Lines 𝐴, 𝐵, and 𝐶 on the distance–time graph shown are parallel. Which of these lines could correspond to the line shown on the speed–time graph? (A) Line 𝐵 only, (B) line 𝐵 and line 𝐶 only, (C) line 𝐴, line 𝐵, and line 𝐶,
(D) line 𝐴 and line 𝐵 only, (E) none of these lines.
In this question, we are asked to relate the information given to us on a
distance–time graph to a speed–time graph.
First, let’s think about the distance–time graph. We have the three lines 𝐴, 𝐵, and 𝐶 drawn on a distance–time graph, where the
horizontal axis represents time and the vertical axis represents distance. The lines are all horizontal. Let us think about what this means. If a line on a distance–time graph is horizontal, it means that all values of time on
the 𝑥-axis correspond to the same distance on the 𝑦-axis.
For example, let’s compare the distance represented by line A at two different times,
𝑡 one and 𝑡 two. These two times correspond to different positions on the 𝑥-axis, but they correspond
to the same position on the 𝑦-axis. This means that at both times, the object represented by line 𝐴 is the same distance
from whatever reference point we’re measuring from. We could do this for any of the times on the 𝑥-axis. The object represented by this line is always at the same distance from the reference
point. So, a horizontal line on a distance–time graph represents an object which is always
at the same distance from the reference point.
On our distance–time graph, lines 𝐴, 𝐵, and 𝐶 are all horizontal. The only difference between them is that they have different vertical positions on
the graph. Line 𝐴 corresponds to the largest value on the 𝑦-axis and so represents an object
at a greater distance from a reference point than line 𝐵 or 𝐶. In fact, line 𝐶 corresponds to zero distance, meaning line 𝐶 represents an object
that is still at the reference point.
We’ve now seen that a horizontal line on a distance–time graph represents an object
that is always at the same distance from the reference point. Let’s think about what this tells us about the speed of the objects represented by
lines 𝐴, 𝐵, and 𝐶.
If the distance of each object from the reference point never changes, this tells us
that none of the objects are moving. They must all be stationary. If an object is stationary, it means that its speed is zero. So the three lines on this distance–time graph all represent objects that have zero
Now let’s consider what the speed–time graph for each of these objects would look
like. On a speed–time graph, time is represented by the horizontal axis and speed is
represented by the vertical axis. In the same way that a horizontal line on a distance–time graph represents a constant
distance, a horizontal line on a speed–time graph represents a constant speed.
We know that lines 𝐴, 𝐵, and 𝐶 from the distance–time graph all represent objects
with zero speed. An object with a speed of zero is represented by a horizontal line that runs along
the horizontal axis at speed zero. Let’s compare this to the speed–time graph given to us in the question.
We can see that the line on this speed–time graph is horizontal, like the graph we
just drew. However, it’s very important to notice that this line does not correspond to zero
speed. A line corresponding to zero speed would run along the 𝑥-axis. But this line corresponds to a value of the speed that is greater than zero. So, the line on this speed–time graph cannot correspond to any of the lines on the
distance–time graph, as all of the lines on the distance–time graph represent
objects with zero speed. The correct answer is option (E), none of the lines.