Physics 30S
Displaying Data with Graphs
A well-designed graph is more than a "picture worth a thousand words." It can give you
more information than either words, columns of numbers, or equations. To be useful,
however, a graph must be drawn properly. In this section we will develop the use of
graphs to display data.
Graphing Data
One of the most important skills to learn in driving a car is how to stop it safely. No car can "stop
on a dime." The faster the car is going, the farther it travels before it stops. If you studied for a
driver's license, you probably found a table in the manual showing how far a car moves beyond
the point at which the driver makes a decision to stop.
One manual shows that a car travels a certain distance between the time the driver decides to 'stop
the car and the time the brakes are applied. This is called the "reaction distance." When the brakes
are applied, the car slows down and travels the "braking distance." Table 1 and the bar graph,
Figure 1, show these distances, together with the total stopping distance for various speeds.
Original speed
Original speed
Reactions
Braking distance
(m/s)
(km/hr)
distance (m)
(m)
10
7.4
9.1
15
11.1
20.5
20
14.8
36.5
25
18.5
57.0
30
22.2
82.1
(Note: convert speeds to km/hr by using magic number of 3.6)
distances vs speed
distance (m)
The first step in analyzing data is
to look at them carefully. Which
variable did the experimenter (the
driver) change? In our example, it
was the speed of the car. Thus
speed is the independent
variable, or manipulated variable.
The other two variables, reaction
distance and braking distance,
changed as a result of the change
in the speed. These quantities are
called dependent variables, or
responding variables.
Total distance
(m)
16.5
31.6
51.3
75.5
104.3
120
100
80
60
40
20
0
10
15
20
25
30
speed (m/s)
How do the distances change for
rxn distance
braking distance
total distance
a given change in the speed?
Notice that the reaction distance increases by about the same amount for each increase in
speed. The braking distance, however, increases much more as the speed increases. The
relationship between the distances and speed can be seen more easily if the data are
plotted on a graph. To avoid confusion, you should plot two graphs, one of reaction
distance and the other of braking distance.
Physics 30S
Plotting Graphs
These steps will help you plot graphs from data tables.
1.
Identify the independent and dependent variables. The independent variable is
plotted on the horizontal, or x-axis. The dependent variable is plotted on the
vertical, or y-axis. In our example, speed is plotted on the x-axis, and distance is
plotted on the y-axis.
2.
Determine the range of the independent variable to be plotted. In the example,
data are given for speeds between 10 and 30 m/s. A convenient range for the xaxis might be 0-30 m/s.
3.
Decide if the origin (0, 0) is a valid data point. When the speed is zero, reaction
and stopping distances are obviously both zero. In this case then, your graph
should include the origin. Spread out the data as much as possible. Let each space
on the graph paper stand for a convenient unit. Choose 2, 5, or 10 spaces to
represent 10 m/s.
4.
Number and label the horizontal axis.
5.
Repeat steps 2-4 for the dependent variable.
6.
Plot your data points on the graph.
7.
Draw the best straight line or smooth curve that passes through as many data
points as possible. Do not use a series of straight-line segments that "connect the
dots."
8.
Give the graph a title that clearly tells what the graph represents.
Linear, Quadratic, and Inverse Relationships
Linear Relationship
distance (m)
Reaction distance vs speed
25
20
15
10
5
0
10
15
20
speed (m/s)
25
30
The graph, Figure 2, of reaction
distance versus speed is a
straight line. That is, the
dependent variable varies
linearly with the independent
variable; there is a linear
relationship between the two
variables.
The relationship of the two
variables shown in this graph can
be written as an equation
y = mx + b, where m and
b are constants called the slope
and y-intercept, respectively.
Each constant can be found from the graph. The slope, m, is the ratio of the
vertical change to the horizontal change. To find the slope, select two points, A and B, as
far apart as possible on the line. The vertical change, or rise, y, is the difference in the
vertical values of A and B. The horizontal change, x, or run, is the difference in the
horizontal values of A and B. The slope of the graph is then calculated as
rise y (18.5  7.4)m 11.1m
m



 0.74s
run x (25  10)m / s 15m / s
Physics 30S
Notice that units have been kept with the variables.
The y-intercept b, is the point at which the line crosses the y-axis and is the y value when the
value of x is zero. In this case, when x is zero, the value of y is 0 m. For special cases like this,
when the y- intercept equals zero, the equation becomes y = mx. The quantity y varies directly
with x.
y
After drawing the graph and obtaining the equation, check to see if the results make
sense. The slope indicates the increased
Negative Slope
reaction distance for an increase in
speed. It has units of
15
meters/(meters/second), or seconds.
Thus it is a time, the reaction time. It is
10
the time your body takes from the
instant the message to stop the car
5
registers in your brain until your foot
0
hits the brake pedal. Thus, the slope is
0
2
4
6
8
a property of the system the graph
-5
describes.
x
The value of y does not always increase with increasing x. If y gets smaller as x gets
larger, then y/x is less than zero, and the slope is negative.
Quadratic Relationship
Braking distance (m) vs speed (m/s)
70
distance
60
50
40
30
20
10
10
15
20
speed
25
30
The graph, Fig. 4, of braking
distance vs. speed is not a straight
line, but rather looks like a
parabola. This indicates that the
braking distance varies as the
square of the original speed. In
other words, if you double your
speed, the braking distance
increases 4 times. The graph’s
slope changes so it cannot be
calculated. You can however,
calculate the slope of the curve at
an instant, by drawing a tangent
line.
Question: If it takes a person 40 m to stop after driving at 50 km/hr, how far will it take to
stop if driving 100 km/hr?
10
Physics 30S
Inverse Relationship
y vs x
Some relationships are such that as one
quantity increases, the other decreases.
This is called an inverse relationship. Its
graph would look like Fig. 5. Again, you
may only determine the instantaneous
slope by applying the tangent line method.
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
10
20
30
40
50
Concept Review
1. Suppose a student doing a lab determines that the y – intercept for a graph of
reaction time vs. speed was 3.4. What would this mean in ‘real life’? Is it
possible? Draw a sketch of what this would look like.
2. In the earlier braking example, at what speed is the reaction distance 10 m?
3. Explain in your own words the significance of a steeper line, or larger slope to the
A graph of reaction distance versus speed is made and the slope is found to be
0.74s. Suppose another experiment is made and the slope is found to be steeper.
What does this mean?
circumference vs. diameter
circumference
(cm)
4. Critical Thinking- Figure 6
shows the relationship
between the circumference
and the diameter of a circle.
a) Could a different circle’s
circumference, be described
by a different straight line?
b) What is the meaning of
the slope? c)What are the
units of the slope?
15
10
5
0
1
2
3
diameter (cm)
Answers
4
5
Physics 30S
Exercises
mass (g) vs. volume (cm3)
300
mass (g)
1) The graph shows the mass of
three substances for volumes
between 0 and 60 cm3.
a) What is the mass of 30
cm3 of each substance?
b) If you had 100 g of each
substance, what would
their volumes be?
c) What are the units of the
slope of the graph?
Describe the meaning of
the slopes of the lines in
this graph.
200
100
0
0
20
40
volume (cm3)
2) During an experiment, a student measured the mass of 10.0 cm3 of ethanol. The
student then measured the mass of 20.0 cm3 of ethanol. In this way the data in the
table were collected.
a) Plot the values given in the table and draw the
Volume
Mass (g)
curve that best fits all points.
(cm3)
b) Describe the resulting curve.
10
7.9
c) Use the graph to write an equation relating the
20
15.8
volume to the mass of the alcohol.
30
23.7
d) Find the units of the slope of the graph. What is
40
31.6
the name given to this quantity?
50
39.6
60
Physics 30S
3) During a demonstration, an instructor placed a 1.0-kg mass on a horizontal table that
was nearly frictionless. The instructor then applied various horizontal forces to the
mass and measured the rate at which the mass gained speed (was accelerated) for
each force applied. The results of the experiment are shown in the table.
a) Plot the values given in the table and draw the curve that best fits all points.
b) Describe, in words, the relationship between force and acceleration according to
the graph.
c) Write the equation relating the force and the acceleration that results from the
graph.
d) Find the units of the slope of the graph.
Force (N)
Acceleration (m/s2)
5
4.9
10
9.8
15
15.2
20
20.1
25
25.0
30
29.9