Video Transcript
The change in velocity of two objects with time is shown in the graph. There are two parts to this question. Do the two objects have the same speed? And, are the two objects moved equal distances from their initial positions?
So we can see in this question, we’ve been given a graph with time on the horizontal axis and velocity on the vertical axis. So, we have two lines drawn on our graph. The blue line shows us how the velocity of some object — let’s imagine this to be a blue object — varies over time. And the red line shows us how the velocity of some other object — so let’s say this is a red object — varies over time as well. So, the first question here asks us whether the two objects have the same speed. Well, since our graph shows us how velocity varies over time, let’s start by recalling the similarities and differences between velocity and speed.
Conceptually, velocity and speed are very similar. They both describe how quickly something is moving. So, we could say they’re both measures of quickness. However, an important difference in the definitions of velocity and speed means that, in certain contexts, they actually have very different meanings. The difference between velocity and speed is that velocity is an example of what’s called a vector quantity, whereas speed is an example of a scalar quantity. Scalar quantities are defined entirely by their magnitude or size, whereas vectors are described by a magnitude and a direction. So, the speed of an object just tells us how quickly an object is travelling, whereas the velocity of an object tells us how quickly and in what direction it’s travelling.
To illustrate the difference between velocity and speed, let’s imagine two objects that are moving away from each other. Let’s say the object on the left is travelling to the left at five meters per second, and the object on the right is moving to the right at a speed of five meters per second as well. Because both objects are travelling at five meters per second, we know they’re travelling at the same speed. However, they have different velocities. This is because velocity has both magnitude and direction. So, even though the magnitudes of their velocities are the same, the directions of their velocities are different, which means their velocities are different.
When we’re using velocities, one of the ways that we can communicate what direction an object is travelling in is by using positive and negative numbers. For example, we could say that the object on the left has a velocity of negative five meters per second, while the object on the right has a velocity of positive five meters per second. And this signifies that the two objects are moving in opposite directions. Note that, in this example, we’re saying that anything moving to the right is positive and anything moving to the left is negative. But we could just as easily define this the other way around. So, if we said that our positive direction was to the left, that means that any objects moving to the left have a positive velocity, and any objects moving to the right have a negative velocity.
This actually highlights another important difference in the way that we represent velocity and speed, that is, that velocities can be positive or negative, whereas speeds can only be positive. This is because speed only has a magnitude, so anything that’s moving will have a positive speed. If we take another look at our graph, we can see how positive and negative values of velocity have been used. Let’s look at the very left-hand side of our graph at the earliest possible time. The heights of the blue and red lines of this time give us the initial velocities of the blue and the red objects. We can see that the velocity of the blue object is positive, whereas the velocity of the red object is negative. This tells us that, initially, the two objects are moving in opposite directions.
But the question is asking us to determine whether these two objects have the same speed as each other. Well, we know that speed is basically the same as velocity but without the direction information. In other words, speed is just the magnitude of the velocity. So in this question, the speed of each object is just given by its velocity but without any negative sign. Of course, in the graph we’ve been given, we don’t actually have any measurements on the velocity axis, so we can’t say exactly what the speeds of the objects are. However, the speed of each object is represented by the vertical distance of its graph from the time axis.
So, looking at the left-hand side of the graph, the initial speed of the blue object is given by this distance, whereas the initial speed of the red object is given by this distance. To make this clearer, let’s make up some values for the velocities. Let’s say the initial velocity of the blue object is two meters per second, and the initial velocity of the red object is negative eight meters per second.
The initial speed of both of these objects is given by the magnitude of their initial velocities, which we can think of as just being the initial velocities but without any negative sign. So, the blue object, in this case, would have an initial speed of two meters per second, whereas the red object would have an initial speed of eight meters per second. Again, this is given by the vertical distance measured on the velocity axis between zero meters per second and the velocity at that time.
So now, let’s return to our original graph without the made-up measurements. And we can think of the speed of each object at any time as being given by the vertical distance between the time axis and the line that represents its velocity. If we look again at the earliest possible time on the far left of the graph, we can see that this vertical distance representing the initial speed of the blue object is much smaller than this distance which represents the initial speed of the red object. Therefore, at this point, the red object has a much higher speed than the blue object. If we look at the rest of the graph, we actually see that the two objects have different speeds virtually all of the time.
So, for example, at this time, we can see that the speed of the red object, which is represented by this distance, is much smaller than the speed of the blue object, which is represented by this distance. Therefore, at this time, the speed of the blue object is much greater than the speed of the red object. Or, for example, if we look at this time, we can see the speed of the blue object is given by this distance. But the speed of the red object is actually zero because the line is touching the time axis.
If we look at this time, we actually find that the speed of the blue object, represented by this height, is the same as the speed of the red object, represented by this height. So actually, at this one single instant, the two objects are travelling with the same speed. However, at all times both before and after this time, the speeds of the two objects are completely different. So, the answer to this first part of the question is no.
Now, let’s take a look at the second part of the question. This time we want to figure out if the two objects are moved equal distances from their initial positions. So, in order to answer this question, we need some way of figuring out the distance travelled from looking at a velocity–time graph. We can recall that the distance travelled is given by the area under a velocity–time graph. Now, there are a couple of very important things that we need to be aware of when we use this rule. Firstly, when we talk about the area under a graph, what we mean is the area between the graph and the horizontal axis — so, in this case, the time axis. So, for example, if we look at our graph, the total distance travelled by the blue object is given by this total area.
Note that, in this case, the shaded area gives us the distance that the blue object has travelled between the start time and the end time shown on the graph. But we could also just as easily work out the distance the blue object travelled in any given time period by just looking at the area of the graph between the start and the end times that we’re interested in. So, for example, the distance that the blue object travels between this time and this time is given by this shaded area.
The other really important thing to remember here is that if this area is above the time axis like this area is, then it tells us that the object has moved some distance in the positive direction. However, if this area is below the time axis, then it tells us that the object has moved some distance in the negative direction. For example, the distance that the red object travels between this time and this time is given by this shaded area. However, because this area is below the time axis, we know that this represents a movement in the negative direction.
To answer this question and find out if the two objects are moved equal distances from their initial positions. We need to consider the total distance that each one of them is moved over the entire duration shown in the graph — in other words, between this time and this time. This blue-shaded area then represents the total distance that the blue object moves. We haven’t been given any measurements on our graph, so we can’t actually calculate how big this area is, which means we don’t actually know the true distance that the blue object has moved. However, for this question, we just need to be able to compare the distances moved by the two objects and decide whether or not they’re equal, which means we don’t actually need any numerical values.
These red-shaded areas represent the distance that the red object moves. However, because this area is below the time axis, that means that it represents movement in the negative direction. And because this area is above the time axis, it represents the distance moved in the positive direction. Because these two areas look like they’re approximately the same, that means that the red object must have moved some distance in the negative direction and then moved approximately the same distance in the positive direction. Which means that it must have ended up in approximately the same position as it started.
In contrast, this relatively large blue area lies entirely above the time axis, which tells us that the blue object moved a relatively large distance in the positive direction. So, we can see that the red object moved away from and then more or less back to its initial position. Whereas the blue object continued to move in the positive direction for the entire time, meaning that it ends up some distance in the positive direction away from its initial position. So, the answer to this part of the question is no. The two objects are not moved equal distances from their initial positions.
