Video Transcript
Speed–time graphs are quite useful
for representing objects in motion. In this lesson, we are going to
focus on objects moving with uniform speed. We will learn how to use speed–time
graphs to find the speed of an object and also how to compare the speeds of multiple
objects. Additionally, we will learn how to
relate the speed–time graph of an object moving with uniform speed to the
corresponding distance–time graph. Let’s start by reviewing the basic
information included in a graph.
Here is a distance–time graph. We know that it is a graph because
it has a horizontal axis and a vertical axis. We know it is a distance–time graph
because the vertical axis is labeled distance, with an appropriate unit for
measuring distance, meters, and the horizontal axis is labeled time, with an
appropriate unit for measuring time, seconds. The straight blue line on the graph
represents the motion of the object. And each point on the line
corresponds to one particular time and one particular distance. It means that the object was at
that distance at that time.
Say, for example, we want to know
what time and distance is represented by this point. First, we draw a straight vertical
line to the horizontal axis. We see that it intersects at the
number four. Now, since the units for the
horizontal axis are seconds, this number four means a time of four seconds. So the time for the point that
we’re interested in is four seconds. To find the distance, we draw a
horizontal line from our point to the vertical axis. We see that it intersects at the
number two. Now the units for the vertical axis
are meters, so this two is two meters. So the time and the distance
represented by our point are four seconds and two meters. That is, after the object had moved
for four seconds, it was at a distance of two meters.
Now that we’ve reviewed some basic
operations for working with graphs using a distance–time graph, let’s move on to
speed–time graphs. A speed–time graph, like any other
graph, has a horizontal axis and a vertical axis. This time, though, the label for
the vertical axis is speed and the label for the horizontal axis is time. Our units for speed are meters per
second, and our units for time are seconds. Now remember, we’re interested in
speed–time graphs for objects moving with uniform speed. So let’s work out what that looks
like.
Uniform speed means that the speed
of the object is constant. So let’s say that our object is
moving with a constant speed of two meters per second. Since the speed is constant, we
know it is the same at every time. So at one second, the speed of the
object is two meters per second. Let’s find that point on the
graph. First, we draw a vertical line that
intersects the horizontal axis at a time of one second. Then we draw a horizontal line that
intersects the vertical axis at a speed of two meters per second. Where these lines intersect is the
point on the graph that represents one second and two meters per second.
Since the speed is constant at two
seconds, three seconds, and four seconds, the speed is also two meters per
second. On the graph, these points are
here, here, and here. All of these points are on the same
horizontal line because the speed is the same at all times. In fact, the entire speed–time
graph for this object is just the horizontal line at the vertical-axis value of two
meters per second. We just drew the speed–time graph
of an object knowing its uniform speed; let’s now do the reverse. We’ll take the speed–time graph of
an object and figure out its speed.
Here, we’ve drawn another line on
our speed–time graph. We can see immediately that this
line is horizontal. Because the line is horizontal,
every point on the line has the same vertical-axis value, one meter per second. So this line represents an object
moving with a constant speed of one meter per second. And we know that the speed must be
constant because the line on the speed–time graph is horizontal. Let’s now compare the two lines
that we’ve drawn, one corresponding to a speed of two meters per second and one
corresponding to a speed of one meter per second.
We can see that the line
corresponding to two meters per second is higher up along the vertical axis than the
line corresponding to one meter second. This makes a lot of sense; values
for speed increase as we move up along the vertical axis. So objects moving with larger
speeds should be represented by lines farther up along the vertical axis. Let’s add one more line to this
speed–time graph. There is no marker where this line
intersects the speed axis. So rather than try to find this
exact value, let’s instead give a qualitative description of the object represented
by this line.
The line is horizontal. And as we saw before, horizontal
lines on speed–time graphs correspond to objects moving with constant speed. We can also see that this line is
farther up along the vertical axis than both the object moving at one meter per
second and the object moving at two meters per second. So this line must correspond to a
uniform speed greater than two meters per second.
Even though we don’t know the exact
value for the speed of this object, just by looking at its speed–time graph, we’ve
been able to determine that it is moving with uniform speed and also compare that
speed to the speed of the other objects that we’ve drawn. One last thing that’s worth
mentioning before we move on is that speed is always positive or zero. And what this means is that the
lines on a speed–time graph can never go below zero on the vertical axis.
Now that we’ve learned how to
interpret speed–time graphs on their own, let’s compare speed–time graphs to
distance–time graphs. The graph on the left is a
distance–time graph, and it is in the shape of a straight line. Recall that straight lines on
distance–time graphs represent uniform speed and the slope of the line is the speed
of the object. We now want to know what the
corresponding speed–time graph for the same object would look like.
Well, the object is moving with
uniform speed, and we know what uniform speed looks like on a speed–time graph. It’s just a horizontal line. This horizontal line on a
speed–time graph also represents an object moving with uniform speed. To make sure that these lines
represent the same object, all we need to do is make sure that the value for the
speed on the speed–time graph, that is, where the horizontal line intersects the
vertical axis, is the same as the value for the slope on the distance–time
graph. Let’s now go the other way from a
horizontal line on the speed–time graph to a straight line on the distance–time
graph.
Here is our horizontal line
representing an object moving with a different speed. Since we haven’t specified a
vertical-axis scale, we don’t know the speed of this object. We do know that the yellow line is
lower down along the vertical axis than the blue line. This means that the object
represented by the yellow line is moving slower than the object represented by the
blue line. Its speed is smaller. On a distance–time graph, an object
moving with a slower uniform speed is represented by a straight line with a
shallower slope, say, this line right here. It makes sense that the slope is
shallower for objects moving with smaller speeds because the gradient of the line
representing the object on a distance–time graph must be exactly the speed of the
object.
It is worth stressing again that
even though we haven’t drawn scales on our axes, the slope of the yellow line on the
distance–time graph is exactly the same value as the vertical-axis value of the
yellow horizontal line on the speed–time graph. Let’s now add a third object to
these graphs. This object will be moving with a
uniform speed in between the two speeds we’ve already represented. This is easy to do on the
speed–time graph. All we need to do is draw a
horizontal line that intersects the vertical axis below the blue line and above the
yellow line. We can pick any vertical-axis value
in this range. Let’s take this one.
The green line is between the blue
and yellow lines, so the speed of the object represented by this line is slower than
the speed of the object represented by the blue line but also faster than the speed
of the object represented by the yellow line. That is, the speed of the object
represented by the green line is somewhere in the middle. The corresponding line on the
distance–time graph would have the same slope as the vertical-axis value of this
line. Since this speed is in the middle
of our other two speeds, the slope of this line would be shallower than the steeper
slope but also steeper than the shallower slope.
This again makes a lot of
sense. In the same amount of time, the
fastest object will move the longest distance. The slowest object will move the
shortest distance. And the object whose speed is in
the middle will move a distance in between the longest and shortest distances. Alright, now that we’ve learned how
to work with speed–time graphs, let’s work through an example.
Which color line on the speed–time
graph shows the motion of the object on the distance–time graph?
We are given two graphs to work
with in this question. On the left, we have the
distance–time graph, which we can tell because it has distance on the vertical
axis. On the right is the speed–time
graph, and we know this one because speed is the label on the vertical axis. Now the distance–time graph has one
line, and this is the motion of the object in the question. What we need to do is figure out
which line, the diagonal red line or the horizontal green line, on the speed–time
graph corresponds to this motion. We can see, though, that neither of
these graphs have scales on their axes, so we won’t be able to approach this problem
quantitatively. We will have to do all of our work
qualitatively.
Okay, so here’s what we know about
the distance–time graph. First of all, it is a straight
line. Now we recall that straight lines
on distance–time graphs represent objects moving with uniform speed. We can also recall that for an
object moving with uniform speed, the slope of the line on the distance–time graph
is the speed of the object. Our key observation is that the
speed of the object is constant. Now we recall that on speed–time
graphs, objects moving with constant speed are represented by horizontal lines. Also on a speed–time graph, the
height of the horizontal line is the constant speed of the object. This is the most important
distinction between speed–time graphs and distance–time graphs showing uniform
speed.
On a distance–time graph for
uniform speed, the speed is the slope of the line. But on a speed–time graph for
uniform speed, the line is always horizontal and its height is the speed. So we just need to identify the
correct horizontal line on the speed–time graph. Well, there’s only one of
those. It’s the green line. The red line is not horizontal. So the horizontal green line on the
speed–time graph must be the line that shows the motion of the object on the
distance–time graph. We can also double-check that the
diagonal red line on the speed–time graph is not the correct answer. The motion that we’re interested in
has uniform speed; that is, the speed is the same at all times.
So let’s look at two different
times on the speed–time graph, say, this time here and this time here. To find the corresponding speeds
represented by the red line at these two times, we first draw a vertical line from
the horizontal axis to the red line. Next, we draw a horizontal line to
the vertical axis. From each of these two intersection
points. Where these horizontal lines
intersect the vertical axis are the speeds at each of the times we were interested
in. In particular, the speed at this
first time is represented by this point on the vertical axis, and the speed at this
second time is represented by this point on the vertical axis. But the two points on the vertical
axis are at different values and therefore represent different speeds.
So what we have found is that the
motion represented by the red line has different speeds at different times. But this means that the red line
cannot be the correct answer because it has different speeds at different times, but
the object we are interested in has uniform speed. That is, it must have the same
speed at all times. So we have both found that the
green line must be the correct answer and also that the red line cannot be the
correct answer.
Great, now that we’ve worked
through this example, let’s review what we’ve learned in this lesson. The first thing we learned in this
lesson is that we can identify a speed–time graph by looking for a vertical axis
labeled speed and a horizontal axis labeled time. We learned that horizontal lines on
a speed–time graph show objects moving with uniform speed. We also saw that the height of
these horizontal lines is the speed of the object. So lines that are higher up on the
vertical axis represent objects moving faster, while lines that are lower down along
the vertical axis show objects that are moving slower.
However, none of these lines, and
indeed no points on the speed–time graph at all, can be below zero along the
vertical axis because speed is always greater than or equal to zero. Finally, we learned how to compare
distance–time graphs and speed–time graphs for uniform speed by comparing the slope
of the straight line on the distance–time graph to the height of the horizontal line
on the speed–time graph.
