases as time passes for both objects, but the increases do not have equal rates.

The lines for both distance–time graphs are curved.

Let us look at an example involving a distance–time graph with both a straight and a curved line.

### Example 5: Comparing the Motion of Two Objects Using a Distance–Time Graph

The red and blue lines show the change in distance moved with time for two objects.

1. Which color line corresponds to the object that moves the greater distance?
2. Which color line corresponds to the object that moves at the greater average speed?
3. Which color line corresponds to the object that has the greater maximum speed?

### Answer

Part 1

It is important to remember that a distance–time graph should not be confused with a diagram showing the path traveled by an object. The red line is clearly longer than the blue line. If the red and blue lines were in a diagram showing the paths traveled by objects, then the red line would indicate a greater distance.

However, on a distance–time graph, the distance traveled is indicated not by the length of a line but by the greatest value on the distance axis for the line. As distance traveled can never decrease with time, the maximum value on the distance for the line equals the value at the end of the line corresponding to the end of the motion of the object. This is shown in the following figure.

We can see that both the red and blue lines end at the same point, and so the lines show equal distances traveled by objects.

Part 2

The average speed of an object in a time interval is the distance that the object moves in the time interval divided by the time interval. We have already seen that the objects travel the same distance, as the red and blue lines end at the same point. This fact tells us that the objects move for the same amount of time. Two objects that travel the same distance in the same time have the same average speed in that time.

Part 3

The average speeds of the objects are equal, but the speeds of the objects are not equal to each other throughout the time interval in which the objects move. The speed of an object is given by where is the speed of the object, is the distance the object travels, and is the time for which the object travels.

For the blue line, the value of is constant. For the red line, though, the object travels greater distance in a time near the end point of its motion than near the start point of its motion. This is shown in the following figure.

The speeds of the objects are equal only at the instant where the red line and blue line are parallel. After that time, the object corresponding to the red line has greater speed than the object corresponding to the blue line. The red line therefore represents the object that has the greater maximum speed.

Alternatively, instead of answering the question this way, it is possible to consider defining a section of the red line corresponding to an arbitrary time interval, less than the full time for which both objects move.

The section of the red line would start and end at arbitrarily selected points on the -axis that were within the full time interval of the motion of the objects.

This section could then be translated in the positive -direction such that the left-hand end of the section was at the same -axis position as the point on the blue line corresponding to the start of the time interval.

Any such section for which the right-hand end of the red line corresponded to a greater increase in distance than a section of the blue line of the same length starting from the same time would demonstrate that within the time interval for the section, the red line must correspond to a greater average speed than the blue line.

Different time intervals could be chosen by trial and error. For any such interval, the speed represented by the blue line would be the same.

We see then that finding even one time interval for which the red line corresponded to a greater average speed than the blue line would show that the red line represented the object with the greater maximum speed.

Let us now summarize what has been learned in these examples.

### Key Points

• The line of a distance–time graph for an object at rest is a horizontal line along the time axis.
• The line of a speed–time graph for an object with a constant speed is a horizontal line that intersects the speed axis of the graph at the value of the speed of the object.
• The line of a distance–time graph for an object with a constant speed is a straight line with a gradient equal to the speed of the object.
• The line of a speed–time graph for an object with a changing speed is a straight line that has a positive gradient for an increasing speed and a negative gradient for a decreasing speed.
• The line of a distance–time graph for an object with a changing speed is a curved line.

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