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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