chapter 3 vectors in physicsnsmn1.uh.edu/rbellwied/classes/spring2013/ch3_notes.pdf · summary of...

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Chapter 3

Vectors in Physics

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Units of Chapter 3• Scalars Versus Vectors

• The Components of a Vector

• Adding and Subtracting Vectors

• Unit Vectors

• Position, Displacement, Velocity, andAcceleration Vectors

• Relative Motion

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3-1 Scalars Versus VectorsScalar: number with units

Vector: quantity with magnitude and direction

How to get to the library: need to know how farand which way

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3-2 The Components of a VectorEven though you know how far and in whichdirection the library is, you may not be ableto walk there in a straight line:

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3-2 The Components of a VectorCan resolve vector into perpendicularcomponents using a two-dimensionalcoordinate system:

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3-2 The Components of a VectorLength, angle, and components can becalculated from each other usingtrigonometry:

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3-2 The Components of a Vector

Signs of vector components:

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3-3 Adding and Subtracting VectorsAdding vectors graphically: Place the tail of thesecond at the head of the first. The sum pointsfrom the tail of the first to the head of the last.

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3-3 Adding and Subtracting Vectors

Adding Vectors Using Components:

1. Find the components of each vector to beadded.

2. Add the x- and y-components separately.

3. Find the resultant vector.

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3-3 Adding and Subtracting Vectors

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3-3 Adding and Subtracting Vectors

Subtracting Vectors: The negative of a vector isa vector of the same magnitude pointing in theopposite direction. Here, .D= A B!

r r r

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Example: Vector A has a length of 5.00 meters and pointsalong the x-axis. Vector B has a length of 3.00 meters andpoints 120° from the +x-axis. Compute A+B (=C).

A x

y

B

120°

C

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adj

opp

cos

sintan

hyp

adjcos

hyp

oppsin

==

=

=

!

!!

!

!

( )

( ) m 50.160cosm00.360cos 60cos

m 60.260sinm00.360sin60sin

!=°!=°!="!

=°

=°=°="=°

BBB

B

BBB

B

xx

y

y

and Ax = 5.00 m and Ay = 0.00 m

A x

y

B

120°60°

By

Bx

Example continued:

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The components of C:( )

m 2.60m 2.60 m 00.0

m 3.50m 1.50 m 00.5

=+=+=

=!+=+=

yyy

xxx

BAC

BAC

x

y

C

Cx = 3.50 m

Cy = 2.60 m

θ

The length of C is:

( ) ( )

m 36.4

m 60.2m 50.322

22

=

+=

+== yx CCC C

The direction of C is:

( ) °==

===

!6.367429.0tan

7429.0m 3.50

m 60.2tan

1"

"x

y

C

C

From the +x-axis

Example continued:

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Example: Margaret walks to the store using the followingpath: 0.500 miles west, 0.200 miles north, 0.300 miles east.What is her total displacement? Give the magnitude anddirection.

x

y

r3

r2

r1

Δr

Take north to be inthe +y direction andeast to be along +x.

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Example continued:

The displacement is Δr = rf − ri. The initial position is theorigin; what is rf?

The final position will be rf = r1 + r2 + r3. The componentsare rfx = −r1 + r3 = −0.2 miles and rfy = +r2 = +0.2 miles.

miles 283.022=!+!=!

yxrrr

°==!

!= 45 and 1tan ""

x

y

r

r

Using the figure, the magnitude anddirection of the displacement are

x

y

ΔrΔry

Δrx

θ

N of W.

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3-4 Unit VectorsUnit vectors are dimensionless vectors of unitlength.

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3-4 Unit VectorsMultiplying unit vectors by scalars: the multiplierchanges the length, and the sign indicates thedirection.

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3-5 Position, Displacement, Velocity,and Acceleration Vectors

Position vector points from the origin tothe location in question.

The displacement vectorpoints from the originalposition to the final position.

frr

r!r

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3-5 Position, Displacement, Velocity,and Acceleration Vectors

Average velocity vector:

(3-3)

So is in the same

direction as .avvr

r!r

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3-5 Position, Displacement, Velocity,and Acceleration Vectors

Instantaneous velocity vector is tangent to thepath:

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3-5 Position, Displacement, Velocity,and Acceleration Vectors

Average acceleration vector is in the directionof the change in velocity:

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3-5 Position, Displacement, Velocity,and Acceleration Vectors

Velocity vector is always in the direction ofmotion; acceleration vector can point anywhere:

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Example (text problem 3.42): At the beginning of a 3 hourplane trip you are traveling due north at 192 km/hour. At theend, you are traveling 240 km/hour at 45° west of north.

(a) Draw the initial and final velocity vectors.

x (east)

y (north)

vivf

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(b) Find Δv.

km/hr 3.2245cos

km/hr 170045sin

!=!°+=!="

!=!°!=!="

ifiyfyy

fixfxx

vvvvv

vvvv

The components are

km/hr 17122

=!+!=!yxvvv

( ) °==!="

"= #

5.71312.0tan1312.0tan1$%

x

y

v

v South ofwest

Example continued:

The magnitude and direction are:

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(c) What is aav during the trip?

t!

!=v

aav

2

av,

2

av,

km/hr 43.7hr 3

km/hr 3.22

km/hr 7.56hr 3

km/hr 170

!=!

="

"=

!=!

="

"=

t

va

t

va

y

y

x

x

The magnitude and direction are:

°==!==

=+=

" 5.7)1310.0(tan1310.0tan

km/hr 2.57

1

av,

av,

22

av,

2

av,av

##x

y

yx

a

a

aaa

South ofwest

Example continued:

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3-6 Relative MotionThe speed of the passenger with respect tothe ground depends on the relative directionsof the passenger’s and train’s speeds:

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3-6 Relative Motion

This also works in two dimensions:

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Example: You are traveling in a car (A) at 60 miles/hour easton a long straight road. The car (B) next to you is travelingat 65 miles/hour east. What is the speed of car B relative tocar A?

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east miles/hour 5

eastmiles/hr 60 east miles/hr 65BA

AGBGBA

AGBGBA

BAAGBG

=

!=

!=

"!"="

"+"="

v

vvv

rrr

rrrFrom the picture:

A

B

A

B

t=0 t>0

ΔrAG

ΔrBG

ΔrBA

Divide by Δt:

+x

Example continued:

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Example: You are traveling in a car (A) at 60 miles/houreast on a long straight road. The car (B) next to you istraveling at 65 miles/hour west. What is the speed of car Brelative to car A?

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A

B

A

B

t=0

ΔrAG

ΔrBG ΔrBA

t>0t>0

estmiles/hr w 125

eastmiles/hr 60estmiles/hr w 65

AGBGBA

AGBGBA

=

!=

!=

"!"="

vvv

rrr

+x

From the picture:

Divide by Δt:

Example continued:

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Summary of Chapter 3

• Scalar: number, with appropriate units

• Vector: quantity with magnitude and direction

• Vector components: Ax = A cos θ, By = B sin θ

• Magnitude: A = (Ax2 + Ay

2)1/2

• Direction: θ = tan-1 (Ay / Ax)

• Graphical vector addition: Place tail of secondat head of first; sum points from tail of first tohead of last

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• Component method: add components ofindividual vectors, then find magnitude anddirection• Unit vectors are dimensionless and of unitlength• Position vector points from origin to location• Displacement vector points from originalposition to final position• Velocity vector points in direction of motion• Acceleration vector points in direction ofchange of motion• Relative motion:

Summary of Chapter 3

13 12 23v = v + vr r r