Information read directly off the horizontal and vertical axis.
REVIEW
The vertical axis units divided by the horizontal axis units.
QUESTIONS
The vertical axis units times the horizontal axis units.
CLICK on a topic or a star or
Welcome, I’m Professor Bob
and I’m here to present
today’s topic,
During the slide presentation
if you want to go to the first slide
just click on me.
There are also forward one page
and back one page buttons on
the bottom of each slide.
We are constantly exposed to
information in graphical
form.
Newspapers, magazines,
TV all use graphs to display
information clearly.
We will look at three different
methods of obtaining
information from a graph.
The first is
information
off
the horizontal and vertical
axis
Vertical Axis
We will refer to information
obtained from the
horizontal and vertical
axis as a
Horizontal Axis
This information you
off the graph. No calculations
are involved.
Let’s look at some
examples.
What month had the
least amount
of rainfall?
Rainfall for Otego, NY
Inches of Rain
2
1
0
July
August
Month
September
What month had the
least amount
of rainfall?
Rainfall for Otego, NY
Inches of Rain
2
1
0
July
August
Month
September
How much rain fell
in September?
Rainfall for Otego, NY
Inches of Rain
2
1
0
July
August
Month
September
What month received
1.75 inches of rain?
Rainfall for Otego, NY
Inches of Rain
2
1
0
July
August
Month
September
The answers to all of
these questions are
Rainfall for Otego, NY
Inches of Rain
2
1
0
July
August
Month
September
The answers are read
Rain
ofRain
Inchesof
Inches
from either the horizontal
Rainfall for Otego, NY
or vertical axis.
2
1
0
July
August
Month
Month
September
Let’s look at how a
applies to linear motion
graphs.
This first graph has TIME
on the horizontal axis
and DISTANCE on the
vertical axis
8
6
4
2
1
2
3
TIME (s)
4
DISTANCE vs. TIME
This is called a
DISTANCE vs. TIME
graph
8
6
4
2
1
2
3
TIME (s)
4
The purple line
represents the distance an
object travels as it moves
along the floor.
DISTANCE vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
DISTANCE vs. TIME
At what time was the
object 6 meters from
its starting point?
8
6
4
2
1
2
3
TIME (s)
4
Start at 6 meters on the
DISTANCE vs. TIME
DISTANCE axis, go horizontally 8
over to the purple line and
then drop down to the
6
TIME-axis.
4
2
1
2
3
TIME (s)
4
DISTANCE vs. TIME
We read the answer,
8
off the TIME AXIS
6
4
2
1
2
3
TIME (s)
4
DISTANCE vs. TIME
How far did the object
travel in 2 seconds?
8
6
4
2
1
2
3
TIME (s)
4
Start at 2 seconds on the
TIME AXIS,
go vertically upward to the purple
line and then over to the
DISTANCE AXIS.
DISTANCE vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
The answer is found
DISTANCE vs. TIME
8
on the
DISTANCE or VERTICAL AXIS
6
4
2
1
2
3
TIME (s)
4
Sometimes there are several
sets of data plotted on the same
graph.
DISTANCE vs. TIME
8
C
B
6
4
A
2
1
2
3
TIME (s)
4
Again, we can ask questions
where the answers are read
off the
horizontal and vertical
axis
DISTANCE vs. TIME
8
C
B
6
4
A
2
1
2
3
TIME (s)
4
DISTANCE vs. TIME
At what time was object A
three meters
from its starting point?
8
C
B
6
4
A
2
1
2
3
TIME (s)
4
DISTANCE vs. TIME
At a time of one second
which object had traveled
the greatest distance?
8
C
B
6
4
A
2
1
2
3
TIME (s)
4
Here is another graph.
8
6
B
4
A
2
1
2
3
TIME (s)
4
When given a graph, the
first thing you should look at
is what is given on the
horizontal and vertical axis.
8
6
B
4
A
2
1
2
3
TIME (s)
4
This graph is a
SPEED vs. TIME
graph
for two different joggers.
SPEED vs. TIME
8
6
B
4
A
2
1
2
3
TIME (s)
4
Which jogger was
traveling the fastest?
SPEED vs. TIME
8
6
B
4
A
2
1
2
3
TIME (s)
4
According to the graph,
jogger A was traveling at 2 m/s
and jogger B was traveling
at 5 m/s.
SPEED vs. TIME
8
6
B
4
A
2
1
2
3
TIME (s)
4
The answer was
SPEED vs. TIME
8
off the vertical axis.
6
B
4
A
2
1
2
3
TIME (s)
4
DISTANCE vs. TIME
The second thing we can get
from a graph is it’s
8
6
4
2
1
2
3
TIME (s)
4
RISE
Slope  RUN
DISTANCE vs. TIME
8
OR
6
4
2
1
2
3
TIME (s)
4
Vertical Axis
Slope 
Horizontal Axis
DISTANCE vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
For this graph:
Distance
Slope 
Time
DISTANCE vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
and since
Distance
 speed
Time
DISTANCE vs. TIME
8
then, the slope of a Distance-Time
graph represents the SPEED of6
the object.
4
2
1
2
3
TIME (s)
4
We can also use the
UNITS
to determine what
the slope represents.
DISTANCE vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
meters
Units for Slope 
seconds
 speed
DISTANCE vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
The SLOPE of a straight line
is constant, therefore since the
slope of a distance-time graph
represents speed, the SPEED
shown here is constant.
DISTANCE vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
The SLOPE of a straight line
is constant, therefore since the
slope of a distance-time graph
represents speed, the SPEED
shown here is constant.
DISTANCE vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
Here the SLOPE is increasing, 8
so the SPEED is increasing.
DISTANCE vs. TIME
6
4
2
1
2
3
TIME (s)
4
Here the SLOPE is decreasing,8
so the SPEED is also decreasing.
DISTANCE vs. TIME
6
4
2
1
2
3
TIME (s)
4
This graph looks identical to
the first distance-time graph
we displayed but now
VELOCITY
is plotted on the vertical axis
8
6
4
2
1
2
3
TIME (s)
4
This is a
VELOCITY vs. TIME
graph
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
Here:
Velocity
SLOPE 
time
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
Or:
m
m
s
Units for Slope 
 2
s
s
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
V
m
or 2  acceleration
t
s
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
Therefore, the
SLOPE of a VELOCITY-TIME
graph represents
ACCELERATION
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
Let’s look at the
DISTANCE-TIME graph
and
VELOCITY-TIME graph
side-by-side.
DISTANCE vs. TIME
VELOCITY vs. TIME
8
8
6
6
4
4
2
2
1
2
3
TIME (s)
4
1
2
3
TIME (s)
4
The third method for obtaining
information from a graph is
by determining the
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
VELOCITY vs. TIME
under the plotted line.
8
6
4
2
1
2
3
TIME (s)
4
Let’s use UNITS to determine
what the
represents.
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
bh

2
The 2 is just a number
and doesn’t have any
associated units.
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
bh
= (s) (m) = (m)

2
(s)
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
If the units for
are (meters)
then the area of a Velocity-Time
graph represents either distance
or displacement.
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
What distance did the object
travel during the first
three seconds?
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
To answer this we need
to find the area from
zero to three seconds.
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
bh

2
m
(3s)(6 )
s

2
 9m
VELOCITY vs. TIME
8
6
4
2
 9m
1
2
3
TIME (s)
4
So, from zero to three
seconds, the object travels
nine meters.
VELOCITY vs. TIME
8
6
4
2
 9m
1
2
3
TIME (s)
4
Let’s try another question.
How far did the object
travel during the third
second?
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
Here we need to find the
AREA
from two to three seconds.
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
We can solve for the area
using the equation for a trapezoid
or by breaking the area down
into a triangle and a rectangle.
VELOCITY vs. TIME
8
6
4
2
1
2
3
TIME (s)
4
VELOCITY vs. TIME
Lets break the area down
into a triangle and a rectangle.
8
6
4
2
1
2
3
TIME (s)
4
Total
Total
Total
VELOCITY vs. TIME
bh
2
LW
(1s)(2m)
s
2
8
(1s)(4m)
s
6
1m
4
1m
4m
Total
4m
2
5m
Total
1
2
3
TIME (s)
4
According to our calculations
of AREA, the object traveled
5 meters during the third
second of travel.
VELOCITY vs. TIME
8
6
1m
4
4m
2
1
2
3
TIME (s)
4
Before trying several problems,
let’s review what we’ve covered
up to this point.
What are the three different
methods of obtaining
information from a graph?
Information read directly off the horizontal and vertical axis.
The vertical axis units divided by the horizontal axis units.
The vertical axis units times the horizontal axis units.
The following questions will refer to this
velocity-time graph
which represents the velocity of a car
over a period of twenty seconds.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
For every question, you should determine if the
question is asking for a
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
For every question, you should determine if the
question is asking for a
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
For every question, you should determine if the
question is asking for
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
What is the speed of the car
at a time of 2 seconds?
This is a direct reading. The answer
is read directly off the vertical
axis.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When is the car traveling
with a constant speed?
This is also a direct reading. We
need to determine when the
velocity is not changing.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When is the car traveling
with a constant speed?
The velocity is constant 4 to 10 seconds
and 14 to 16 seconds.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
Between 0 and 4 seconds is the
car speeding up or slowing down?
This is another direct reading.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
Between 0 and 4 seconds is the
car speeding up or slowing down?
First look at the velocity at 0 seconds.
Then the velocity at 4 seconds.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
Between 0 and 4 seconds is the
car speeding up or slowing down?
The velocity changed from 0 to 10 m/s
therefore the car was speeding up.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When does the car have a
velocity of +5 m/s?
Another direct reading.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When does the car have a
velocity of +5 m/s?
Determine the times when the purple
line crosses the +5 m/s line.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When does the car have a
velocity of +5 m/s?
The car has a velocity of +5 m/s at
a time of 2 s and 11.3 s.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When is the car slowing down?
Here we need to find when the SPEED
of the car is decreasing.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When is the car slowing down?
The first section starts at 10 seconds.
The car slows down from 10 m/s
to 0 m/s.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When is the car slowing down?
The second section starts at 16 seconds.
The car slows down from 5 m/s
to 0 m/s.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When is the car slowing down?
Remember, velocity is a vector. It is a
combination of SPEED and direction.
The - sign just shows the direction
the car is traveling.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When is the car accelerating?
On a velocity-time graph
represents acceleration. Therefore,
the car is accelerating where the
slope is not zero.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When is the acceleration positive?
Here we are looking for a positive
The slope is positive 0 to 4 s
and 16 to 20 s.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When does the car have the
greatest acceleration?
Here we are looking for the steepest
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When does the car have the
greatest acceleration?
Acceleration is also a vector. Positive
and negative slopes only indicate
direction, not magnitude.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
When does the car have the
greatest acceleration?
Therefore the acceleration is the greatest
where the slope is the greatest,
regardless of the direction.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
What is the acceleration of the car
between 10 and 14 seconds?
Here we need to determine the
between 10 and 14 seconds.
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
V
V V
 2 1
t
t 2  t1
What is the acceleration of the car
between 10 and 14 seconds?
 5 ms  (10 ms )
14s  10s
The acceleration for the entire
time period of 10 to 14 seconds
is -3.75 m/s2.
m
 3.75 2
s
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
What is the acceleration of the car at
12.7 seconds?
We know the acceleration between
10 s and 14 seconds is -3.75 m/s2
so
m
 3.75 2
s
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
What is the acceleration of the car at
12.7 seconds?
even though the car has an instantaneous
velocity of zero at a time of 12.7 s,
it is still
accelerating at a - 3.75 m/s2.
m
 3.75 2
s
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
What is the acceleration of the car at
12.7 seconds?
Also, you can not determine the slope of a
point but need to find the slope of the
line that goes through that point.
m
 3.75 2
s
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
What is the acceleration of the car at
12.7 seconds?
Therefore, it is necessary to use two points
and determine the slope of the line that goes
through the point in question.
m
 3.75 2
s
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
How far does the car travel during the
first four seconds?
To obtain distance or displacement from a
velocity-time graph we need to find:
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
How far does the car travel during the
first four seconds?
So, we need to solve for the
the first 4 seconds?
for
VELOCITY vs. TIME
+10
+5
TIME (s)
0
-5
-10
4
8
12
16
20
How far does the car travel during the
first four seconds?
(4s)(10 m
s)
2
bh
2
20m
VELOCITY vs. TIME
+10
+5
20m
0
-5
-10
4
TIME (s)
8
12
16
20
How far does the car travel during the
time interval of 4 to 10 seconds?
This time we need to solve for the
of a rectangle.
VELOCITY vs. TIME
+10
+5
20m
0
-5
-10
4
TIME (s)
8
12
16
20
How far does the car travel during the
time interval of 4 to 10 seconds?
(6s)(10 m
s)
LW
60m
VELOCITY vs. TIME
+10
+5
20m
0
-5
-10
4
60m
8
TIME (s)
12
16
20
How far does the car travel during the
time interval of 10 to 12.7 seconds?
Again, find the
VELOCITY vs. TIME
+10
+5
20m
0
-5
-10
4
60m
8
TIME (s)
12
16
20
How far does the car travel during the
time interval of 10 to 12.7 seconds?
(2.7s)(10 m
s ) 13.5m
2
bh
2
VELOCITY vs. TIME
+10
+5
20m
0
-5
-10
4
60m
8
13.5m
12
TIME (s)
16
20
What is the car’s displacement for the
first 12.7 seconds of travel?
We need to find the area from 0 to 12.7
seconds.
VELOCITY vs. TIME
+10
+5
20m
0
-5
-10
4
60m
8
13.5m
12
TIME (s)
16
20
What is the car’s displacement for the
first 12.7 seconds of travel?
(20m) (60m) (13.5m)
 93.5m
VELOCITY vs. TIME
+10
+5
20m
0
-5
-10
4
60m
8
13.5m
12
TIME (s)
16
20
What is the car’s displacement for the
first 12.7 seconds of travel?
(20m) (60m) (13.5m)
 93.5m
VELOCITY vs. TIME
+10
 93.5m
+5
TIME (s)
0
-5
-10
4
8
12
16
20
What is the car’s displacement for the last
7.3 seconds of travel?
Again we need to solve for area but this
time let’s use the equation for a trapezoid:
VELOCITY vs. TIME
+10
 93.5m
+5
TIME (s)
0
-5
-10
4
8
12
16
20
What is the car’s displacement for the last
7.3 seconds of travel?
Equation for a trapezoid:
(b1  b2)
2 h
(7.3s  2s)
m
(5 )
2
s
VELOCITY vs. TIME
+10
 93.5m
+5
TIME (s)
0
-5
-10
4
8
12
16
20
What is the car’s displacement for the last
7.3 seconds of travel?
 23.25m
VELOCITY vs. TIME
+10
 93.5m
+5
TIME (s)
0
-5
-10
4
8
12
 23.25m
16
20
How far did the car travel from zero
to 20 seconds?
We are looking for the total distance.
= 93.5 m +23.25 m = 116.75 m
VELOCITY vs. TIME
+10
 93.5m
+5
TIME (s)
0
-5
-10
4
8
12
 23.25m
16
20
How far did the car travel from zero
to 20 seconds?
Distance is a scalar quantity so direction
isn’t included.
VELOCITY vs. TIME
+10
 93.5m
+5
TIME (s)
0
-5
-10
4
8
12
 23.25m
16
20
At a time of 20 seconds, how far is
the car from it’s starting point?
This question is asking for
displacement, therefore direction is
important.
VELOCITY vs. TIME
+10
 93.5m
+5
TIME (s)
0
-5
-10
4
8
12
 23.25m
16
20
At a time of 20 seconds, how far is
the car from it’s starting point?
The car traveled 93.5 meters in the
positive direction then 23.25 meters in
the negative direction.
VELOCITY vs. TIME
+10
 93.5m
+5
TIME (s)
0
-5
-10
4
8
12
 23.25m
16
20
At a time of 20 seconds, how far is
the car from it’s starting point?
= +93.5 m -23.25 m
= +70.25 m
VELOCITY vs. TIME
+10
 93.5m
+5
TIME (s)
0
-5
-10
4
8
12
 23.25m
16
20
We worked mainly with two
different graphs; a
graph
and a
graph
8
VELOCITY vs. TIME
8
6
4
2
DISTANCE vs. TIME
1
6
2
TIME (s)
4
2
1
2
TIME (s)
3
4
3
4
A direct reading on the
VELOCITY vs. TIME
graph
8
gives us
distance and time.
8
6
4
2
DISTANCE vs. TIME
1
6
2
TIME (s)
4
2
1
2
TIME (s)
3
4
3
4
A direct reading on the
VELOCITY vs. TIME
graph
8
gives us
velocity and time.
8
6
4
2
DISTANCE vs. TIME
1
6
2
TIME (s)
4
2
1
2
TIME (s)
3
4
3
4
The SLOPE of a
VELOCITY vs. TIME
graph
8
is
6
4
8
2
DISTANCE vs. TIME
1
6
2
TIME (s)
4
2
1
2
TIME (s)
3
4
3
4
The SLOPE of a
VELOCITY vs. TIME
graph
8
is
6
4
8
2
DISTANCE vs. TIME
1
6
2
TIME (s)
4
2
1
2
TIME (s)
3
4
3
4
The AREA of a
VELOCITY vs. TIME
graph
8
gives us
6
4
8
2
DISTANCE vs. TIME
1
6
2
TIME (s)
4
2
1
2
TIME (s)
3
4
3
4
VELOCITY vs. TIME
We are not limited to these
two graphs so:
8
6
4
8
2
DISTANCE vs. TIME
1
6
2
TIME (s)
4
2
1
2
TIME (s)
3
4
3
4
Whenever you are given
a graph remember the three
different methods of obtaining
information from a graph.
Information read directly off the horizontal and vertical axis.
The vertical axis units divided by the horizontal axis units.
The vertical axis units times the horizontal axis units.