Video Transcript
Which of the following most
correctly describes the motion represented by the following distance–time graph? The time intervals shown are
equal. Is it (A) the solid and dashed
lines represent the same uniformly accelerated motion? Is it (B) the solid line represents
uniform motion, and the dashed line represents accelerated motion? (C) The solid line represents
accelerated motion, and the dashed line also represents accelerated motion but with
a different value of acceleration. Or (D) the solid line represents
accelerated motion, and the dashed line represents uniform motion.
The distance–time graph shows a
line that has been drawn in two sections. One section of the line has been
drawn as a solid line, and the other section of the line has been drawn as a dashed
line.
Looking at the solid section of the
line, we see that it curves smoothly. A line curving this way represents
a gradual increase in speed of an object. For this section of the line, we
can draw tangent lines that touch the line at different time values. We see that the slope of each
tangent increases as the time value increases. This indicates that the speed of
the object is increasing or, in other words, that the object is accelerating.
Now let’s look at the dashed part
of the line. There are two time intervals
defined on this part of the graph. These time intervals are shown by
the diagram in the question to be equal. We can see though that the
distances traveled in these two time intervals are not equal.
If we mark the start and end
distances for each time interval as 𝑑 sub one and 𝑑 sub two for the first time
interval, which starts at time 𝑡 sub one and ends at 𝑡 sub two, we can calculate
the average speed for the first time interval. If we mark the start and end
distances for the second time interval as 𝑑 sub three and 𝑑 sub four, which starts
at time 𝑡 sub three and ends at 𝑡 sub four, we can calculate the average speed for
the second time interval.
The first time interval shows a
smaller distance traveled than the second interval, which also tells us that the
object is traveling at a greater average speed during the second time interval than
in the first time interval. In other words, the object
increased in speed. We can see then that the object
also accelerates in the section of the graph where the line is dashed. When we draw tangents to the dashed
line, shown in green, the gradient of these tangents, representing the speed of the
object, also increases as time increases, which demonstrates that the speed of the
object is increasing in this part of the graph.
So the object is accelerating in
both sections of the graph, but is the acceleration equal in both sections? The detail that tells us that the
acceleration changes when the first section ends and the second section begins is
that the line does not curve smoothly at the time that first section ends and the
second section begins. The time that this occurs is shown
as 𝑡.
We can see how the distance would
need to have changed with time if the line had curved smoothly after 𝑡 in the same
way that it did before 𝑡. And this line does not match the
line in the question. We can see how the distance would
need to have changed with the time if the line had curved smoothly before 𝑡 in the
same way that it did after 𝑡. This line does not match the line
in the question either.
So for the accelerations in the two
sections to be equal, the line would have to curve smoothly in the same way before
and after 𝑡. The line shown in the question does
not do this, and so the acceleration must have changed at 𝑡.
We see then that the acceleration
represented by the first section of the line does not equal the acceleration
represented by the second section of the line. The correct answer is (C). The solid line represents
accelerated motion, and the dashed line also represents accelerated motion but with
a different value of acceleration.
