The changes in the speeds of three objects are shown in the graph.
Here, you can see we have a graph with time on the horizontal axis and the speed of an object on the vertical axis. So, we have three lines drawn on this graph, where the red line represents how the speed of object one changes over time. The blue line represents how the speed of object two changes over time. And the yellow Line represents how the speed of object three changes over time.
The first part of this question asks, which object has the greatest initial speed? A) one, B) two, C) three, or D) they all have the same initial speed.
So, we’re being asked to compare the initial speeds of these three objects and determine which of them is the greatest or if they’re all the same. The initial speed of an object refers to its speed at the beginning of the graph. So, that means we’re looking at the left-hand edge of the graph. We often see that this is the point where the time is zero, but in this case, it doesn’t actually matter that it hasn’t been labelled. The initial speeds of each of the three objects are given by the vertical height of the line that represents their motion on the left-hand side of the graph.
We look at the left-hand side of the graph because this is where the time value is lowest. It represents the time when we started taking measurements. So, it’s when we say the objects were travelling at their initial speeds. So, for example, to find the initial speed of object one, we would go upward from this point. And when we reach the line for object one, the height of the line at that point gives us the initial speed. So, this is the initial speed of object one.
Similarly, this gives us the initial speed of object two. And this gives us the initial speed of object three. We can see that the line that represents the motion of object three starts higher than the other lines, which tells us that the object with the greatest initial speed is object three. So, now, we can move on to the second part of the question.
This time we’re asked, which object has the greatest final speed? A) One, B) two, C) three, or D) they all have the same final speed.
So, in the first part of this question, we found the initial speed of the objects by looking at the beginning of the graph, which is on the left. But here, we’re looking for the final speed. To find the final speed, we go as far to the right as possible as this is the latest time plotted on the graph. So, we’re looking here. The heights of each graph at this point tell us the final speeds of each of the objects. So, the final speed of object one is given by the height of this point. The final speed of object two is given by the height of this point. And the final speed of object three is given by the height of this point.
Now, it’s often the case that we can go horizontally along from each of these points, and we can read off the values of their final speed on the vertical axis. Now, in this question, we haven’t actually been given any values on either the vertical or the horizontal axis. So, we can’t say what the final speeds actually are. However, if we had been given values on the vertical axis, then the final speed of object one would be given by this value, the final speed of object two would be given by this value, and the final speed of object three would be given by this value.
For this question though, just like previously, we only need to be able to compare their final speeds. We can see that because the line for object two is higher at the end of the graph than the other lines, then object two must have the greatest final speed. So, for this question, which object has the greatest final speed, the answer is two. Now, let’s move on and look at the third part of the question.
Which object has the greatest average speed? A) one, B) two, C) three, or D) they all have the same average speed.
In the previous parts of this question, we compared the speeds of the objects at specific moments. So, we first looked at their initial speeds by looking here. And then, we looked at their final speeds by looking at the other end of the graph. But finding the average speed of an object is generally much more difficult than finding the speed at any specific moment. Fortunately, there is a technique we can use to easily find the average speed of an object from a graph of its speed against time, but only if the speed of the object changes at a constant rate.
That means we can only use this technique if the graph of an object’s speed against time is a straight line. And fortunately, in this question, that is the case for all three of our objects. The way that we find the average speed of these objects is to look at their speeds at the time which is halfway between the start time and the end time of their journeys. So, this represents the time in which their journey started. And this represents the time in which their journey ends. And halfway in-between the two is where we can find the average speeds.
So, that means that the height of this point represents the average speed of object two, the height of this point represents the average speed of object three, and the height of this point represents the average speed of object one. Once again, if we had labels on the speed axis, then going along horizontally from these points would enable us to read off the actual average speeds. Unfortunately, in this case, we can’t do that. But we can still compare the average speeds of these objects by looking at the relative heights of each of these points.
It’s really important to note that you can only use this technique where we look at the middle time of a journey to work out the average speed when the speed-time graph is a straight line, which means that the speed is changing at a constant rate. For example, if we had a speed-time graph that looked like this, where initially the speed decreases then stays the same for a period before increasing, then the speed in the middle of the time axis is not the average speed. Likewise, if we have a curved speed-time graph like this, working out the average speed actually becomes quite complicated.
But in our graph, we can see, by looking in the middle of the time axis, that the object with greatest average speed is object one. We can also see that object two has the lowest average speed and the average speed of object three is somewhere in-between. So, the answer to this question, which object has the greatest average speed, is one. So, now, we can look at the fourth and final part of the question.
This time we’re asked, which object was not moving. A) one, B) two, C) three, or D) all the objects were moving.
To answer this question, let’s start by thinking about what we mean by moving as opposed to not moving. We can only say that an object is not moving if it has a speed of zero. This means that the question here is equivalent to asking which object has a speed of zero. In fact, in order for an object to be considered not moving, it would have to have a speed of zero for the entire time. So, to answer this question, we’ll need to look at our graph and determine if any of these objects have a constant speed of zero for the entire time.
If we look at our graph, we can see that the blue line representing the motion of object two and the yellow line representing the motion of object three are both sloped. That means they both show a change in speed as time passes. In fact, we’ve already shown how the initial speeds of both of these objects vary from the final speeds of both of these objects. So, because the speeds are changing, we know that both of these objects are definitely moving, so we can eliminate options B and C.
However, it’s slightly more complicated to determine whether object one has a speed of zero the entire time or not. And the reason for this is that the vertical axis, which shows speed, doesn’t actually have any measurements on it. On many graphs, zero on the vertical axis is located at the point where the vertical axis meets the horizontal axis, in this case, at the bottom of the graph. So, it’s tempting to assume that this is the case with this graph.
However, this doesn’t actually have to be the case. It could just as well be true that the point representing zero on the vertical axis is located here, or even here. And at this point is indeed the point where speed is equal to zero. Then, that would mean that object one is actually not moving because it stays at a constant speed of zero for the entire time.
However, one important property of speed tells us that this isn’t the case. Speed is a scalar quantity, which means only its size, or magnitude, matters. It doesn’t matter whether an object is moving up, down, forward, or backwards. If it’s moving, then its speed will be positive. That means that speed can’t take negative values. Since we can see the objects two and three both have speeds below this point, we know that this point can’t represent zero speed because no object can have a speed below zero.
So, this means the object one does not have a speed of zero. Instead, it has some positive constant value of speed for the entire time. So, this means we can eliminate option A as well. The correct answer is D, all the objects were moving.