### Video Transcript

Two distance–time graphs show
objects moving with uniform speeds. How do the speeds of the objects
compare?

In this question, we’ve been given
two distance–time graphs for two different objects, and we want to compare the
speeds of the objects. Is the speed of the object shown by
the blue or the red line greater? Or are they equal? To answer this, let’s first take a
look at our graphs and pick out some important information. Because they’re both distance–time
graphs, each of them has a vertical axis showing the distance in meters that the
object has traveled and a horizontal axis showing the time in seconds that the
object has traveled for. Overall, these graphs look very
similar, but notice that they actually have different scales along their axes.

Look at the graph on the left. Here on either axis, the side
length of each grid square represents 10 units. So the distance axis reads zero
meters, 10 meters, 20 meters, and so on, and the time axis reads zero seconds, 10
seconds, 20 seconds, and so on. But the graph on the right uses
different scales. On this graph, the side length of
each grid square represents one unit on either axis. So this distance axis reads zero
meters, one meter, two meters, and so on, and the time axis reads zero seconds, one
second, two seconds, and so on.

It will be important to keep these
differences in mind when we compare the two graphs. Now to answer this question, we
need to compare the speeds of the two objects represented by the lines on these
graphs. Recall that on a distance–time
graph, the speed of an object is equal to the gradient of the line that represents
its motion. Therefore, to compare the speeds of
the objects, we’ll calculate and compare the gradients of the two lines.

Now, we might recall that when
multiple lines are drawn on graphs that use axes with the same scales, we can
compare the gradients of the lines just by looking at them. For example, if we wanted to
compare the gradient of this blue line to, say, this pink line, we would know that
the pink line represents a faster-moving object since the pink line is steeper than
the blue one. But we would only know that for
sure because the pink and blue lines are plotted using the same scales on the
distance and time axes.

For lines on graphs that have
different scales though, like the two graphs in this question, we can’t necessarily
compare their gradients visually. It’s much safer to calculate the
gradient for each line and then compare those values. To do this, we should recall that
we measure the gradient of a line between two points on the line and that the
gradient is equal to the change in distance divided by the change in time between
those points.

We already know that the gradient
of the line corresponds to the speed of the object. So we’ll use this formula as our
formula for speed. Also note that because both of
these objects are moving at uniform or constant speeds, both of the lines on the
graphs have constant gradients. This means that for each graph, we
can choose to measure the gradient between any two points along the line of
motion.

Now that we understand how to use
this formula, let’s apply it to each graph, and we’ll use the subscripts blue and
red to express which line we’re dealing with. Let’s start with the graph on the
left with the blue line of motion and measure the object’s speed or gradient between
this point and this point.

First, let’s find the change in
distance between the points. We can see that this point
corresponds to a distance of zero meters, and this point corresponds to a distance
of 40 meters. Therefore, the change in distance
between these two points is equal to 40 meters minus zero meters, which is just 40
meters. Next, we’ll find the change in time
between the two points. We can see that this point here
corresponds to a time of zero seconds, and this point corresponds to a time of 40
seconds. So the change in time between these
two points equals 40 seconds minus zero seconds or just 40 seconds.

Now, we can substitute these values
into the equation for the speed of the object. So for the blue line, we have a
speed of 40 meters divided by 40 seconds, which simplifies to one meter per
second. This is the speed of the object
represented by the blue line. Now, we just need to repeat this
process for the graph on the right with the red line of motion.

Let’s choose these two points to
measure the speed between. And we’ll start by working out
their change in distance. We can see that this point
corresponds to a distance of zero meters, and this point corresponds to a distance
of four meters. So the change in distance between
the points equals four meters minus zero meters, which is just four meters.

Next, we’ll find the change in time
between these two points. This point here corresponds to a
time of zero seconds, and this point here corresponds to a time of four seconds. Thus, the change in time between
these two points equals four seconds minus zero seconds, or just four seconds. Now, let’s substitute these values
into the equation for the speed of the object represented by the red line, and we
get a speed of four meters divided by four seconds, which simplifies to one meter
per second.

So we’ve calculated that both of
the objects move at a speed of one meter per second. And therefore, when asked how the
speeds of the objects represented by the lines on the graphs compare, we know that
the speeds of the objects are equal.