Video Transcript
Do the speeds corresponding to the
lines shown on the following distance–time graph change value in the same ratio for
any two adjacent lines?
Let’s begin by clarifying exactly
what this question is asking.
We’re shown four different colored
lines plotted on a distance–time graph. Each of the four lines represents a
different speed. Using ratios, we want to compare
the different speeds represented by each set of two adjacent lines, or lines that
are next to each other. Then, we’ll compare those ratios to
each other. Since there are four lines plotted
on this graph, there are three sets of two adjacent lines.
To help us stay organized, let’s
designate and label the three sets. We’ll call the orange and green
line the first set of adjacent lines. The green and blue line is set
number two, and the blue and red lines make up set three. For each set, we’ll calculate a
ratio of the different speeds represented by its two lines. Then, we’ll see whether any of the
three ratios we come up with are the same. So let’s get to work calculating
the speed represented by each line.
To begin, we should recall that the
speed of an object is given by the formula 𝑣 equals Δ𝑑 over Δ𝑡, where 𝑣 is the
speed of the object, Δ𝑑 is the distance the object travels, and Δ𝑡 is the time for
which the object travels. And remember that for a
distance–time graph, like we have here, Δ𝑑 and Δ𝑡 are represented by changes in
value along the vertical axis and horizontal axis, respectively. In other words, the speed is
represented by the slope of each straight line on the distance–time graph. We’ll first calculate the speed of
each line separately. And then once we have those values,
we can go to work devising ratios from them.
Let’s start with the orange
line. Remember that we can choose to
measure the changes in values between any two points on a straight line. So let’s just choose two points
that are convenient. Notice that all of these plotted
lines pass through the origin. Since zero is a very convenient
value to use in calculations, we’ll use the origin as the starting point.
It’s also easy to see that the
orange line passes through this point here, which corresponds to a distance of eight
meters and a time of two seconds. Since we’re using the origin as the
starting point, we can write the orange line’s change in distance over change in
time as eight meters minus zero meters over two seconds minus zero seconds. Of course, since we’re subtracting
zero in both the numerator and denominator, this makes the math a bit easier. This is precisely why we chose the
origin as one of our two points for calculating the slope, rather than any other
point that the line happened to pass through. The expression is equal to eight
meters over two seconds, which simplifies to four meters per second. This is the speed represented by
the orange line.
With this in mind, let’s move on to
the green line. We already know that we’ll use the
origin as the starting point. So, for the other point, let’s
choose this one here, which corresponds to a distance of six meters and a time of
two seconds. Substituting these values into the
speed formula and canceling out the minus zero terms from the origin, we have a
speed of six meters over two seconds, or three meters per second. This is the speed represented by
the green line.
Next, for the blue line, let’s use
the origin and this point here, which corresponds to four meters and two
seconds. Substituting these values into the
formula gives a speed of two meters per second for the blue line.
Finally, for the red line, let’s
use the origin and this point, which is two meters and two seconds, giving a speed
of one meter per second.
We now have a value of speed for
each of the plotted lines on the graph. Let’s now refer to the different
sets of two adjacent lines and calculate a ratio for each set using the speeds we
just calculated.
For set one, we’ll write the ratio
as the speed of the orange line divided by the speed of the green line. Substituting in these values, we
have four meters per second divided by three meters per second, which equals about
1.33. Next, let’s move on to the ratio
for set two. The speed of the green line divided
by the speed of the blue line is three meters per second divided by two meters per
second, or 1.5. Finally, for the third set of
adjacent lines, we have a ratio of two meters per second over one meter per second,
which equals just two.
We are now ready to answer this
question. Do the speeds corresponding to the
lines change value in the same ratio for any two adjacent lines? Well, we can see that the three
ratios we just devised all have different values, so the answer must be no. For each set of two adjacent lines
on the graph, we calculated the ratio of their different speeds and then compared
all the resulting ratios. Since none of them are equal, our
final answer is no. The speeds corresponding to the
lines do not change value in the same ratio for any two adjacent lines.
