GRAPHICAL ANALYSIS OF MOTION
1
GRAPHICAL ANALYSIS OF MOTION
motion graphs come in three formats
2
GRAPHICAL ANALYSIS OF MOTION
motion graphs come in three formats
A. Position vs. time graph
3
GRAPHICAL ANALYSIS OF MOTION
motion graphs come in three formats
A. Position vs. time graph
B. Velocity vs. time graph
4
GRAPHICAL ANALYSIS OF MOTION
motion graphs come in three formats
A. Position vs. time graph
B. Velocity vs. time graph
C. Acceleration vs. time graph
5
GRAPHICAL ANALYSIS OF MOTION
motion graphs come in three formats
A. Position vs. time graph
B. Velocity vs. time graph
C. Acceleration vs. time graph ( this graph is of little
use in our course)
6
GRAPHICAL ANALYSIS OF MOTION
motion graphs come in three formats
A. Position vs. time graph
B. Velocity vs. time graph
C. Acceleration vs. time graph ( this graph is of little
use in our course)
Each graph is a function of time
7
GRAPHICAL ANALYSIS OF MOTION
all formats use a Cartesian System for graphing
6
5
position
(m)
4
3
2
1
0
0
1
2
3
4
time (sec)
5
6
7
8
GRAPHICAL ANALYSIS OF MOTION
all formats use a Cartesian System for graphing
time is always on the X axis (abscissa)
independent variable
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GRAPHICAL ANALYSIS OF MOTION
all formats use a Cartesian System for graphing
time is always on the X axis (abscissa)
independent variable
position, velocity, and acceleration are always on
the Y axis (ordinate)
dependent variable
10
Position vs. Time Graph
6
5
position
(m) 4
3
2
1
0
0
1
2
3
4
time (sec)
5
6
7
11
Velocity vs. Time Graph
6
5
velocity
( m/s) 4
3
2
1
0
0
1
2
3
4
time (sec)
5
6
7
12
Position vs. Time Graph
This is a graph of position vs. t for an object
moving with constant velocity. The velocity is
the slope of the y-t curve.
13
Position vs. Time Graph
Information provided
a. Measured Position at a specific
time
b. Estimated Position between two
known points
c. Slope of line gives average velocity
between any two Cartesian points
d. Slope of line can determine
instantaneous velocity
14
One-Dimensional Displacement and
Velocity: Vector Quantities
Different ways of visualizing uniform velocity:
15
One-Dimensional Displacement and
Velocity: Vector Quantities
This object’s velocity is not uniform.
a. the average velocity between any two
points can be found by finding the slope
of a line drawn between any two points
16
One-Dimensional Displacement and
Velocity: Vector Quantities
This object’s velocity is not uniform.
a. The average velocity between any two
points can be found by finding the slope
of a line drawn between any two points
b. The red line indicates a positive slope
and therefore a positive velocity
17
One-Dimensional Displacement and
Velocity: Vector Quantities
This object’s velocity is not uniform.
a. The average velocity between any two
points can be found by finding the slope
of a line drawn between any two points
b. The red line indicates a negative slope
and therefore a negative velocity
• the negative indicates direction
• the object may be moving towards the
origin
18
One-Dimensional Displacement and
Velocity: Vector Quantities
This object’s velocity is not uniform.
a. The average velocity between any two
points can be found by finding the slope
of a line drawn between any two points
b. The red line is horizontal between these
two points
• The average velocity is zero during this
time period
• the object the object is moving, but its
position is unchanged between points
2&5
19
One-Dimensional Displacement and
Velocity: Vector Quantities
This object’s velocity is not uniform.
a. The average velocity between any two
points can be found by finding the slope
of a line drawn between any two points
b. The red line is tangent to a point on the
curved line
c. This tangent line indicates an
instantaneous velocity at a particular
time
20
One-Dimensional Displacement and
Velocity: Vector Quantities
This object’s velocity is not uniform.
a. The average velocity between any two
points can be found by finding the slope
of a line drawn between any two points
b. The red line is horizontal to a point on the
curved line
c. This tangent line indicates the
instantaneous velocity is zero at this
particular time
21
Graphical Analysis of Linear Motion
On the left we have a graph of velocity vs. time for an object with varying velocity; on
the right we have the resulting position vs. time curve. The instantaneous velocity is
tangent to the curve at each point.
22
Position vs. Time Graph
6
5
position
(m) 4
3
2
1
0
0
1
2
3
4
time (sec)
5
6
7
23
Position vs. Time Graph
6
5
position
(m) 4
3
2
1
0
0
1
2
3
4
5
6
time (sec)
Find the position of this object at times 0.0, 2.0 sec, 5.0 sec, 5.5 sec
7
24
Position vs. Time Graph
6
5
position
(m) 4
3
2
1
0
0
1
2
3
4
time (sec)
Find the slope and velocity of this object
5
6
7
25
Position vs. Time Graph
6
5
position
(m) 4
3
2
1
0
0
1
2
3
4
time (sec)
5
6
7
Pick two points on the line
26
Position vs. Time Graph
6
5
position
(m) 4
3
2
1
0
0
1
2
3
4
5
time (sec)
Determine the Cartesian point values of the two points
6
7
27
Position vs. Time Graph
6
5
(5.9, 4.9)
position
(m) 4
3
2
( 1.9,2.0)
1
0
0
1
2
3
4
5
time (sec)
Determine the Cartesian point values of the two points
6
7
28
Position vs. Time Graph
• Finding Velocity on a position vs. time graph when
object has a constant velocity
• Draw straight line between the two points being
analyzed (1.9 sec and 5.9sec)
• Determine the slope of the line
• Slope = rise/run = ΔY/ ΔX
= (4.9- 2.0)m/(5.9- 1.9)sec
= 2.9m/4.0sec
= 0.725m/s
= 0.73m/s
The average velocity between times 1.9 & 5.9 sec is
0.73m/s
29
Position vs. Time Graph
of an object with a variable velocity
6
5
position
(m) 4
3
2
1
0
0
1
2
3
4
time (sec)
5
6
7
30
Position vs. Time Graph
of an object with a variable velocity
6
5
position
(m) 4
3
2
Find average velocity of object between 0.0 and 4.0 sec
1
0
0
1
2
3
4
time (sec)
5
6
7
31
Position vs. Time Graph
of an object with a variable velocity
6
5
position
(m) 4
3
2
Find average velocity of object between 0.0 and 4.0 sec
1
0
0
1
2
3
4
time (sec)
5
6
7
32
Position vs. Time Graph
of an object with a variable velocity
6
(4.0, 5.5)
5
position
(m) 4
3
( 0.0,3.0)
2
Find average velocity of object between 0.0 and 4.0 sec
1
0
0
1
2
3
4
time (sec)
5
6
7
33
Position vs. Time Graph
of an object with a variable velocity
•
•
•
•
•
Finding Average Velocity on a position vs. time graph
Draw straight line between the two points being analyzed
Determine the slope of the line
Slope = average velocity of the object
Slope = rise/run = ΔY/ ΔX
= (5.5- 3.0)m/(4.0- 0.0)sec
= 2.5m/4.0sec
= 0.625m/s
= 0.63 m/s
Average velocity at between 0.0 sec and 4.0 sec is 0.63m/s
34
Position vs. Time Graph
of an object with a variable velocity
6
5
position
(m) 4
3
2
Find instantanious velocity of object at 2.0 sec
1
0
0
1
2
3
4
time (sec)
5
6
7
35
Position vs. Time Graph
of an object with a variable velocity
6
5
position
(m) 4
3
2
Find instantanious velocity of object at 2.0 sec
place tangent line at 2.0 sec point
1
0
0
1
2
3
4
time (sec)
5
6
7
36
Position vs. Time Graph
of an object with a variable velocity
6
5
position
(m) 4
3
2
Find instantanious velocity of object at 2.0 sec
find two points on the tangent line
1
0
0
1
2
3
4
time (sec)
5
6
7
37
Position vs. Time Graph
of an object with a variable velocity
6
(3.5, 6.2)
5
position
(m) 4
(3.8, 0.0)
3
•
Find instantanious velocity of object at 2.0 sec
find two points on the tangent line
label the points
1
0
0
1
2
3
4
time (sec)
5
6
7
38
Position vs. Time Graph
of an object with a variable velocity
6
(3.5, 6.2)
5
position
(m) 4
(0.0, 3.8)
3
•
Find instantanious velocity of object at 2.0 sec
find two points on the tangent line
label the points
determine slope of tangent line
1
0
0
1
2
3
4
time (sec)
5
6
7
39
Position vs. Time graph
• Find instantaneous velocity of an object with variable
velocity
• Find slope of the tangent line
• tangent = slope
( 0.0s,3.8m) & (3.5s,6.2m)
• find the slope of the tangent line
• Slope = rise/run = ΔY/ ΔX
= (6.2-3.8)m/(3.5-0.0)sec
= 2.4m/3.5sec
= 0.6857m/s
= 0.69 m/s
Instantaneous velocity at 2.0 sec = 0.69m/s
40
Position vs. Time Graph
of an object with a variable velocity
6
5
position
(m) 4
3
2
Find instantaneous velocity of object at 5.5 sec
1
0
0
1
2
3
4
time (sec)
5
6
7
41
Position vs. Time Graph
of an object with a variable velocity
6
5
position
(m) 4
3
2
Find instantaneous velocity of object at 5.5 sec
1
0
0
1
2
3
4
time (sec)
5
6
7
42
Position vs. Time Graph
of an object with a variable velocity
6
5
position
(m) 4
3
2
Find instantaneous velocity of object at 5.5 sec
place a tangent line at this point
1
0
0
1
2
3
4
time (sec)
5
6
7
43
Position vs. Time Graph
of an object with a variable velocity
6
5
position
(m) 4
(5.0,4.0)
3
(6.2,3.0)
2
Find instantaneous velocity of object at 5.5 sec
determine two points on the tangent line
1
0
0
1
2
3
4
time (sec)
5
6
7
44
Position vs. Time Graph
of an object with a variable velocity
•
•
•
•
Find slope of the tangent line
tangent = slope
( 5.0s,4.0m) (6.2s,3.0m)
find the slope of the tangent line
Slope = rise/run = ΔY/ ΔX
= (3.0-4.0)m/(6.2-5.0)sec
= -1.0m/1.2sec
= -0.833m/s
= -0.83 m/s
Instantaneous velocity at 5.5 sec = -0.83 m/s
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Position vs. Time graph
Instantaneous velocity at 5.5 sec = -0.83 m/s
The negative sign indicates the direction the object is
moving.
The object is moving in the negative X direction
The object is not moving at a speed less than zero
46
Graphical Analysis of Linear Motion
On the left we have a graph of velocity vs. time for an object with varying velocity; on
the right we have the resulting position vs. time curve. The instantaneous velocity is
tangent to the curve at each point.
47
Velocity vs. Time Graph
6
5
velocity
( m/s) 4
3
2
1
0
0
1
2
3
4
time (sec)
5
6
7
48
Velocity vs. Time Graph
6
5
Velocity
( m/s) 4
3
2
1
0 0
-1
1
2
3
4
time (sec)
5
6
7
49
Velocity vs. Time Graph
• Information That can be obtained from a velocity
time graph
a. Measured Velocity at a specific time
b. Estimated velocity between two known points
c. Slope of line gives average acceleration between two
points
d. Slope of line can give instantaneous acceleration
e. Area under the curve can determine displacement of
object
50
Velocity vs. Time Graph
6
Find the velocity at times 1.0sec, 2.0sec, 4.5 sec, 7.0 sec
Velocity 5
( m/s)
4
3
2
1
0 0
-1
1
2
3
4
time (sec)
5
6
7
51
Velocity vs. Time Graph
6
5
4
Find average acceleration from
a. 0.0 sec to 2.5sec
b. 0.0 sec to 5.0 sec
c. 2.5 sec to 5.0 sec
d. 3.5 sec to 7.0 sec
Velocity 3
( m/s)
2
1
0 0
1
2
3
4
time (sec)
5
6
7
-1
52
Velocity vs. Time Graph
6
5
4
Find average acceleration from
a. 0.0 sec to 2.5sec
b. 0.0 sec to 5.0 sec
c. 2.5 sec to 5.0 sec
d. 3.5 sec to 7.0 sec
Velocity 3
( m/s)
2
(2.5, 2.2)
1
(0,0) 0 0
1
2
3
4
time (sec)
5
6
7
-1
53
Velocity vs. Time Graph
Average acceleration from 0.0 sec to 2.5 sec
aver velocity =
= (2.2-0)m/2.5 sec
= + 0.88m/s2
54
Velocity vs. Time Graph
6
5
4
Find average acceleration from
a. 0.0 sec to 2.5sec
b. 0.0 sec to 5.0 sec
c. 2.5 sec to 5.0 sec
d. 3.5 sec to 7.0 sec
Velocity 3
( m/s)
2
(5.0,1.6)
1
(0,0) 0 0
1
2
3
4
time (sec)
5
6
7
-1
55
Velocity vs. Time Graph
Average acceleration from 0.0 sec to 5.0 sec
aver acceleration =
= (1.6-0)m/s/5.0 s
= + 0.320m/s2
= + 0.32 m/s2
56
Velocity vs. Time Graph
6
5
4
Find average acceleration from
a. 0.0 sec to 2.5sec
b. 0.0 sec to 5.0 sec
c. 2.5 sec to 5.0 sec
d. 3.5 sec to 7.0 sec
Velocity 3
( m/s)
2
(2.5, 2.1)
(5.0, 1.6)
1
0 0
1
2
3
4
time (sec)
5
6
7
-1
57
Velocity vs. Time Graph
Average acceleration from 2.5 sec to 5.0 sec
aver acceleration =
=
= (1.6-2.10)m/s/(5.0 -2.5)sec
= -0.50m/s/2.5s
= - 0.20m/s2
58
Velocity vs. Time Graph
6
5
4
Find average acceleration from
a. 0.0 sec to 2.5sec
b. 0.0 sec to 5.0 sec
c. 2.5 sec to 5.0 sec
d. 3.5 sec to 7.0 sec
Velocity 3
( m/s)
2
1
0 0
1
2
3
4
time (sec)
5
6
7
-1
59
Velocity vs. Time Graph
• Average velocity from 3.5 seconds to 7.0 seconds -1.2857m/s2
= -1.3 m/s2
 In the segment from 3.5 s to 7.0 s the object is slowing until it
reaches zero velocity.
Its acceleration is negative
 After the graph crosses into the negative Y segment it gains speed in
the negative direction (going backwards)
its acceleration is negative
its speed is increasing
its velocity is growing increasingly negative
60
Velocity vs. Time Graph
The area under the curve has the units
velocity X time
Area = m/s X sec = meters ( length)
Area under the curve = the displacement
of the object in the velocity time graph
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