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GRAPHICAL ANALYSIS OF MOTION. GRAPHICAL ANALYSIS OF MOTION. m otion graphs come in three formats. GRAPHICAL ANALYSIS OF MOTION. m otion graphs come in three formats Position vs. time graph . GRAPHICAL ANALYSIS OF MOTION. m otion graphs come in three formats Position vs. time graph

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Graphical analysis of motion1
GRAPHICAL ANALYSIS OF MOTION

motion graphs come in three formats


Graphical analysis of motion2
GRAPHICAL ANALYSIS OF MOTION

motion graphs come in three formats

  • Position vs. time graph


Graphical analysis of motion3
GRAPHICAL ANALYSIS OF MOTION

motion graphs come in three formats

  • Position vs. time graph

  • Velocity vs. time graph


Graphical analysis of motion4
GRAPHICAL ANALYSIS OF MOTION

motion graphs come in three formats

  • Position vs. time graph

  • Velocity vs. time graph

  • Acceleration vs. time graph


Graphical analysis of motion5
GRAPHICAL ANALYSIS OF MOTION

motion graphs come in three formats

  • Position vs. time graph

  • Velocity vs. time graph

  • Acceleration vs. time graph ( this graph is of little use in our course)


Graphical analysis of motion6
GRAPHICAL ANALYSIS OF MOTION

motion graphs come in three formats

  • Position vs. time graph

  • Velocity vs. time graph

  • Acceleration vs. time graph ( this graph is of little use in our course)

    Each graph is a function of time


Graphical analysis of motion7
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 5 6 7

time (sec)


Graphical analysis of motion8
GRAPHICAL ANALYSIS OF MOTION

all formats use a Cartesian System for graphing

time is always on the X axis (abscissa)

independent variable


Graphical analysis of motion9
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


Position vs time graph
Position vs. Time Graph

6

5

position

(m) 4

3

2

1

0

0 1 2 3 4 5 6 7

time (sec)


Velocity vs time graph
Velocity vs. Time Graph

6

5

velocity

( m/s) 4

3

2

1

0

0 1 2 3 4 5 6 7

time (sec)


Position vs. Time Graph

This is a graph of positionvs. t for an object moving with constant velocity. The velocity is the slope of the y-tcurve.


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


One dimensional displacement and velocity vector quantities
One-Dimensional Displacement and Velocity: Vector Quantities

Different ways of visualizing uniform velocity:


One dimensional displacement and velocity vector quantities1
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


One dimensional displacement and velocity vector quantities2
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


One dimensional displacement and velocity vector quantities3
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


One dimensional displacement and velocity vector quantities4
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


One dimensional displacement and velocity vector quantities5
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


One dimensional displacement and velocity vector quantities6
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


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.


Position vs time graph1
Position vs. Time Graph

6

5

position

(m) 4

3

2

1

0

0 1 2 3 4 5 6 7

time (sec)


Position vs time graph2
Position vs. Time Graph

6

5

position

(m) 4

3

2

1

0

0 1 2 3 4 5 6 7

time (sec)

Find the position of this object at times 0.0, 2.0 sec, 5.0 sec, 5.5 sec


Position vs time graph3
Position vs. Time Graph

6

5

position

(m) 4

3

2

1

0

0 1 2 3 4 5 6 7

time (sec)

Find the slope and velocity of this object


Position vs time graph4
Position vs. Time Graph

6

5

position

(m) 4

3

2

1

0

0 1 2 3 4 5 6 7

time (sec)

Pick two points on the line


Position vs time graph5
Position vs. Time Graph

6

5

position

(m) 4

3

2

1

0

0 1 2 3 4 5 6 7

time (sec)

Determine the Cartesian point values of the two points


Position vs time graph6
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 6 7

time (sec)

Determine the Cartesian point values of the two points


Position vs time graph7
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


Position vs time graph of an object with a variable velocity
Position vs. Time Graphof an object with a variable velocity

6

5

position

(m) 4

3

2

1

0

0 1 2 3 4 5 6 7

time (sec)


Position vs time graph of an object with a variable velocity1
Position vs. Time Graphof 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 5 6 7

time (sec)


Position vs time graph of an object with a variable velocity2
Position vs. Time Graphof 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 5 6 7

time (sec)


Position vs time graph of an object with a variable velocity3
Position vs. Time Graphof 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 5 6 7

time (sec)


Position vs time graph of an object with a variable velocity4
Position vs. Time Graphof 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


Position vs time graph of an object with a variable velocity5
Position vs. Time Graphof 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 5 6 7

time (sec)


Position vs time graph of an object with a variable velocity6
Position vs. Time Graphof 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 5 6 7

time (sec)


Position vs time graph of an object with a variable velocity7
Position vs. Time Graphof 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 5 6 7

time (sec)


Position vs time graph of an object with a variable velocity8
Position vs. Time Graphof 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 5 6 7

    time (sec)


Position vs time graph of an object with a variable velocity9
Position vs. Time Graphof 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

    1 determine slope of tangent line

    0

    0 1 2 3 4 5 6 7

    time (sec)


Position vs time graph8
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


Position vs time graph of an object with a variable velocity10
Position vs. Time Graphof 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 5 6 7

time (sec)


Position vs time graph of an object with a variable velocity11
Position vs. Time Graphof 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 5 6 7

time (sec)


Position vs time graph of an object with a variable velocity12
Position vs. Time Graphof 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 5 6 7

time (sec)


Position vs time graph of an object with a variable velocity13
Position vs. Time Graphof 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 5 6 7

time (sec)


Position vs time graph of an object with a variable velocity14
Position vs. Time Graphof 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


Position vs time graph9
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


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.


Velocity vs time graph1
Velocity vs. Time Graph

6

5

velocity

( m/s) 4

3

2

1

0

0 1 2 3 4 5 6 7

time (sec)


Velocity vs time graph2
Velocity vs. Time Graph

6

5

Velocity

( m/s) 4

3

2

1

0 0 1 2 3 4 5 6 7

-1 time (sec)


Velocity vs time graph3
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


Velocity vs time graph4
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 2 3 4 5 6 7

-1 time (sec)


Velocity vs time graph5
Velocity vs. Time Graph

6 Find average acceleration from

a. 0.0 sec to 2.5sec

5 b. 0.0 sec to 5.0 sec

c. 2.5 sec to 5.0 sec

4 d. 3.5 sec to 7.0 sec

Velocity 3

( m/s)

2

1

0 0 1 2 3 4 5 6 7

time (sec)

-1


Velocity vs time graph6
Velocity vs. Time Graph

6 Find average acceleration from

a. 0.0 sec to 2.5sec

5 b. 0.0 sec to 5.0 sec

c. 2.5 sec to 5.0 sec

4 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 5 6 7

time (sec)

-1


Velocity vs time graph7
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


Velocity vs time graph8
Velocity vs. Time Graph

6 Find average acceleration from

a. 0.0 sec to 2.5sec

5 b. 0.0 sec to 5.0 sec

c. 2.5 sec to 5.0 sec

4 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 5 6 7

time (sec)

-1


Velocity vs time graph9
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


Velocity vs time graph10
Velocity vs. Time Graph

6 Find average acceleration from

a. 0.0 sec to 2.5sec

5 b. 0.0 sec to 5.0 sec

c. 2.5 sec to 5.0 sec

4 d. 3.5 sec to 7.0 sec

Velocity 3

( m/s) (2.5, 2.1)

2

(5.0, 1.6)

1

0 0 1 2 3 4 5 6 7

time (sec)

-1


Velocity vs time graph11
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


Velocity vs time graph12
Velocity vs. Time Graph

6 Find average acceleration from

a. 0.0 sec to 2.5sec

5 b. 0.0 sec to 5.0 sec

c. 2.5 sec to 5.0 sec

4 d. 3.5 sec to 7.0 sec

Velocity 3

( m/s)

2

1

0 0 1 2 3 4 5 6 7

time (sec)

-1


Velocity vs time graph13
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


Velocity vs time graph14
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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