Chapter 3
80 minutes
Two Dimensional Motion and Vectors
Section 1:
Scalar and Vectors
Learning Objective:



Students will be able to distinguish between scalars and vectors.
Students will add and subtract vectors using graphics
Students will understand that vectors may be multiplied or divided by
scalars and vectors
5 min. Bell Ringer:
Think of examples of Vectors and Scalars that you may have see in real
life situations. List them on a separate piece of paper. You will need this later.
10 min. Scalar and Vector Anticipation Guide
Separate into critical thinking pairs and complete the before section
Students return to seats when finished
15 min. Teach Scalars
Definition:
Scalar: a physical quantity that has magnitude with no direction. (5 Volts)
Q: What are some examples of scalars?
Using equity cards choose several students for examples
A: Voltage or difference in potential, speed, mass, charge, temperature, time
(Exception to time in special relativity)
Definition:
Vector: a physical quantity that has magnitude and direction. (5 m east.)
Q: What are some examples of Vectors?
(Connection to last chapter and displacement)
A: displacement, acceleration, velocity, others include momentum, electric and
magnetic fields.
Both scalars and vectors have symbols Vectors are represented in bold text such
as the letter A; where as the scalar would be represented by the letter A and is
purely a numerical value with units.
Chapter 3
Two Dimensional Motion and Vectors
5 min practice
Determine if the following are scalars of vectors
5 meters
10 kg
2∙Sin(π/2)
30° C
5 m/s
1 Newton
10 Joules
5 Watts
65 mph
12knts by 030°
Discuss correct answers with students
20 min Teach Vector Addition and Subtraction using the graphic method
Using transparencies or graph paper demonstrate vector addition
Fig 1. Note that in the example vector addition is
accomplished by transferring the B vector from the
base of A to the head of A and drawing the
resultant vector C.
This can be done for many examples. Using graph
paper and the Elmo demonstrate this procedure for
the students. The concept that vectors can be
moved to a new position parallel to the previous
location is integral to this process.
Perform several examples using various angled
vector.
The process is similar for vector subtraction. First
we invert the vector that is being subtracted (B).
Next we move the vector to the head of vector A
parallel to the negative vector. Finally we draw the
vector from the base of A to the head of –B shown
in blue (B) in Fig 2.
Perform several examples using various angled
vector.
Chapter 3
Two Dimensional Motion and Vectors
20 min Graphic Problem Practice
The students will complete in class practice using the handouts provided.
Handouts:
Vector Addition and Subtraction
Vector Scalar Multiplication
5 min Complete Anticipation Guide
Discuss the final results of the anticipation guide
Home Work
Complete Section 1 Review
Have Students complete Home Work for Vectors and Graphical Method
Section 2 Preparation
Read: Vector Operation pg 86 of text.
Vectors and Scalars
Names ________________
Before
After
True _____ False____
Scalars have magnitude
only
True _____ False____
True _____ False____
Scalars can be added
algebraically
True _____ False____
True _____ False____
Scalars can be multiplied
by vectors resulting in a
scalar
True _____ False____
True _____ False____
Vectors have magnitude
and direction
True _____ False____
True _____ False____
Vectors can be added
algebraically resulting in
a scalar
True _____ False____
True _____ False____
Vectors can be multiplied
and divided
True _____ False____
True _____ False____
The product of two
scalars is a vector
True _____ False____
True _____ False____
Vectors can be combined
algebraically using
graphical and
mathematical methods
True _____ False____
True _____ False____
True _____ False____
True _____ False____
Subtraction of vectors
uses the negative of both
vectors
Addition of vectors is
commutative where A+B
= B+A resulting in a new
vector with a new
magnitude
Mathematical vector
summation uses the
Pythagorean Theorem to
resolve for the resultant
True _____ False____
True _____ False____
True _____ False____
Vector Addition & Subtraction
1.
2.
3.
4.
Vector A is 10 squares right and 6 up.
Vector A is 9 squares left and 11 down.
Vector A is 10 right and 10 down.
Vector A is 8 squares up and 3 Right.
Use pencil for this exercise.
Vector B is 9 up and 1 right
Vector B is 8 squares right and 10 down.
Vector B is 10 squares left and 4 up
Vector B is 9 squares down
(Add)
(Add)
(Sub)
(Sub)
Scalar Vector Multiplication
1.
2.
3.
4.
A = 3 right, 3 up, multiply by 2.
A = 2 left, 3 down, multiply by 3.
A = 5 down, 2 left, multiply by 2.
A = 10 up 2 left, divide by 2.
B = 4 left, 4 up multiply by 2
B = 5 right, 1 up, multiply by 2
B = 3 up, 3 right, multiply by 3
B = 9 down, 6 right, divide by 3
(Add)
(Sub)
(Add)
(Add)
Chapter 3
80 minutes
Two Dimensional Motion and Vectors
Section 2:
Vectors Operations
Learning Objective:




Identify appropriate coordinate system for solving problems with
vectors
Apply the Pythagorean Theorem and tangential function to calculate
the magnitudes and direction of a resultant vector
Resolve vectors into components using the sine and cosine functions
Add vectors that are not perpendicular
5min Bell Ringer
Think of a way in which we can determine a better description of both
vertical and horizontal movement Hint: (Think back to section 1 and graphical
analysis of vectors).
10 min Brainstorm with your critical thinking partner ideas in which to describe vectors
mathematically using graphs from section 1
Q: Ask students there ideas about graphical descriptions using math.
A: At least one student should have come up with the Pythagorean Theorem
15 min Teach Vector Analysis and Vector Components.
Vectors can be broken down into component form using graphical analysis and
the Pythagorean Theorem.
Vector Components: projection of a vector along the axes of a coordinate
system
Example 1:
From the example shown it can be see that
each vector can be broken down into a right
triangle on the x y plane such that the sum
of components of the triangle are equal to
the resultant vector. The following example
is given.
9x + 8y
-3x + 6y
6x + 14y
The Magnitude of each vector is determined
by the Pythagorean Theorem. Where x2 + y2
= r2. or √x2 + y2 = r. The angle of the vector
is described as the Tan -1 (y/x).
Chapter 3
Two Dimensional Motion and Vectors
Considering further using a table format
X
9
-3
6
A Components
B Components
C Components
Y
8
6
14
Let’s calculate the magnitude of the resultant C and its angle in reference to the x
plane.
_______
√ 62 + 142 = 15.2 as a Magnitude now Tan-1 (14/6) = 66.8° or 1.16 radians
Remember the conversion for degrees to radians and vs.
π /180°
In Trigonometry the Tangent of the angle of a right triangle is the opposite side
over the adjacent or O/A. Finding the Arctangent or inverse Tangent of this
function results in the angle between the Hypotenuse and the Adjacent.
Complete several examples using the Elmo
10 min Practice A
Complete the practice problems on page 89
15 min Resolving Vectors into Components
A quick way to remember the rules for finding the Sine, Cosine and Tangent is as
follows:
SOH-CAH-TOA
Sine = Opposite / Hypotenuse
Cosine =Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
This allows for finding the Sine Cosine and Tangent functions of a right triangle.
Sample Problem B
Find the components of a helicopter traveling at 95 km/h at an angle of 35
degrees to the ground
Vy = 95km/h∙Sin 35° =
54km/h = y component
Vx = 95km/h∙Cosn 35° =
78km/h = x component
Checking our work with the Pythagorean Theorem
Chapter 3
Two Dimensional Motion and Vectors
2
V =
Vy + Vx2 =
2
__________________
we find that 95km/h = √ (54km/h)2 + (78km/h)2
And as such it does
9025 = 9000
Had we included no rounding we would have found that the actual figures
would agree more closely
10 min Practice B
Complete Practice B problems in the text.
5 min Adding Vectors that are not perpendicular
From Example 1, we have already done this but let’s practice another
On the 1st day of her hike Mary walks at a -45.0° for 5.66km. The next day she
walks 6.32 km at 71.6°
5.66 km θ = -45.0 °
6.32 km θ = 71.6 °
6.32 km θ = 18.4 °
Cos θ
4
2
6
15 min Practice C
Home Work
Complete Section 2 Review
Complete Home Work Problems for Vector Operations
Section 2 Preparation
Read: Projectile Motion pg 95 of text.
Sinθ
-4
6
2
Chapter 3
80 minutes
Two Dimensional Motion and Vectors
Section 3:
Projectile motion
Learning Objective:



Recognize examples of projectile motion
Describe the path of a projectile as a parabola
Resolve vectors into their components and apply kinematic equations
to solve problems of projectile motion
5min Bell Ringer
Think about examples of parabolic motion you may have seen recently or
in the past.
10 min Brainstorm with your critical thinking partner about the shape of the curve of
projectile motion do you recognize the shape? Does this shape relate to a mathematical
equation of some kind?
15 min Teach Projectiles Launched Horizontally
Q: What is the shape of the curve of an object that is traveling in projectile
motion?
A: It is the shape of a parabola and has the characteristic quadratics equation that
describes it mathematically.
Many factors can affect the curve, air resistance, initial velocity, and other forces.
Trajectory: The path that a projectile follows
Vertical Motion of a Projectile From Rest
Vyf = ayΔt
Vyf2 = 2ayΔy
Δy = ½ay(Δt)2
Horizontal Motion of a Projectile
Vx = Vxi = constant
Δx = VxΔt
Chapter 3
Two Dimensional Motion and Vectors
Example Problem D
A Rock is kicked of the St. George Bridge
Given:
Δy = -321 m
Unknown:
Vi =Vx =?
Δx = 45.0m
ay = -9.81m/s2
We choose a coordinate system with positive x to the right and positive y points
up.
Air resistance is neglected in this problem
So we choose Δx = VxΔt
And we choose Δy = ½ay(Δt)2
This is due to no initial vertical velocity.
1st Solve for Δt
_____
______________
Δt = √2Δy/ay = √2∙-321m/-9.81m/s2 = 8.09s
2nd Solve for Vx
Δx/Δt = 45m/8.09s = 5.56m/s
10 min Practice D
Complete practice problems D
15 min Teach Projectiles Launched at an Angle
Horizontal Component
ViCosθ = Vxi = constant
Δx = ViCosθΔt
Vyf = ViSinθ+ayΔt
Vyf2 = Vi2(Sinθ) 2+2ayΔy
Δy = (ViSinθ)Δt + ½ay(Δt)2
Chapter 3
Two Dimensional Motion and Vectors
Problem E
Perform Problem E with students in class, or
Proceed to lab for the lab on projectile motion
Home Work
Complete Section 3 Review
Complete Home Work Problems for projectile motion
Section 2 Preparation
Read: Relative Motion pg 102 of text.
Section 4 Relative Motion
Objectives


Describe situations in terms of frame of reference
Solve Problems involving relative velocity
Have student complete practice problems E
This is just the application of a new idea to concepts that are already learned
Home Work
Complete Section 4 Review
Complete Problems for Relative Motion