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GRAPHICAL ANALYSIS OF MOTIONPowerPoint Presentation

GRAPHICAL ANALYSIS OF MOTION

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GRAPHICAL ANALYSIS OF MOTION

motion graphs come in three formats

GRAPHICAL ANALYSIS OF MOTION

motion graphs come in three formats

- Position vs. time graph
- Velocity vs. time graph

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 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 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 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 MOTION

all formats use a Cartesian System for graphing

time is always on the X axis (abscissa)

independent variable

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

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

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

Different ways of visualizing uniform velocity:

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 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 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 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 Quantities a. The average velocity between any two points can be found by finding the slope of a line drawn between any two points

This object’s velocity is not uniform.

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 Quantities a. The average velocity between any two points can be found by finding the slope of a line drawn between any two points

This object’s velocity is not uniform.

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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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