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? And do the two objects have the same speed?
Taking a look at our graph, we can see that it shows displacement against time. We can see that both objects start with an initial displacement of zero, which increases at a constant rate up until this time. After that, they diverge, with the object represented by the blue line continuing to increase its displacement at a constant rate and the object represented by the dashed red line decreasing its displacement at a constant rate.
So do the two objects have the same velocity? Here, we need to recall the definition of velocity, which is that velocity is equal to displacement divided by time. So on this graph, the velocity is equal to the vertical value divided by the horizontal value, which is the same as the slope of the graph. Now it’s helpful to consider the graph in two segments. If we first just look at this first segment, we can see that the motion of the two objects is identical. The two objects have the same change in displacement over the same time interval and therefore have the same velocity.
But what about the second interval? Here, the object represented by the blue line has a positive slope, and the object represented by the dashed red line has a negative slope. So over this time interval, the two objects must have different velocities. Therefore, the answer to the question “Do the two objects have the same velocity?” is no.
Now let’s consider “Do the two objects have the same speed?” To answer this question, it might be helpful to consider a scenario which might give rise to this graph. Let’s consider a person starting from home with an initial displacement of zero. They then walk to a shop which is one kilometer away, at which point they have a displacement of one kilometer. And the distance they have traveled is also one kilometer.
Now let’s say they walk a further kilometer to a second shop. At this point, their displacement from their origin point of home will now be two kilometers and the distance traveled is also two kilometers. But what if instead of traveling on to that second shop, they had returned home. At this point, they’ve returned to their point of origin, so the displacement is zero. However, they’ve still walked to the shop and back, so the distance traveled is two kilometers.
This is because displacement is a vector, which means it has both magnitude and direction and can therefore be negative, whereas distance is a scalar, which means it has magnitude only and therefore can only increase even if you return to your starting point.
Now let’s relate this back to our graph and recall that speed is equal to distance divided by time. Again, in this first segment, we can see that both objects have the same motion and therefore the same speed. But what happens in this second time segment? At the start of this time interval, we can see that both objects have the same displacement. By the end of the interval, the object represented by the blue line has increased its displacement and the object represented by the red dashed line has decreased its displacement back down to zero.
The amount by which the displacement has changed in both cases is given by these gray arrows, which we’re told are the same length. And remember that distance cannot decrease. So if the displacement decreases for the object represented by the red line by the same amount that it increases for the object represented by the blue line, then the distance has increased by the same amount in both cases. Therefore, the change in distance for both objects is the same, and we’re looking at the same time interval. And so over this interval, the two objects also have the same speed. Therefore, the answer to the question “Do the two objects have the same speed?” is yes.