Video Transcript
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.
