The change in displacement of two objects with time is shown in the graph. The gray arrows in the diagram are the same length. Do the two objects have the same velocity? Do the two objects have the same speed?
All right, taking a look at this graph, we see that it plots displacement against time. And we have the displacement versus time of two objects, one represented by the red line and the other by the blue line. And we can see that these lines are separated by these three arrows. The problem statement calls them gray arrows. And it says that all three have the same length. So, in other words, this distance here is the same as this distance here is the same as this distance here. Knowing all this, we want to figure out whether these two objects depicted in this graph have the same velocity and whether they have the same speed.
Let’s begin with this question of velocity. We can recall that velocity in general is defined as displacement divided by time. Going back to our graph, this means if we consider the displacement of these two objects divided by the time it takes them to be displaced that far, then that ratio or that fraction, displacement divided by time, is equal to the object’s velocity. And on our graph, displacement over time is equal to the slope or the gradient of these lines.
Now, just as a convention for simplicity, let’s say that this red line represents an object that is in fact red in color and that the blue line represents an object that is blue in color. So when we talk about the red object, we’re referring to this line. And we say the blue object, we’re talking about this one.
So anyway, in order to answer this question about whether the velocity of these two objects is the same, we now see that to figure this out, we’ll want to compare the slopes of these lines. In particular, we’ll want to compare the slopes of the red and the blue lines over this time period and also over this one. We divide our analysis up into two parts because we can see the slopes of these lines both change over these time intervals.
So considering this first time interval, we can see that our red object starts out with a displacement of zero and then over this time span achieves a total displacement represented by the length of one of these arrows. Over that same time interval, our blue object ends up with a displacement of zero. But it starts with a negative displacement here. And because the two lines start out separated by one of our three equal-length arrows, we know that that initial displacement of the blue object is as much below zero as the final displacement of the red object is above zero. This is just another way of saying that this arrow and this arrow have the same length.
So over this particular time interval, we can call it 𝑡 one, our red object gains in displacement by the length of one of these arrows and likewise our blue object gains in displacement by the same amount, which means that the displacement of the red object is the same as that of the blue object. They start out in different positions, but their displacement over this time interval is the same.
And since we’re considering this displacement over the same amount of time for both objects, that means that in this fraction, both the numerator and the denominator are the same for the red and the blue object over 𝑡 one. So we can say that over this first time interval, what we’ve called 𝑡 one, the red object and the blue object have the same velocity.
Now, let’s consider the second time interval. We can call this 𝑡 two. Once again, we’ll be looking at the slope or the gradient of the red and the blue lines over this interval. And once more, these lines start out separated by a certain distance, the height of one of our arrows, and they end up separated by that same distance. The change in displacement, then, of our red object over this second time interval can be represented by this arrow. And the change in displacement of the blue object can be represented by this one. These arrows are the same length, which means that both the red and the blue object have the same displacement but not the same position over this time interval. We can see that one of them, the blue object, ends up with a negative displacement and the other, the red object, ends up with a displacement of zero.
Now, notice that over the second time interval, the gradients or the slopes of these lines are both negative, whereas over the first time interval, they were both positive. Despite this change, we can see that the red and the blue object change in the same way over each of these intervals. That is, the slope of this segment of the red line is the same as the slope of this segment of the blue line and that the slope of this segment of the red line is the same as the slope of this segment of the blue line.
Based on our equation for velocity, we can see that this means that over the time interval 𝑡 two, our objects also have the same velocity. Now, as we consider these two different time intervals, note that these velocities are not the same. So if we wanted to, we could give them subscripts, 𝑣 one and 𝑣 two, to clarify that they’re different. Despite this difference, we can say that over any time interval over the time shown in our graph, including the entire graph, the velocities of these two objects are the same. And so that’s our answer to our first question. These two objects do have the same velocity.
The next question asks, do they have the same speed? At this point, we can recall that velocity is a vector quantity having both magnitude or size as well as direction. Now, it turns out that another name for the magnitude or the size of a velocity is speed. So we could say that velocity has speed and direction. Speed is always positive. And it never takes into account the direction of the object that’s in motion.
Now let’s consider this. Because these two objects have the same velocity, that means that both of their magnitudes, as well as their directions, are the same. Both those conditions must be met in order to say that they have the same velocity. But then, if they have the same magnitude, which they do, then they must have the same speed since speed is synonymous with the magnitude of velocity. So we can quickly answer yes to the second question because any time two objects have the same velocity, they must also have the same speed.