Ch 2 Velocity~Motion in One Dimension~

Scalar versus Vector

• Scalar – quantity that only has magnitude

• Vector – quantity that has magnitude and direction.

• Three friends drive their four wheelers a distance of 100.0 m.• Will they end up in the same place? Why or why not.• No, 100.0 m is a scalar and does not specify a specific direction.• Now suppose they drive 175.0 m due East.• Will they end up in the same place? Why or why not.• Yes, 100.0 m East is a vector. It has direction

Scalar versus Vector

Distance• Distance is the measure of separation between two objects.

• It is given the variable “d” and is measured in meters (m).

• If the positions are known, distance is calculated as follows.

• The distance between the buoys be?

• The distance from the shore to each buoy is.

• Distance is a scalar quantity.

2 1d x x

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2 1d x x 16 3d m m 13d m

13m

3m and 16m

16m

3m

Displacement• When an object is displaced, it is moved from an initial

position (x1) to a final position (x2).

• Displacement (variable x) is a measure of the change in position of an object after it has moved.

2 1d x x

Example Displacement - Distance

• John travels east along a straight highway and passes mile marker 260. John continues until mile marker 150 and then doubles back to mile marker 175.

• What is Johns displacement from marker 260?• +85 miles

• What is the total distance John traveled?• 135 miles

Example Displacement - Distance

• John travels west along a straight highway and passes mile marker 260. John continues until mile marker 150 and then doubles back to mile marker 175.

• What is Johns displacement from marker 260?• -85 miles

• What is the total distance John traveled?• 135 miles

Average Velocity• Average Velocity – the change in position

of an object over a given time interval.

xv

t

2 1

2 1

x x

t t

Finial positionInitial position

Initial time

Final time

Vector or Scalar??

• A ball moves at 12m/s and coasts up a hill with a uniform acceleration of -1.6m/s2.

Example

• Distance traveled for the 1st 6s.

• Distance traveled for the 1st 9s.

Simultaneous equations

3 7 95x y

11 2 88x y

• Solve the following equations for both x and y

x = 6

Y = 11

Simultaneous equations• An F-15 on patrol traveling at 103m/s is ordered to over

take and observe what is believed to be a hostile MIG jet fighter flying at 323Km/hr in the same direction as the F-15. The MIG is 3.8Km ahead of the F-15.

• How far (Km) will the F-15 travel before over taking the MIG?

2 1

2 1

v v

t t

Average Acceleration• The rate at which the Velocity

changes.– Scalar or vector?

va

t

Average acceleration

Change in velocity

Change in time

What are the units of acceleration?

2ms

Average Acceleration• Can an object speed up and have a

negative average acceleration?

– YES!

• Let’s see an example how!

Practice Problem – Avg Acceleration

• Simon rolls backwards faster and faster down his driveway. He starts at -2.0 m/s and is moving at -9.0 m/s 2.0 s later. What is his average acceleration?

12

12

tt

vva

02

)2(9

25.3s

m

Negative a!

AHHH!!!!

Average Acceleration• Can an object slow down and have a

positive average acceleration?

– YES!

• Let’s see an example how!

Practice Problem – Avg Acceleration

• Simon rolls up the other side of the ramp and slows down from-9.0m/s to -2m/s in 2s. What is his average acceleration?

12

12

tt

vva

2 ( 9)

2 0

23.5ms

AHHH!!!!

-250

100150

250

0 10 20 30 40 50 60 70 80 90

Time (s)

0-50

-100-150-200

Po

siti

on

(m

)

50

200

Position vs. Time Graph

What is the position of the object at t = 10s?

What is the position of the object at t = 40s?

What is the position of the object at t = 60s?

Position vs. Time Graph

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-15-10

-505

1015

2025

0 10 20 30 40 50 60 70 80 90

Po

siti

on

(m

)

Time (s)

Position vs. Time Graph

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-15-10

-505

1015

2025

0 10 20 30 40 50 60 70 80 90

Po

siti

on

(m

)

Time (s)

riseSlope

run

2 1

2 1

x xSlope

t t

_

2 1

2 1

x xv

t t

_

v Slope

Position vs. Time Graph

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-15-10

-505

1015

2025

0 10 20 30 40 50 60 70 80 90

Po

siti

on

(m

)

Time (s)

_

v Slope

+ Slope + VelocityMoving forwards

- Slope - VelocityMoving backwards

0 Slope 0 VelocityNot moving

Position vs. Time Graph

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-15-10

-505

1015

2025

0 10 20 30 40 50 60 70 80 90

Po

siti

on

(m

)

Time (s)

_

v Slope

During what time period(s) is the object moving forward?

During what time period(s) is the object moving backwards?

During what time period(s) is the object not moving?

Instantaneous Velocity

Position vs. Time

010

2030

4050

6070

8090

100

0 1 2 3 4 5 6 7 8 9 10

Time (s)

Pos

ition

(m)

Draw a tangent line at the point that corresponds to that instant in time

Find the slope of that tangent line at 5.0s, 2.0s, and 9.0s.

Rise - Δd

Run - Δt

Position Time Graphs Velocity Time Graphs

Velocity vs. Time Graph

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0 10 20 30 40 50 60 70 80 90

Time (s)V

elo

cit

y (m

/s)

Position vs. Time Graph

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0 10 20 30 40 50 60 70 80 90

Po

siti

on

(m

)

Time (s)

Position–Time graphs lets us calculate velocity.

Velocity–Time graphs lets us calculate displacement (x).

Area Calculations• In order to calculate area, you will need to

know how to find the area of different shapes.

Trapezoid

11 22A b b h

1b

2b

h

Triangle

12A bh

b

h

Rectangle/Square

A bh

h

b

Velocity vs. Time Graph

-25

-20

-15

-10

-5

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90

Time (s)

Ve

loc

ity

(m/s

)

Displacement equals the area between the curve and the x-axis. (d = Area)

Find the displacement between 10s-25s.

15 *(15 )msx s

225x m

A bh

“Negative” Area• Area above the x-

axis indicates positive displacement.

• Area below the x-axis indicates negative displacement.

• Negative velocity means negative displacement

Velocity vs. Time Graph

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10152025

0 10 20 30 40 50 60 70 80 90

Time (s)

Ve

loc

ity

(m/s

)

Velocity vs. Time Graph

-25

-20

-15

-10

-5

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90

Time (s)

Ve

loc

ity

(m/s

)

Displacement equals the area under the curve (d = Area)

Find the distance traveled between 40-55s

12 (15 )( 15 )msx s

112.5x m

Velocity vs. Time Graph

-25

-20

-15

-10

-5

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90

Time (s)

Ve

loc

ity

(m/s

)

Displacement equals the area under the curve (d = Area)

Find the distance traveled between 25-65s

12 (15 )(15 )msx s

150x m

12 40 25 15 m

ss s

65 55 15 mss s

Velocity vs. Time Graph

-25

-20

-15

-10

-5

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90

Time (s)

Ve

loc

ity

(m/s

)

Displacement equals the area under the curve (d = Area)

Find the displacement traveled between 10-55s

225x m

1 (15 )(15 )msx s

225 112.5 112.5x m m m

3

1(15 )( 15 )

2msx s

12 2 (15 )(15 )msx s

Velocity vs. Time Graph

-25

-20

-15

-10

-5

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90

Time (s)

Ve

loc

ity

(m/s

)

Find the displacement between 40-75s

337.5x m

1 1(15 )( 15 / ) 10 ( 15 / ) (10 )( 15 / )

2 2x s m s s m s s m s

337.5m left

Velocity-TimeVelocity vs. Time Graph

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0 1 2 3 4 5 6 7 8 9

Time (s)

Vel

oci

ty (

m/s

) Δv

Δt

What is the average acceleration of the object over the first 2 s?

21002

020s

ma

Slope of velocity-time graph is the average acceleration!!

12

12

tt

vv

t

va

Velocity-Time Graphs

Velocity vs. Time Graph

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10152025

0 1 2 3 4 5 6 7 8 9

Time (s)

Velo

city

(m/s

)

During what time period(s) does the object have a positive acceleration?

During what time period(s) does the object have a negative acceleration?

During what time period(s) is the object not accelerating?

Velocity-Time Graphs

Velocity vs. Time Graph

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10152025

0 1 2 3 4 5 6 7 8 9

Time (s)

Velo

city

(m/s

)

During what time period(s) is the sign of the velocity and acceleration opposite?

Constant Acceleration

• Acceleration that does not change in time is uniform or constant acceleration.

• On a Velocity-Time Graph, constant acceleration is a straight line

Velocity – Time Graph

The slope is the Acceleration

Velocity - Time Graph

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6 7

Time (s)

Vel

oci

ty (

m/s

)

Acceleration vs. Time

0

2

4

6

8

10

12

0 0.5 1 1.5 2 2.5 3 3.5 4

Time (s)

Acce

lerati

on (m

/s^2)

Velocity of an Object with Constant Acceleration

• Constant Acceleration = Uniform Acceleration

Velocity vs. Time Graph

0

10

20

30

40

50

60

0 1 2 3 4

Time (s)

Velo

city

(m/s

)

What would the graph of a versus t look like?

Velocity of an Object with Constant Acceleration

D vs t v vs t a vs t

Distance v. Time

0

1

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1

Time (s)

Dis

tan

ce (

m)

Zero Acceleration, Constant Speed

We can use small triangles to visualize the distance traveled per increment of time.

The same amount of distance is covered in the same amount of time.

The speed of the object remained constant.

Negative AccelerationWe can use small triangles to visualize the distance traveled per increment of time.

The second triangle is much smaller that the first triangle.

The larger size means that the body traveled more distance in the same increment of time

The object was moving slower during the second increment.

The object experienced a negative acceleration.

Distance v. Time

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1

Time (s)

Dis

tan

ce (

m)

Positive Acceleration

Distance v. Time

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Time (s)

Dis

tan

ce (

m)

We can use small triangles to visualize the distance traveled per increment of time.

The second triangle is much larger that the first triangle.

The larger size means that the body traveled more distance in the same increment of time

The object was moving faster during the second increment.

The object experienced a positive acceleration.

Instantaneous Speed• You raced your 4-wheeler

over a 30 mile long track in 1.5 hours.

• What is your average speed?• 20mph

• Watch the animation. Is the speed always 20mph?

t

x

tt

xxv

12

12

Graphs of Motion• Label the three distance v. time graphs below as

either accelerating positive, accelerating negative, or zero acceleration.

Distance v. Time

0

1

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1

Time (s)

Dist

ance

(m)

Distance v. Time

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Time (s)

Dist

ance

(m)

Distance v. Time

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1

Time (s)

Dist

ance

(m)

Velocity versus Time• Label the three velocity versus time graphs below as either

– accelerating positive– accelerating negative – zero acceleration.

• Simply read the values directly from the graph.

Velocity v. Time

0

2

4

6

8

10

12

0 0.5 1 1.5

Time (s)

Velo

city

(m/s

)

Velocity v. Time

0

1

2

3

4

5

6

0 0.5 1 1.5

Time (s)

Velo

city

(m/s

)

Velocity v. Time

0

2

4

6

8

10

12

0 0.5 1 1.5

Time (s)

Velo

city

(m/s

)