### Video Transcript

Lines š“, šµ, and š¶ on the distanceātime graph shown are parallel. Which of these lines could correspond to the line shown on the speedātime graph? (A) Line šµ only, (B) line šµ and line š¶ only, (C) line š“, line šµ, and line š¶,
(D) line š“ and line šµ only, (E) none of these lines.

In this question, we are asked to relate the information given to us on a
distanceātime graph to a speedātime graph.

First, letās think about the distanceātime graph. We have the three lines š“, šµ, and š¶ drawn on a distanceātime 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 distanceātime graph is horizontal, it means that all values of time on
the š„-axis correspond to the same distance on the š¦-axis.

For example, letās compare the distance represented by line A at two different times,
š” one and š” two. These two times correspond to different positions on the š„-axis, but they correspond
to the same position on the š¦-axis. This means that at both times, the object represented by line š“ is the same distance
from whatever reference point weāre measuring from. We could do this for any of the times on the š„-axis. The object represented by this line is always at the same distance from the reference
point. So, a horizontal line on a distanceātime graph represents an object which is always
at the same distance from the reference point.

On our distanceātime graph, lines š“, šµ, and š¶ are all horizontal. The only difference between them is that they have different vertical positions on
the graph. Line š“ corresponds to the largest value on the š¦-axis and so represents an object
at a greater distance from a reference point than line šµ or š¶. In fact, line š¶ corresponds to zero distance, meaning line š¶ represents an object
that is still at the reference point.

Weāve now seen that a horizontal line on a distanceātime graph represents an object
that is always at the same distance from the reference point. Letās think about what this tells us about the speed of the objects represented by
lines š“, šµ, and š¶.

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 distanceātime graph all represent objects that have zero
speed.

Now letās consider what the speedātime graph for each of these objects would look
like. On a speedātime 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 distanceātime graph represents a constant
distance, a horizontal line on a speedātime graph represents a constant speed.

We know that lines š“, šµ, and š¶ from the distanceātime 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. Letās compare this to the speedātime graph given to us in the question.

We can see that the line on this speedātime graph is horizontal, like the graph we
just drew. However, itās very important to notice that this line does not correspond to zero
speed. A line corresponding to zero speed would run along the š„-axis. But this line corresponds to a value of the speed that is greater than zero. So, the line on this speedātime graph cannot correspond to any of the lines on the
distanceātime graph, as all of the lines on the distanceātime graph represent
objects with zero speed. The correct answer is option (E), none of the lines.