0901 Quiz

1.  A graph of v vs. t consists of a straight line segment from (4 sec, 6 cm/s) to (9 sec, 18 cm/s).  Sketch this segment on a graph of v vs. t.

• What is the slope of the segment?

The slope is

rise / run

= (18 cm/s - 6 cm/s) / (9 sec - 4 sec)

= 12 cm/s / (5 s)

= 2.4 (cm/s)/s.

• What is the meaning of the rise of this segment?

The rise is 12 cm/s, which is the change in the vertical coordinate of the graph.  The vertical coordinate measures velocity, so the rise is change in velocity.

• What is the meaning of the run of this segment?

The run is 5 s, which is the change in the horizontal coordinate of the graph.  The horizontal coordinate measures clock time, so the run is change in clock time.

• What is the meaning of the slope of the segment?

The slope is rise / run, which in this case is change in velocity / change in clock time.  This is the definition of rate of change of velocity with respect to clock time, or acceleration.

• What is the midpoint velocity?

Since the graph is linear the midpoint velocity is halfway between initial and final velocities (by the symmetry of the segment about the midpoint), so midpoint velocity is (6 cm/s + 18 cm/s) / 2 = 12 cm/s.

• What is the meaning of the midpoint velocity?

The midpoint velocity is the average velocity (by the symmetry of the graph about the midpoint, since the v vs. t graph is linear).

• How far does the object travel during the time interval represented by the graph?

Displacement is average vel * change in clock time, so the object travels (12 cm/s) * 5 s = 60 cm.

2.  A graph of position vs. t consists of two straight line segments, one from (4 sec, 8 cm) to (9 sec, 68 cm) and another from this point to (12 sec, 140 cm).

• What is the slope of the first segment?

The slope of the first segment is

slope = rise / run

= (68 cm - 8 cm) / (9 sec - 4 sec)

= 60 cm / (5 sec)

= 12 cm/s.

• What is the meaning of the rise of this segment?

Since the vertical coordinate measures position, the rise is the change in the position of the object; in this case the rise is 60 cm which means that the position changed by +60 cm.

• What is the meaning of the run of this segment?

The run of the segment is 5 sec, which is the change in horizontal coordinate.  The horizontal coordinate measures clock time, so the run is change in clock time.

• What is the meaning of the slope of this segment?

The slope of the segment is rise / run, which for this example means change in position / change in clock time, or average rate of change of position with respect to clock time.

In this situation we find that this average rate is 60 cm / (5 s) = 12 cm/s.  The average rate of change of position with respect to clock time is also called the average velocity of the object.

• How far does the object travel during the time interval represented by this segment?

The change in position is 60 cm, which corresponds to the rise of the graph.

• Answer the same questions for the second segment.

The slope of the second segment is (140 cm - 68 cm) / (12 sec - 9 sec) = 72 cm / (3 sec) = 24 cm/s.

Using the same sequence of interpretation we find that this is the average velocity on the second segment.

3.  For the graph of #1, what does the graph of position vs. t look like for the time interval described by the graph?

The displacement between t = 4 sec and t = 9 sec was found to be 60 cm.  On a position vs. t graph the displacement is the rise.  So we know that between t = 4 sec and t = 9 sec, the position vs. clock time graph has a rise of 60 cm.

We know from the v vs. t graph that the velocity increases from 6 cm/s to 18 cm/s between these clock times.   The velocity of the object is the slope of the position vs. clock time graph.  So the slope of our position vs. clock time graph has to increase from 6 cm/s to 18 cm/s.

Questions to Ask when Interpreting a Graph

A graph of A vs. B has quantity A on the vertical axis, quantity B on the horizontal axis.

Any graph you sketch should be labeled in the following manner:

• The title of the graph will contain the identification A vs. B, with A and B being the appropriate names of the things being graphed.
• Each axis should be labeled and the units should be included in the label (e.g., for a graph of velocity v vs. clock time t, the vertical axis might be labeled 'v (cm/s)' and the horizontal axis 't (sec)').

When you analyze a graph you typically ask the following questions:

• What quantity is graphed on the vertical axis and what does this quantity tell you about the situation being graphed?
• What quantity is graphed on the horizontal axis and what does this quantity tell you about the situation being graphed?
• What does the highest point of the graph represent in terms of the situation?
• What does the lowest point of the graph represent in terms of the situation?
• What does the rise between two points of the graph mean in terms of the situation?
• What does the run between two points of the graph mean in terms of the situation?
• What does the slope of the graph represent in terms of the situation?
• Where is the slope greatest, where is the slope least, and what does this tell you about the situation
• Where is the slope zero, and what does this tell you about the situation?

• What does the midpoint height of a line segment on the graph represent and what does this tell you about the situation?
• What does the width of the trapezoid beneath the line segment represent and what does this tell you about the situation?
• What does the area of the trapezoid beneath the line segment represent and does this quantity tell you anything important about the situation, and if so what