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.
So, we can see that we’ve been given a diagram here on which two different objects have been represented. The first is the object represented with the blue line, which we’ll call object one. And the second is the object represented by the pink line, which we’ll call object two. Now, specifically, the graph is showing us the displacement over time of object one and object two. In other words, displacement is shown on the vertical axis and time is shown on the horizontal axis.
Now, we can see that for object one, for example, as time progresses, in other words as we move from left to right on the horizontal axis, the displacement stays exactly the same. The displacement does not, for example, increase, which we would see if the line was going upwards as we move towards the right. And it doesn’t decrease either, which we would see if the line went downwards as we move towards the right. No, the line is a perfectly flat line and it stays at exactly the same displacement as time progresses. And actually, the same is true for the pink object, for object two. The displacement does not change with time because the line representing the displacement over time of object two is a perfectly flat straight line.
Now, let’s, first of all, recall that displacement is defined as the shortest distance between two points. So, in this case, the blue line representing object one is showing us the displacement between some random point, which happens to be at the origin of our graph. So, let’s say that, in real space, this is that point. We’ll call that point 𝑜, for origin. And object one, which we’re being told that, as time progresses has a constant displacement; the displacement is not changing. In other words then, the shortest distance between the origin and object one is not changing as time progresses. This is telling us that object one is stationary; it’s not moving.
Now, the other thing to note is that displacement is a vector quantity. This means it has a magnitude, or size, and a direction. In other words, if we’re measuring from the origin, from this zero point here that we’ve labelled randomly, then the displacement to object one has some magnitude that is given by the value on the vertical axis read off here, whatever that value may be, and a direction. The direction is, in this case, towards the right. But that’s just arbitrary because we’ve drawn the diagram that way. The important thing, though, is that because displacement is a vector quantity, this means it can have a negative value, just like the displacement of object two shown on this graph.
We’re being told that the displacement of object two is, whatever this value is read off on the displacement axis. But it’s a negative value, which means object two is displaced in the opposite direction to object one. In other words, if displacements towards the right starting at the origin are positive, then displacements towards the left starting at the origin are negative. And additionally, the question tells us that the two gray arrows in the diagram are the same length.
Now, here’s one gray arrow. And here’s the other one, which is telling us that the magnitude, or size, of the displacement of object one is the same as the magnitude, or size, of the displacement of object two. In other words, object one and object two are both equally far away from this random point that we’ve labelled the origin. Except that object one is in one direction, whatever that direction may be, and object two is in the opposite direction because it has a negative displacement. But anyway, so now that we’ve looked at all of that, what we need to do is to answer the first part of the question.
We’ve been asked that the two objects have the same velocity.
Well, to answer this question, we can recall that velocity is defined as the rate of change of displacement, or in other words how much the displacement of an object changes over a given unit of time. In symbols then, we can say that the velocity of an object, lower case 𝑣, is equal to the change in displacement, Δ, representing change, and 𝑠, representing displacement, divided by the time interval, Δ𝑡, in which this change of displacement occurs. And so, this equation is telling us that the velocity of an object is equal to how much its displacement changes divided by the time taken for that displacement change to occur.
But if we look carefully, we’ll realise that on the vertical axis of our graph, we’ve got displacement, and on the horizontal axis, we’ve got time. Therefore, the change in displacement divided by the change in time, is the same thing as the slope, or gradient, of our displacement–time graph. Hence, in order to be able to work out the velocity of object one and object two to see if they have the same velocity, we need to work out the slope, or gradient, of each line representing the displacement of object one and object two over time.
At this point though, we can see that the line representing object one is perfectly flat. In other words, we can say that its slope is equal to zero. And actually the same is true for object two. The slope is equal to zero because over any given period of time, as shown in the graph, the change in displacement for each object is zero. The displacement of each object is not increasing and is not decreasing either. It’s staying exactly the same. Therefore, we can say that the velocity of object one, we’ll call this 𝑣 subscript one, is equal to zero because the slope of the blue line is zero. And additionally, we’ll say that the velocity of the second object, object two, is also equal to zero for the same reason.
Now, at this point, we might remember that velocity is a vector quantity; it has a magnitude and direction. However, when the velocity of an object is zero, it’s not moving in any particular direction. Therefore, any two objects that have zero velocity will have the same velocity as each other. It’s zero. They’re not moving anywhere in any direction. And hence, when the question asks us, do the two objects have the same velocity, our answer to that question is yes.
Moving on to the next part of the question then, this asks us, do the two objects have the same speed?
Now, to answer this, we can recall that speed can be defined as the rate of change of the distance moved by an object. In other words, speed is a scalar because the distance moved by an object doesn’t necessarily take into account the direction in which an object is moving. And hence, speed only has a magnitude, or size. In fact, we can think of speed as also being the magnitude, or size, of the velocity of an object. So, for example, if we had some random object, let’s say this is our object, moving at five metres per second towards the right. Then, we could say that the speed of the object was simply five metres per second. That’s the magnitude, or size, of this velocity.
And luckily for us, we’ve found the velocities of both object one and object two in the previous part of the question. We found that each object had a velocity of zero. Therefore, the magnitude, or size, of each velocity is also zero. And this tells us that the speed of each object is zero. This makes sense because, once again, neither object is moving at all. Each object is stationary. And hence, our answer to this part of the question is yes, the two objects have the same speed.