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Regents / Honors Physics

Unit 3

Vectors

velocity

acceleration

velocity

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Unit Packet Contents

Unit Objectives 3

Notes 1: Vectors Introduction 5

Guided Practice: Graphical Addition 11

TTQ’s :Set 1 13

Notes 2: Forces and Vectors 17

Guided Practice: Free Body Diagrams 21

Notes 3: Components of Vectors 23

Guided Practice: Components of Vectors. 27

Notes: Concurrent Forces on a Slope 29

Concept Development: Force Vectors and Friction 31

Guided Practice: Friction Forces 35

TTQ’s Set 2 37

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Name ______________________ Regents / Honors Physics

Date ____________

Unit Objectives: Vectors

At the end of this unit you will be able to:

1. Define the terms vector and scalar and give examples of vector and scalar quantities.

2. Describe the terms velocity, speed, displacement and distance as vector or scalar quantities.

3. Draw scale diagrams of vectors and use them to add vectors graphically.

4. Draw free body diagrams which includes a coordinate axis for an object with multiple forces

acting on it.

5. Discuss how a vector may be considered to be made up as components of other vectors.

6. Demonstrate use of trig functions to find components of vectors.

7. Resolve a vector into x- and y- components on a set of coordinate axes.

8. Add vectors algebraically using x- and y- components.

9. Explore examples where combinations of forces may be separated into x- and y- components.

10. Use x- and y- components of forces to describe systems involving friction on an incline.

Galileo's experiment worked because the air is sufficiently thin. Who knows what he would have concluded if we lived in a thicker medium...

What would you have thought, Galileo, If instead you dropped cows and did say, "Oh! To lessen the sound Of the moos from the ground, They should fall not through air but through mayo!"

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Name________________________ Regents / Honors Physics

Date_____________

Notes: Vectors Introduction

Objectives

1. Define the terms vector and scalar and give examples of vector and scalar quantities.

2. Describe the terms velocity, speed, displacement and distance as vector or scalar quantities.

3. Draw scale diagrams of vectors and use them to add vectors graphically.

Vectors and Scalars

What is the definition of the term quantity?

Give 3 examples of physical quantities.

Sometimes when we use physical quantities in nature to describe the ______________________ of

an object or system it is important to describe the ____________________ that the quantity is

acting.

For example if you wanted to take a trip from Johnson City to Washington DC it is not only

important that your displacement has ______________________ it is also

important that you are _________________________.

Quantities where direction is important are called _______________________.

o Think of some examples of other vector quantities.

Some quantities we want to include direction for ___________________ . . .

o Traveling to DC it’s important to say we traveled 300 miles to the

south

o Both the magnitude which means __________ and

________________ or which way are important.

o We refer to this as a _______________________ which is the

difference between your _______________________________

positions.

The word magnitude means “HOW BIG”. That’s why we say every vector quantity has magnitude and direction.

traits or status

direction

enough miles

heading south

vector quantities

some purposes

purpurposespur

poses

how far

direction

displacement

starting and ending

http://www.youtube.com/wat

ch?v=A05n32Bl0aY

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. . . and ________________ direction for other purposes.

o Let’s say a JC wildcat cross country jogger runs from the school _______________ to the

bottom of Reynolds Rd then turns around and runs ___________________ back up then

stops to rest.

o If we consider her displacement, which is the _______________ between starting and ending

positions, we would say she ran ________________ roughly south.

o Clearly she’s more concerned with the __________________ and not with the direction.

o Instead of displacement we refer to her __________________ and we say that she ran 3100

meters and disregard direction.

For some quantities it makes ____________________to have a direction associated with them.

o It’s just goofy to say the temperature in the classroom is 71° F

___________________________

Quantities where we disregard the direction or it makes no sense to include direction are called

___________________________.

Working with Vectors

A vector quantity is represented by a _________________________.

o The _____________ of the arrow represents the _________________ of the measured

value

o The direction represents the ____________________ of the measured value.

o Draw a vector to represent a 300 mile trip south from JC to Washington DC

o Billy is pushing his mini-van to the gas station pushing east with a 20 Newton force.

Draw a vector to represent his force acting east.

disregard

1500 meters

difference

1600 meters

100 meters

total run

distance

no sense

pointing to the north

scalar quantities

drawn arrow

length magnitude

direction

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I can be a scalar or a vector

Note that all vector quantities also have corresponding __________________________

Examples:

Vector Quantity Corresponding Scalar quantity

Velocity

Displacement

Force

Acceleration

Sample Question 1: Velocity is to speed as displacement is to __________________

Adding Vectors: Graphical Method

Frequently it is appropriate to consider the result of combining _____________________ vectors.

Hunter gets out of his car and takes a walk through the woods. The first segment of his trip is 1200

meters east. He then turns and walks 500 meters to the north. Draw vectors to determine how far he

is from his car.

The result of this combination of vectors can be determined and is referred to as the _____________

of vector a and vector b or also known as the resultant

A vector sum is determined through a variety of methods but it ________________ the algebraic

sum of the vectors’ magnitudes.

In a plain language description of this we would say that s is the ________________________ and

a and b are the ___________________________

scalar quantities

distance

speed

force magnitude

distance

acceleration magnitude

2 or more

vector sum

IS NOT

vector sum

component vectors

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Graphical Addition Triangle Method

The following procedure can be followed to add vectors graphically.

o On a sheet of paper, lay out the vector a to some convenient _______________ and at the

___________________

o Lay out vector b to the same scale, with its __________ at the ____________ of vector a

again at the proper angle.

o Construct the vector sum s by drawing an arrow from the _____________________ to the

______________________.

Speed vs. Velocity

A bicyclist rides her bike 300 meters north along Main St. then turns and rides 400 meters west along

Third St. The total trip take her 6.8 minutes.

What is the total distance (recall distance is scalar) traveled?

What is her total displacement (displacement is vector)?

What is the average speed (speed is scalar) of her trip?

What was ther average velocity (velocity is vector)?

scale

proper angle

tail head

tail of a

head of b

700 m

d1 = 300 m

d2 = 400 m

dnet = 500 m

speed = distance / time speed = 700 m / 6.8 min speed = 102.9 m/min velocity = displacement / time velocity = 500 m / 6.8 min velocity = 73.5 m / min

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Graphical Addition: Moving a vector

Frequently vectors are conveniently represented as originating from the ____________________

Often this is because they are on a set of _________________________

Graphical addition may be accomplished by _______________________ one of the vectors.

After the vector move it must be:

________________________ the original so that ______________________ isn’t changed

the __________________________ as the original so that ___________________ isn’t

changed.

coordinate axes

same point

moving

parallel

same length

direction

magnitude

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http://www.youtube.com/watch?v=iN3zBbSTEf4

Sample question 3: The following vector diagram represents the

velocity vectors for an airplane that is traveling at a northwest

heading while there is a wind blowing from west to due east. Find

the resulting magnitude and direction of the airplane.

vplane

vwind

vwind’

vheading

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Name ________________________ Reg / Honors Physics

Guided Practice: Graphical Addition For each of the following find the resultant. 1. A person walks 40.0 meters east then 100.0 meters south. Draw a vector diagram to scale and find

the total distance and the displacement of the person. Dist = 140 m Disp = 108 m 2. A motorboat heads due west at 10.0 m/s. Draw a scale vector diagram of the velocity vectors. The

river has a current of 6.0 m/s due south. What is the resultant velocity of the boat? 3. For each of the following vector diagrams draw vector C which is the vector sum of A + B. Draw in

the resultant with a straight edge and include an arrowhead to indicate the resultant’s direction. Give the magnitude and the direction relative to vector A. (Scale: 1 cm = 20 Newtons)

40 m

108m 100m

10.0 m/s

6.0 m/s

11.7 m/s

A

B B’

A

B

B’

A

B

B’

A + B

A + B

A + B

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A

B

B’

A

B

B’

A

B

A B

B A

B’

A B

B’

B

A

B

A

B’

A + B

A + B

A + B

A + B

A + B

A + B A + B A + B

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TTQ’s (Typical Test Questions)

Documented Thinking

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Documented Thinking

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Documented Thinking

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Notes: Forces and Vectors

Objective:

1. Draw free body diagrams which includes a coordinate axis for an object with multiple forces

acting on it.

Free Body Diagrams

A free body diagram is a diagram that shows all of the ______________ acting on a particular object

as ____________________

The first step of drawing a free body diagram is to define the _______________ on which forces are

acting.

The force vectors are then drawn originating from __________________ that represents the object.

The FBD can be used to determine if there is a non-zero ________________ acting on the object.

Example 2: A 35 kg monkey is standing on top of a grand piano while the zookeeper is trying to capture

him to take him back to the zoo. The zookeeper is pulling with a force of 325 Newtons on the monkey.

There is a force of friction resisting the pull which is 310 Newtons.

a. Draw a free body diagram showing the forces acting on the monkey

b. Are the forces on the monkey balanced or unbalanced?

c. Is the monkey accelerating? If so what is the rate of acceleration.

.

Forces vectors

object

a point

net force

Fg

FN

Fz Ff

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Example 1: Three teams are in an unusual 3 way tug-o-war. Team A is pulling with a force

of 300 Newtons, Team B is pulling with a force of 212 Newtons and Team C is pulling with

a force of 212 Newtons. The angles that the teams make relative to one another are given.

Determine who is winning.

90°

135 135

A

B

C

A

300N

C

212N B

212N

B’

212N B+C

1. Draw a FBD showing forces

acting on the knot in the center of

the rope; vectors A, B and C to

scale with measured angles

2. Perform a vector move e.g. draw

vector B’ same length and angle

as B starting at the tip of vector C.

3. Draw vector B+C and measure

length.

4. Use measured length of B+C and

scale to determine magnitude of

B+C combined force vector

5. Note that B+C is equal in

magnitude and opposite direction

to vector A.

6. Nobody wins the T.O.W.

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Example 2: A 2750 kg rocket is lifting off and is accelerating at a rate of 35 m/s2.

a. Draw a free body diagram showing the forces acting on the rocket during liftoff.

b. What is the net force acting on the rocket?

c. What is the magnitude of the thrust force exerted by the rocket’s engine.

Fengine

Fg

a = 35 m/s2

m = 2750 kg

Fnet = m a

Fnet = (2750 kg) (35 m/s2)

Fnet = 96 250 kg m/s2

Fnet = 96 250 N

Fnet = Fengine - Fg

Fengine = Fg + Fnet

Fg = m g = (2750 kg) (9.8 m/s2)

Fg = 26 950 N

Fengine = 26 950N + 96 250 N

Fengine = 123 200N

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Guided Practice: Draw free body diagrams for each of the following:

b. If the mass of the crate is 45 kg, what is the

rate of acceleration of the crate?

b. If the mass of the crate is 45 kg, what is the

rate of acceleration of the crate?

Fn

Fg

Fn

Fg

Fh Ff

Fn

Fg

FT Ff

Fn

Fg

Fh

Note that Fn is equal in

magnitude to Fh + Fg

F1 = 165N

F2=225N

Fnet = F1 + F2

= 165N + 225N

= 395 N

a = Fnet / m

a = 395N / 45 kg

a = 8.78 m/s2

F1 = 165N F2=225N

Fnet = F2 – F1

= 225N – 165N

= 60N

a = Fnet / m

a = 60N / 45 kg

a = 1.33 m/s2

F1 = 165N

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F2=225N

F1 = 165N

Fnet

FR

Fg

FR

Fg

Fe

Fg

Fnet2 = F1

2 + F22

Fnet2 = (165 N)2 + (225 N)2

Fnet = 279 N

Constant speed means

forces are balanced and

FRand Fg should be drawn

as equal length vectors

Constant speed means

forces are balanced and

FRand Fg should be drawn

as equal length vectors

The rocket accelerating

upward means FE should

be greater than Fg

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Name________________________ Regents/Honors Physics

Date_____________

Notes: Components of Vectors

Objectives:

1. Represent vectors as a combination of components.

2. Resolve a vector into x- and y- components.

3. Add vectors algebraically using x- and y- components.

Vectors Can Have Components Consider the following vector: Show four ways that this vector can be represented as components of

other vectors.

Lets look more closely at the last example:

Draw the x and y axis as shown and label the horizontal and vertical components of vector V

V V

V

V

V1

V2 V3

V4

V1

V2

V1

V2

V1

V2

V

x

y

Vx

Vy

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Resolving Vectors on Coordinate Axes

Any vector may be placed on a set of _______________________________.

Vectors may be _________________ around a set of coordinate axes as long as:

o The vector in its new position is ___________________ to the vector in its original

position.

o The new vector is the ____________________ as the original.

A convenient way of drawing coordinate axes around a vector is so that the _________________ of

the axes is at the ____________ of the vector.

With the vector drawn this way the x- and y- components of the vector are sometimes thought of as

the ___________________ of the original vector on the x- and y- axes respectively.

If the ______________ and the ___________________ of the vector are known then the

magnitudes of the x- and y- components can be found by using ____________________.

On the above diagram:

Draw in the x- component (or the x- projection) of vector V.

Draw in the y-component (or the y- projection) of vector V.

coordinate axes

moved

parallel to

same length

origin

tail

projection

x

magnitude angle

trig functions

SOHCAHTOA Or

A

OTan

H

ACos

H

OSin

V

y

Vx

Vy Vy

’

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Label the angle of vector V as 35°

Calculate the x-component of vector V using the _______________________

V

V

H

ACos x

Calculate the y-component of vector V using the _______________________

V

V

H

OSin

y

If the angle between the vector and the

_______________________ is always used the

following two relationships will always apply.

ASinA

ACosA

y

x

Sample Exercise: Given the following vector diagram indicate what the x- and y- components are;

cosine function

sine function

V

y

Vx = ___________

Vy = ___________

V

Vx = ___________

Vy = ___________

V

y

Vx = ___________

Vy = ___________

positive x-axis

V

Vx = ___________

Vy = ___________

65°

A in the equation

means it can be used

for ANY vector

VCos0 = V

VSin0 = 0 VSin90 = V

VCos90 = 0

VCos180 = -V

VSin180 = 0 VCos65 = 0.423V

VSin65 = 0.906 V

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Example 1: A girl is pulling her sister on a sled on level ground. The girl pulls with a force of 16

newtons on the rope which makes an angle of 40 º with the ground, and keeps the sled moving at

constant speed.

a. Draw a free body diagram showing the forces acting on the sled.

b. Find the magnitude of the horizontal and vertical components of the girl’s forces acting on the

sled.

c. What is the magnitude of the friction force acting on the sled?

V

y

Vx = ___________

Vy = ___________

V

Vx = ___________

Vy = ___________

15°

30°

Fgirl

FgirlX

FgirlY

Fg

Ff

FN

Constant speed so . . .

Ff = FgirlX = Fgirl Cosθ

Ff = 16 N Cos 40̊

Ff = 12.3 N

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Name_________________________________ R/H Physics

Date__________________

Guided Practice: Components of Vectors

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Notes: Concurrent Forces on a Slope

Objectives:

1. Use x- and y- components of forces to describe systems involving friction on an incline.

Components of forces

Recall the methods of combining forces by _________________________ into components

Start with a _______________________

Use a _________________________ to show the forces acting on the object in a convenient

orientation over the x-y- axis

The free body diagram will show the __________________ only and as the name suggests it is free

of _______________ or bodies.

Resolve _________________________ for each force by using __________________

Breaking them

Coordinate axes

Free body diagram

Forces

objects

Into components Trig functions

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Static Equilibrium

Consider a block resting on the surface of a ___________ that

is prevented from sliding by its friction force.

Since the block is at rest, it is not _____________________

and therefore the net force acting on the block must be

___________

Therefore the sums of the ____________________________

separately must also be zero.

First let’s label the forces on the diagram and then draw a free

body diagram showing the forces in the above situation.

The easiest way to analyze these

forces is to superimpose a set of x- y-

coordinate axes ___________ to the

direction of the hill.

Note that the friction force and the

normal force are directly on top of

the ____________________

respectively.

Assuming the cart has a mass of

500. grams find its weight.

Now that we know the magnitude of

the weight force (Fg), find the x and

y components of the weight.

Since forces are balanced the forces

on the y-axis and x-axis must be

balanced.

Therefore on the x-axis the

____________ force must be equal

to __________ so it equals______ N.

On the y-axis the _____________

force must be equal to ________ so

it equals _________N

FREE BODY DIAGRAM (FBD)

accelerating

zero

x and y components

parallel

x and y axes

ramp

Ff

Fg

FN

m = 500 g = 0.500 kg

Fg = mg

Fg = (0.500kg) ( 9.8 m/s2)

Fg = 4.9 N

Fgx = Fg Cos θ

Fgx = (4.9 N) (Cos 65̊ )

Fgx = 2.07 N

Fgy = Fg Sin θ

Fgy = (4.9 N) (Sin 65̊ )

Fgy = 4.44 N

Ff = Fgx = 2.07 N

FN = Fgy = 4.44 N

friction

Fgx 2.07

normal

4.44

Fgy

Fgx

Fgy

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Name__________________________ Regents Physics

Date____________________

Concept Development: Force Vectors and Friction

1. Consider the block resting on the incline as shown.

a. Draw the force vector for the weight.

b. Determine the magnitude of the force vector and label the vector on the diagram

c. Draw an x-y axis with the x-axis parallel to the slope of the incline

d. Draw a dotted reference line from the tip of the Fg vector to the x-axis so that it is

perpendicular to the x-axis.

e. Draw a dotted reference line from the tip of the Fg vector to the y-axis so that it is

perpendicular to the y-axis.

f. Draw the x and y components of the weight force.

g. Draw the force vector for the friction force. How do you know how long to draw this force?

h. Draw the force vector for the normal force. How do you know how long to draw this force?

i. For a steeper incline, the component parallel to the incline is (greater) (the same) or (less)

j. For a steeper incline the component perpendicular to the incline is (greater) (the same) or

(less)

1 Kg

Fg

Fg = mg

Fg = (1kg) ( 9.8 m/s2)

Fg = 9.8 N

Fgx

Fgy

Ff

Fn

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2. Billy is riding a go-cart coasting down a hill at constant speed. Billy and the cart combined have a

mass of 150 kg. The hill is at a 40° angle to the horizontal ground.

a. Are the forces acting on Billy and the cart balanced or unbalanced?

b. Draw a free body diagram (FBD) showing all of the forces acting on Billy and the go-cart.

c. Draw an x-y coordinate axis on your FBD and draw the x and y components of the Fg force.

d. What is the magnitude of the weight force?

e. Find the x and y components of the weight force.

f. What is the magnitude of the friction force?

g. What is the magnitude of the normal force?

Balanced

Fg

Fn

Ff

Fg = mg

Fg = (150kg) ( 9.8 m/s2)

Fg = 1470 N

Fgx = Fg Cos θ

Fgx = (1470 N) (Cos 50̊ )

Fgx = 944.9 N

Fgy = Fg Sin θ

Fgy = (1470 N) (Sin 50̊ )

Fgy = 1126 N

Ff = Fgx = 944.9 N

FN = Fgy = 1126 N

Remember if the given angle is the

angle of incline – use the

complimentary angle (90 –θ) to

calculate x and y components

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3. Rose is sledding down an ice-covered hill inclined at an angle of 15° with the horizontal. If Rose

and the sled have a combined mass of 54.0 kg, what is the force pulling them down the hill?

The force pulling Rose down the hill is the component of gravity

that is parallel to the hill so . . .

Fgx = Fg Cos θ

Fgx = m g Cos θ

Fgx = (54.0 kg) (9.8 m/s2) Cos 15̊

Fgx = 511.2 N

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Name_________________________ Regents Physics

Date____________________

Guided Practice: Friction Forces

1. Ezekiel is riding his skateboard at constant velocity down a 22° slope as depicted in the diagram.

Ezekiel and his skateboard combined have a mass of 77 kg.

a. Are the forces acting on Ezekiel and the skateboard balanced or unbalanced?

b. Explain how you know this.

c. Draw a free body diagram showing all forces acting on Ezekiel and his skateboard.

d. Calculate the weight force of Ezekiel and the skateboard.

e. Determine the x and y components of the weight force

f. What is the magnitude of the normal force acting on the skateboard by the pavement?

g. What is the magnitude of the friction force resisting his motion?

h. Calculate the coefficient of friction between the skateboard and the pavement.

22°

Ff

Fg

Fn Fgx = Fg Cos θ

Fgx = (754.6 N) (Cos 68̊ )

Fgx = 282.7 N

Fgy = Fg Sin θ

Fgy = (754.6 N) (Sin 68̊ )

Fgy = 699.7 N

Ff = Fgx = 282.7 N

FN = Fgy = 699.7 N

Fg = mg

Fg = (77kg) ( 9.8 m/s2)

Fg = 754.6 N

Ff = µ Fn

282.7 N = µ (699.7N)

µ = 282.7 N / 699.7 N

µ = 0.404

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2. Skye is trying to make her 70.0 kg Saint Bernard go out the back door but the dog refuses to walk. If

the coefficient of sliding friction between the dog and the floor is 0.50, how hard must Skye push in

order to move the dog with a constant speed?

3. Rather than taking the stairs, Pierre gets from the second floor of his house to the first floor by

sliding down the banister that is inclined at an angle of 30.0 ° to the horizontal.

If Pierre has a mass of 45 kg and the coefficient of sliding friction

between Pierre and the banister is 0.20, what is the force of friction

impeding Pierre’s motion down the banister?

If the banister is made steeper (inclined at a larger angle), will this have

any effect on the force of friction? If so, what?

Fa = Ff = µFn

Fa = (0.50) (686N)

Fn = Fg = mg

Fn = (70kg) (9.8 m/s2)

Fn = 686 N

Ff = µFn

Ff = (0.20) (382 N)

Ff = 76.4 N

Fn = Fgy = mg sin 60̊

Fn = (45kg) (9.8 m/s2)

Fn = 382 N

A steeper incline will result in a smaller Fgy, therefore

a smaller Fn, therefore a smaller Ff

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TTQ’s (Typical Test Questions)

Documented Thinking

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Documented Thinking

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Documented Thinking

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Documented Thinking