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Question Video: Using a Velocity–Time Graph to Compare the Speeds and Velocities of Two Objects Physics

The change in velocity of two objects with time is shown in the graph. Which one of the following statements about the speeds and velocities of the two objects is correct? [A] Their speeds are the same, but their velocities are different. [B] Their velocities are the same, but their speeds are different. [C] Both their speeds and velocities are different. [D] Both their speeds and velocities are the same.

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Video Transcript

The change in velocity of two objects with time is shown in the graph. Which one of the following statements about the speeds and velocities of the two objects is correct? (A) Their speeds are the same, but their velocities are different. (B) Their velocities are the same, but their speeds are different. (C) Both their speeds and velocities are different. (D) Both their speeds and velocities are the same.

In this question, we are given a velocity–time graph and asked to identify the correct statement about the speeds and velocities of the two objects whose motion is shown on this graph. We can recall that velocity is a vector quantity, which means that it has both a magnitude and a direction. In order for two objects to have the same velocity as each other, they must have velocities with equal magnitudes and equal directions.

On a velocity–time graph, the height of a line on the velocity axis at a particular point tells us the value of that object’s velocity at that particular time value. Velocity values above the horizontal axis are positive, meaning the object is moving in the positive direction, while velocity values below the horizontal axis are negative.

Looking at the graph, we can see that the object represented by the blue line starts out with some positive value of velocity. At this same initial instant, the object represented by the red line starts out with some negative velocity value. Since the red line begins further below the horizontal axis and the blue line begins above it, that is, since this distance here is larger than this one here, then the magnitude of the initial velocity of the red object is larger than that of the blue object.

We have found then that initially the red and blue objects move in opposite directions with different magnitudes of their respective velocities. Therefore, they do not have equal velocities. The same must also be true at any later point in time. Since the two lines on the graph appear to be parallel, this means that they have the same slope as each other. That is, the two velocities change at the same rate. Since they start out as different velocity values, then the two velocities will remain different at all later values of time.

Before this instant, the two objects have different directions. After this point, the red object has begun moving in the positive direction, that is, the same direction as the blue object. However, the magnitude of the red object’s velocity is still not the same as that of the blue object’s velocity since the lines still have different heights on the velocity axis at any given instant in time.

We have found then that the two objects have different velocities. That means we can eliminate the answer choices (B) and (D), which claim that the velocities are the same.

Now we need to consider the speed of the two objects. Let’s clear some space on screen to do this. Let’s recall that, unlike velocity, speed is a scalar quantity. That means it has a magnitude but no associated direction. An object’s speed is equal to the magnitude of its velocity. It is possible for two objects with different velocities to have equal speeds. For example, if two objects move in opposite directions, but the magnitudes of their velocities are equal, then they will have the same speed despite having different velocity values.

However, that is not the case here. We have seen that, in general, the magnitudes of the velocities of the red and blue objects are not equal. Therefore, the speeds are not equal. So we can say that the speeds of these two objects are not equal for the duration of time shown on this graph.

We could, in fact, use the information from the velocity–time graph to draw a speed–time graph for the two objects. Since speed is the magnitude of the velocity, then positive velocity values will look the same on a speed–time graph, while negative velocity values will be replaced by the equivalent positive values.

The blue object has a positive velocity at all values of time shown on the graph. So the speed–time graph for the blue object will look just the same as its velocity–time graph. The red object has a positive velocity after this point. So the speed–time graph for this object will look just the same as its velocity–time graph after this point. Before this point, the red object’s velocity is negative, and the speed is equal to the magnitude of this negative velocity, which is a positive value. So the speed of the red object in this interval looks like this.

Now, at this instant here, where the red and blue lines cross, the two objects instantaneously have the same speed. But at all other time values, their speeds are different. So in general then, we have found that the two objects have different speeds.

We can therefore eliminate answer option (A), which claims that their speeds are the same.

Since we’ve seen that the velocities and speeds are both different, we can identify the correct answer as option (C). Both their speeds and velocities are different.

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