### Video Transcript

In this video, we will learn how to
determine speeds from distance–time graphs. Let’s begin by recalling what a
distance–time graph shows us. A distance–time graph is a graph
that measures distance on the 𝑦-axis against time on the 𝑥-axis. If the graph is labeled with units
and a numerical scale for both of the two axes, then we can read off the distance
and the time for any given point plotted on the graph.

For example, let’s consider this
graph here. We’ll suppose that we want to read
the distance and time values for this point. To find the time value, we draw a
vertical line down from the point until we get to the time axis. So that’s the horizontal or
𝑥-axis. We then read the value on this time
axis at the point where the vertical line meets it. In this case, that value is
four. And we can see that the time axis
is in units of seconds. So the time at this point is equal
to four seconds.

To find the distance value, we go
back to this point and draw a horizontal line from it until we get to the distance
axis. So that’s the vertical or
𝑦-axis. Reading the value on the distance
axis at the point where this line meets it, we find that this value is two. We can see that the distance axis
has units of meters. So we have read from the graph that
the distance at this point is two meters.

Okay, so now that we’ve reminded
ourselves what a distance–time graph is, let’s think about the physical quantity
speed. We can recall that the speed of an
object is defined as the distance traveled by that object per unit of time. Let’s suppose that we have an
object that is moving at a constant speed that we’ll label as 𝑣. If we know that the object moves a
distance 𝑑 in a time 𝑡, then the speed 𝑣 of that object is given by 𝑑 divided by
𝑡.

Since a distance–time graph gives
us information about both distance and time, then it also tells us about speed. We may recall that a line drawn
through points on a distance–time graph represents a speed. In particular, a straight line
means a constant speed. The object travels equal amounts of
distance in equal amounts of time. Let’s add some units and numerical
values to the axes of this graph to see an example of this. First off, let’s look at what
happens between a time of zero seconds and a time of one second. We can see that at zero seconds,
the object has moved a distance of zero meters. Let’s label this pair of time and
distance values with a subscript one. Then, at a time of 𝑡 one equal to
zero seconds, the object has moved a distance of 𝑑 one equal to zero meters.

Next, we’ll look at the point on
the graph with a time value of one second. Let’s label this with a subscript
two so that we have 𝑡 two is equal to one second. We want to find the distance that
the object has moved at this time 𝑡 two. We’ll label this distance as 𝑑
two. To find the value of 𝑑 two, we
draw a vertical line up from the time of 𝑡 two, which is one second, until this
vertical line meets the line plotted on the graph. This point on the graph then
corresponds to this time of 𝑡 two equal to one second.

Then, to find the distance 𝑑 two,
we draw a horizontal line from this point across to the distance axis. The line hits the axis at a value
of one, and the distance axis has units of meters. So the value of 𝑑 two is equal to
one meter. In other words, between a time 𝑡
one equal to zero seconds and a time 𝑡 two equal to one second, the object moves
from a distance 𝑑 one equal to zero meters to a distance 𝑑 two equal to one
meter.

The time taken between this point
and this point on the graph is equal to the final time, 𝑡 two, minus the initial
time, 𝑡 one. We’ve labeled this time taken as
Δ𝑡. Δ is a Greek letter that we
typically use to represent a change in a quantity. In this case, that quantity is
time. Δ𝑡 is the change in time between
the time 𝑡 one and the time 𝑡 two. Subbing in our values for 𝑡 one
and 𝑡 two, we have that Δ𝑡 is equal to one second minus zero seconds, which is one
second. In the same way, we can define the
change in distance moved between this point and this point as Δ𝑑 is equal to 𝑑 two
minus 𝑑 one. Subbing in the values for 𝑑 one
and 𝑑 two gives us that Δ𝑑 is equal to one meter minus zero meters, which is one
meter.

Now, we saw earlier that if we
divide the distance moved by an object by the time taken to move that distance, this
gives us the object’s speed. In our case, Δ𝑑 is the distance
moved between this point and this point on the graph. And Δ𝑡 is the time between these
same two points. So between these two points on the
graph, the object has moved a distance of Δ𝑑 in a time of Δ𝑡. This means that we can write the
speed of the object as the distance moved, Δ𝑑, divided by the time taken, Δ𝑡. For the section of the graph that
we’ve looked at, we have Δ𝑑 is equal to one meter and Δ𝑡 is equal to one
second. Subbing these values into this
expression for the speed, we find that 𝑣 is equal to one meter divided by one
second. This works out as a speed of one
meter per second.

Now, we said that a straight line
on a distance–time graph represents a constant speed. This graph here is clearly a
straight line. So let’s check that it does in fact
show a constant speed. We’ve already found the speed
between zero seconds and one second. This speed was one meter per
second. Let’s now check that we get the
same value using the part of the graph between two seconds and three seconds. First, we’re going to need to clear
some space to do this.

Okay, so to calculate the speed
between two seconds and three seconds, we’ll use the exact same approach as we did
to find this speed between zero seconds and one second. The first step is to find the
distance moved at each of the two time values. We draw a vertical line up from the
time of two seconds until we get to the plotted line. Then, we draw a horizontal line
across to the distance axis. We can then read from the graph
that at a time of two seconds, the object has moved a distance of two meters. We’ll label this time as 𝑡 one and
this distance as 𝑑 one.

Now, we need to do the same thing
for the time of three seconds. We find that at three seconds, the
object has moved a distance of three meters. We’ll label this second pair of
values with a subscript two. Just as we did before, we can
calculate the distance moved between this point on the graph and this point. This distance moved, Δ𝑑, is equal
to 𝑑 two minus 𝑑 one. Subbing in our values, we have that
Δ𝑑 is equal to three meters minus two meters, which is one meter.

Similarly, the time interval
between the two points, Δ𝑡, is equal to 𝑡 two minus 𝑡 one. Again, subbing in our values, we
have that Δ𝑡 is equal to three seconds minus two seconds, which is one second. Let’s now use this equation to
calculate the speed of the object between the two points. When we sub in our values for Δ𝑑
and Δ𝑡, we have that the speed 𝑣 is equal to one meter divided by one second. This works out as a speed of one
meter per second.

So we have looked at two different
parts of the graph and calculated the same speed of one meter per second in each
case. In fact, no matter what part of the
graph we used, we would get the same result. This graph represents an object
moving with a constant speed of one meter per second. To work out this speed, what we did
was to calculate the slope of the line on the graph.

The slope of a straight line, which
is also known as the gradient of the line, is defined as the change in the vertical
coordinate between two points on that line divided by the change in the horizontal
coordinate between the same two points. This is exactly what we calculated
to find the speed 𝑣. Δ𝑑 is the change in distance or
vertical coordinate. And we divided this by Δ𝑡, which
is the change in time or horizontal coordinate. So the slope of a line on a
distance–time graph tells us the speed of the object. We calculated that this orange line
has a slope or speed of one meter per second.

Now, let’s consider this pink line
that we have added. Since it’s a straight line, then we
know that it represents a constant speed. This means we can use any part of
the line to work out what the speed is. Let’s consider the part between
zero seconds and one second. We’ll label these times as 𝑡 one
and 𝑡 two, respectively. We can see from the graph that at
zero seconds, the object has moved a distance of zero meters. We’ll call this distance 𝑑
one. At one second, the object has
traveled a distance of two meters. We’ll call this distance 𝑑
two.

We want to find the slope of the
line or the speed of the object, which is equal to the change in distance, Δ𝑑,
divided by the change in time, Δ𝑡. Δ𝑑 is equal to 𝑑 two minus 𝑑
one, while Δ𝑡 is 𝑡 two minus 𝑡 one. Subbing in our values and
evaluating the expressions, we find that Δ𝑑 is two meters and Δ𝑡 is one
second. Then, 𝑣 is equal to the slope Δ𝑑
divided by Δ𝑡. For our values, this is two meters
divided by one second. This works out as two meters per
second.

So on this distance–time graph, we
found that the orange line represents a speed of one meter per second, while the
pink line represents a speed of two meters per second. The pink line has a greater slope,
and so it represents a greater speed. Without any calculation, we can see
that the pink line looks steeper than the orange line. For any two straight lines drawn on
the same distance–time graph, the steeper line has the greater slope and so
represents the greater speed.

The same logic applies when we have
a distance–time graph like this, where the line has different sections with
different steepnesses. With no calculations, we can look
at this graph and see that the line is steepest between zero seconds and one
second. This segment of the graph therefore
has the greatest slope. And so we can say that the object
has the greatest speed between zero seconds and one second. However, for two lines drawn on
different distance–time graphs, then the slope of the lines is not necessarily the
same as how steep the lines look.

For example, let’s consider these
two distance–time graphs. The left-hand one is the same as
the first graph that we looked at. We already worked out that this has
a slope of one meter per second. The line on the right-hand graph
looks steeper than the line on the left-hand graph. Let’s now calculate the slope of
this right-hand line. We can recall that the slope is the
change in vertical coordinate, distance, divided by the change in horizontal
coordinate, time. Let’s consider the part of the
graph between these two points on the time axis. Reading off the scale on the time
axis, we see that the first point is at a time of zero seconds and the second point
is at a time of 10 seconds. The time interval Δ𝑡 between the
two points is equal to the second time, 10 seconds, minus the first time, zero
seconds. This works out as 10 seconds.

We can see from the graph that at
zero seconds the distance moved is zero meters and that at 10 seconds the distance
moved is three meters. This means that the distance Δ𝑑
moved in the time Δ𝑡 is equal to three meters minus zero meters. This works out as three meters. So the slope of this graph, Δ𝑑
divided by Δ𝑡, is equal to three meters divided by 10 seconds, which is 0.3 meters
per second.

Despite the fact that the
right-hand distance–time graph looks steeper than the left-hand graph, the slope of
the right-hand graph has a smaller value than the slope of the left-hand one. So the right-hand graph represents
a smaller speed. The reason that this was possible
is that the two graphs have different scales on the time axis. When we use this equation to
calculate the slope, we’re using the numerical values from the axes. So this slope takes the axis scales
into account. However, this isn’t the case when
we just look at how steep the line looks. So we need to be extra careful when
comparing lines drawn on different sets of axes. And, in general, whenever we’re
calculating a slope, it’s important to take care to check the scales on each of the
two axes.

Okay, let’s finish up by
summarizing what we have learnt. First off, we reminded ourselves
that a distance–time graph plots distance on the vertical axis against time on the
horizontal axis and that a straight line on a distance–time graph represents an
object that is moving with a constant speed. We then saw that the slope of a
straight line on a graph is defined as the change in the vertical coordinate between
two points divided by the change in horizontal coordinate between the same two
points.

In this case, the vertical
coordinate is distance and the horizontal coordinate is time. So the slope of a line on a
distance–time graph is the change in distance, Δ𝑑, divided by the change in time,
Δ𝑡. We learnt that the slope is equal
to the speed of the object. We then saw that for two lines
drawn on the same distance–time graph, the steeper line has the greater slope and so
represents the greater speed. However, the same is not
necessarily true for lines drawn on different distance–time graphs because the two
graphs could have different axis scales to each other.