Lesson Explainer: Graphing Velocity Physics

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

Distance and speed are scalar quantities and so only have a magnitude. Distance and speed can only have positive values. Values plotted on distance–time graphs and speed–time graphs, therefore, only occur within the part of the graph bounded by positive values of both axes of the graph, as shown in the following figure.

Displacement and velocity are vector quantities and so have a direction as well as a magnitude. For motion along a line, one direction along the line corresponds to positive values and the opposite direction corresponds to negative values. Points plotted on displacement–time and velocity–time graphs, therefore, occur within the part of the graph bounded by positive values of time, as shown in the following figure.

Positive values of displacement or velocity correspond to points above the time axis and negative values to points below the time axis.

We can consider a velocity–time graph showing two objects that move in opposite directions at the same speed.

The following figure shows a speed–time graph that represents either of these objects.

Now, let us consider an object that has an initial velocity of 1 m/s that slows down, changes direction, and has a final velocity of 1 m/s. The following figure shows the velocity–time graph and the speed–time graph for the object.

We see that at the instant at which the velocity is zero, the speed is zero.

We also see that the gradient of the line of the velocity–time graph has the same magnitude as that of either of the lines of the speed–time graph.

Where the velocity is approaching a value of zero, the gradient of the speed–time graph is negative. Where the velocity is diverging from a value of zero, the gradient of the speed–time graph is positive.

This relationship between the velocity and the speed is also true if the initial velocity is negative and the final velocity is positive, as shown in the following figure.

We can see from this that it does not matter whether the velocity approaches a value of zero from an initially positive or negative value; in either case, this results in a negative gradient for the line of the speed–time graph.

We can see also that it does not matter whether the velocity diverges from a value of zero toward a positive or negative value; in either case, this results in a positive gradient for the line of the speed–time graph.

Let us now look at an example in which the velocity and speed of objects are compared by inspection of a velocity–time graph.

Example 1: Comparing the Speeds and Velocities of Objects Using a Velocity–Time Graph

The change in velocity of two objects with time is shown in the graph.

  1. Do the two objects have the same speed?
  2. Are the two objects moved equal distances from their initial positions?

Answer

Part 1

For each of the lines on the velocity–time graph, we can draw the equivalent line on a speed–time graph, as shown in the following figure.

We can now compare the speeds of the two objects. The speeds of both objects change, but we can see that they do not change by the same amount in the same time. The following figure shows the average speeds for both objects as dashed horizontal lines.

The average speeds are clearly not equal.

Part 2

The distance moved by an object equals the product of speed and time, which is equal to the area under the line in a speed–time graph. This area is shown for both objects in the following figure.

These areas are clearly not equal, so the objects do not move the same distance.

The relationship between speed–time graphs and velocity–time graphs is very similar to the relationship between distance–time graphs and displacement–time graphs.

Consider the following graphs.

The velocity–time graph and displacement–time graph each show two objects that move at the same speed in opposite directions. The speed of these objects is the same as the speed shown in the speed–time graph and the distance–time graph.

Let us look at an example involving interpreting a displacement–time graph for two moving objects.

Example 2: Comparing the Speeds and Velocities of Objects Using a Displacement–Time Graph

The change in the displacement of two objects with time is shown in the graph. The gray arrows in the diagram are the same length.

  1. Do the two objects have the same velocity?
  2. Do the two objects have the same speed?

Answer

Part 1

The displacements of the objects diverge from zero at the same rate. This is clear from the fact that the gray arrows that correspond to the magnitudes of the displacements of the objects at the same instant are the same and that both lines are straight.

One line diverges from zero toward positive displacement values, and the other line diverges from zero toward negative displacement values. This means that the objects move in opposite directions. The displacements of the objects are, therefore, in opposite directions.

Velocity is the time rate of change of displacement. Velocity is, therefore, a vector quantity, and so the definition of a velocity includes its direction. As the objects have displacements in different directions, the velocities are not the same.

Part 2

The magnitude of the displacement of an object is the distance moved by the object in a single direction. The magnitudes of the displacements of the objects are equal. These displacements occur in equal times. Speed is the time rate of change of distance, and so the speeds of the objects are equal.

Let us now look at another example.

Example 3: Comparing the Speeds and Velocities of Objects Using a Displacement–Time Graph

The change in the displacement of two objects with time is shown in the graph. The gray arrows in the diagram are the same length.

  1. Do the two objects have the same velocity?
  2. Do the two objects have the same speed?

Answer

Part 1

The displacements of the objects change with time identically for the first half of the motion of the objects. After that instant, the direction of the displacement of the object shown by the red dashed line reverses. So, the objects do not have the same velocity.

Part 2

The motions of the objects correspond to the horizontal lengths of the arrows shown in the following figure.

This diagram can be amended to show only the horizontal lengths of the arrows.

Note that the red line is split into two lines to make it easy to visualize the motion, but the actual motion of the object is on the same line; the object just reverses its direction of motion.

We see from this that the total horizontal lengths of the arrows are equal. This means that the distances moved by the objects are equal.

The objects move equal distances in equal times, and so they have equal speeds.

Let us now look at another example.

Example 4: Comparing the Speeds and Velocities of Objects Using a Displacement–Time Graph

The change in displacement of two objects with time is shown in the graph. The lines plotted on the graph are parallel.

  1. Do the two objects have the same velocity?
  2. Do the two objects have the same speed?

Answer

Part 1

The displacements of the objects change with time identically as the blue and red line are parallel. The initial displacements are unequal, however.

The motions of the objects correspond to the arrows shown in the following figure.

In the graph above, O is the point from which displacement is measured and 𝑠 is the magnitude of half the displacement of any of the objects.

The change in the displacements of the objects due to their motion is their final displacements minus their initial displacements.

The blue arrow represents a change of displacement given by 02𝑠=2𝑠.

The red arrow represents a change of displacement given by 𝑠𝑠=2𝑠.

The displacements are the same in both cases.

The objects have the same displacement in the same time, so they must have the same velocity.

Another way to think of this is to recall that the gradients of the lines are of equal magnitudes and signs. The gradient of a displacement–time graph equals a velocity. Equal gradients, therefore, correspond to equal velocities.

Part 2

Since the objects did not change their direction of motion, then the speed is equal to the magnitude of velocity. The velocities of the objects are equal, and so their speeds must be equal.

The motion of an object that has a changing velocity is represented on a velocity–time graph by a nonhorizontal line. On a displacement–time graph, the motion of an object moving in such a way is represented by a curved line.

Now, let us look at another example.

Example 5: Identifying the Region of a Displacement–Time Graph in Which an Object Decreases in Speed

The change of an object’s displacement with time is shown in the displacement–time graph. In which region of the graph is the speed of the object decreasing?

Answer

The graph uses the symbol 𝑑 for displacement rather than the more commonly used symbol 𝑠. The question states that this is a displacement–time graph, however, so the vertical axis measures displacement.

An easy mistake to make in this example is to notice that the line curves toward zero displacement in the region DE and state that this must be the region in which speed is decreasing. In fact, the region in which speed is decreasing is BC. Let us consider why this is so.

If we assume that the object moves along a line, then the motion of the object is initially in the positive direction along the line until the instant C. Between C and D, the displacement does not change. From D onward, the displacement is in the negative direction.

The motion of the object corresponds to the following arrows in the positive and negative directions, with the position of the object at the instants labeled on the graph shown along each arrow.

At instants C and D, the object is at rest. Between instant D and instant E, the object is increasing speed in the negative direction, starting from rest.

It is between instants B and C that the object is coming to rest, hence decreasing its speed.

It is worth noting that if we did not assume that the object moved along a line, then this displacement–time graph would be consistent with the horizontal motion of an object shown in the following figure.

In this case, it would be correct to say that the horizontal velocity of the object, rather than its speed, decreased between B and C.

Let us now summarize what we have learned in this explainer.

Key Points

  • Points plotted on displacement–time and velocity–time graphs occur within the part of the graph bounded by positive values of time.
  • For motion of an object along a line, the reversal of the direction of the motion of the object corresponds on a displacement–time graph to the changing of the sign of the slope of the line representing the motion and corresponds on a velocity–time graph to the changing of the place of the line representing the motion (whether it is drawn above or below the time axis).
  • Where the velocity of an object is approaching a value of zero, the gradient of the speed–time graph of the motion of the object is negative. Where the velocity is diverging from a value of zero, the gradient of the speed–time graph is positive. It does not matter whether the velocity approaches zero from the negative or the positive direction.

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