Martha Casquete

Read Chapter 3 (Vectors) HW3 Set due next Tuesday, 9/17 Pg. 68 – 70: 1, 4, 11, 16, 26, 32, 43 (8th edition)

Question/Observation Thursdays

Research Q/O Tuesdays with HW (due date

Tuesdays)

Coordinate Systems

Vector and Scalar Quantities

Properties of Vectors

Component of Vector and Unit Vectors

Distance and displacement

Average velocity and average speed

Instantaneous velocity and speed

Acceleration

Free fall acceleration:

a = g =- 9.80 m/s2

Would you risk your life driving drunk?

Intro

2

002

1gttvxx

6

In 2-D : describe a location in a plane

• by polar coordinates : distance r and angle • by Cartesian coordinates :

distances x, y, parallel to axes

with: x = rcosθ y = rsinθ x

y

r

( x , y )

0 x

y

• Since we know the Cartesian coordinates, we can find r and :

• x

ytan

22 yxr

A fly lands on one wall of a room. The lower left corner of the wall is selected as the origin of a two-dimensional Cartesian coordinate system. If the fly is located at the point having coordinates (2.00, 1.00) m , (a) how far is it from origin? (b) What is its location in polar coordinates?

Physical quantities are classified as scalars, vectors, etc.

Scalar : described by a real number with units

examples: mass, charge, energy . . .

Vector : described by a scalar (its magnitude) and a direction in space

examples: displacement, velocity, force . . .

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Vectors have direction, and obey different rules of arithmetic.

Scalars : ordinary or italic font (m, q, t . . .) Vectors : - Boldface font (v, a, F . . .) - arrow notation

Pay attention to notation :

“constant v” and “constant v” mean different things!

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.) . . F ,a ,v(

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Magnitude : a scalar, is the “length” of a vector.

e.g., Speed, v = |v| (a scalar), is the magnitude of velocity v

3.3 Properties of Vectors

Direction

Commutative Law Associative Law of addition

ABBA

CBACBA

)()(

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CBA

e.g.

A B

Triangle Method Parallelogram Method

A

B

A

B

BAC

BAC

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CBA

e.g.

A

B

A

B

BAC

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Two students are moving a refrigerator. One pushes with a force of 200 newtons, the other with a force of 300 newtons. Force is a vector. The total force they (together) exert on the refrigerator is:

a) equal to 500 newtons b) equal to newtons c) not enough information to tell

22 300200

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Two students are moving a refrigerator. One pushes with a force of 200 newtons (in the positive direction), the other with a force of 300 newtons in the opposite direction. What is the net force ?

a)100N

b)-100N

c) 500N

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A A

2

3

A

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Multiplication: scalar vector = vector

Later in the course, we will use two other types of multiplication: the “dot product” , and the “cross product”.

Follow the steps for Example 2.3 from your book (pg. 60)

define the axes first

are scalars

axes don’t have to be horizontal and vertical

the vector and its components form a right triangle with the vector on the hypotenuse

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) (and , , zyx vvv

x

y

vy

vx

v

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cos vvx

x

y

vy

vx

v

yv

sin vvy

xv

22

yx vvv

x

y

v

v1tan

Choose the correct response to make the sentence true: A component is

(a) always

(b) never

( c ) sometimes

…..larger than the magnitude of the vector.

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19

---Specify a given direction Define coordinate unit vectors i, j, k along the x, y, z axis.

z

y

x

i

j

k

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A vector can be written in terms of its components: A

kAjAiAA zyx

i

j

A

Ax i

Ay j

Ay j

Ax i

A

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Ax

Ay A

By

Bx

B

By

Bx

B

Ay

Ax

A

C

Cx

Cy

If A + B = C , then:

zzz

yyy

xxx

BAC

BAC

BAC

Three scalar equations from one vector equation!

Tail to Head

BA

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CBA

In components (2-D for simplicity) :

jiji )( )( yxyyxx CCBABA

The unit-vector notation leads to a simple rule for the components of a vector sum:

Eg: A=2i+4j B=3i-5j

A+B = 5i-j A - B = -i+9j

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Magnitude : the “length” of a vector. Magnitude is a scalar.

In terms of components: On the diagram, vx = v cos vy = v sin

x

y

vy

vx

v

e.g., Speed is the magnitude of velocity: velocity = v ; speed = |v| = v

22|| yx vv v

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Example 2

While exploring a cave, a spelunker starts at the entrance and moves the following distances in a horizontal plane. She goes 75.0 m north, 250 m east, 125m at an angle θ = 30˚ north of east, and 150 m south. Find her resultant displacement from the cave entrance.

vector quantities must be treated according to the rules of vector arithmetic

vectors add by the triangle rule or parallelogram rule (geometric method)

a vector can be represented in terms of its Cartesian components using the “unit vectors” i, j, k these can be used to add vectors (algebraic method)

if and only if:

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A

BAC

zzzyyyxxx BACBACBAC