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?
This question asks about the speeds
represented by different lines on a distance–time graph. The line on a distance–time graph
for an object with a constant speed is a straight line with a slope equal to the
speed of the object. Remember that speed 𝑣 is equal to
change in distance Δ𝑑 divided by change in time Δ𝑡. And this is the slope of the line
on a distance–time graph.
Comparing Δ𝑑 over Δ𝑡 for each
color line on the graph, we see that all lines are plotted over the same range of
time values. For all the lines, Δ𝑡 is given by
10 seconds minus zero seconds, which is 10 seconds. For any color line, the distance
traveled can be compared to the distance traveled shown by another line that is
adjacent to that line. Saying that a line is adjacent to
another line means that these lines are closer to each other than they are to other
lines. This means that we would say that
the blue line is adjacent to the red line, the red line is adjacent to the green
line, and the green line is adjacent to the yellow line.
Let’s just look at the parts of the
lines between times of zero and eight seconds, that is, during the time interval Δ𝑡
equals eight seconds. We see that if we do this, then the
distance traveled in this eight-second interval is two meters for the yellow line,
four meters for the green line, six meters for the red line, and eight meters for
the blue line.
The values of Δ𝑑 and Δ𝑡 can be
used to find the speeds represented by each of the lines. Substituting the values into the
speed equation in each case and evaluating the expressions, we find that the speeds
are 0.25 meters per second for the yellow line, 0.5 meters per second for the green
line, 0.75 meters per second for the red line, and one meter per second for the blue
line.
Let’s now clear some space on
screen so we can calculate the ratios of these speeds. We can calculate the ratio between
the speeds of adjacent lines by dividing the greater speed by the lesser speed in
each case. The ratio between the green and the
yellow lines is 0.5 meters per second over 0.25 meters per second, which works out
as two. The ratio between the red and the
green lines is 0.75 meters per second over 0.5 meters per second, which is 1.5. And the ratio between the blue and
red lines is one meter per second over 0.75 meters per second, which comes out as
1.3 recurring.
These three values we have
calculated are not the same. This means that the ratio of the
speed represented by the green line to the speed represented by the yellow line does
not equal the ratio of the speed represented by the red line to the speed
represented by the green line. And neither of these equal the
ratio of the speed represented by the blue line to the speed represented by the red
line. That is, we have found that the
speeds represented by each pair of adjacent lines do not change in the same
ratio. Our answer to this question then is
no.
