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10/11/2013Today I will use trigonometry to solve right triangles.
Warm up-Define and give an example of:Scalar QuantityVector Quantity
Part 1 - Vectors
Chapter 3 Projectile Motion
TrigonometryWhen working with right triangles, if we
know two sides, we can find the other using the Pythagorean theory
A2 + B2 = C2 hypotenuse
Trigonometrya2 + b2 = c2
(8 m)2 + (x)2 = (12 m)2
64m2 + x2 = 144m2
x2 = 80m2
x = 8.9 m 12 m
8 m
? m
Trigonometry!Right Triangles
hypotenuse
θ
opposite
ad
jace
nt
TrigonometryTrigonometric Functions
SINE - COSINE – TANGENT
Formulas-
hypotenuse
opposite sin
hypotenuse
adjacent cos
adjacent
opposite tan
TrigonometryHow can I remember these
equations?
hypotenuse
opposite sin
hypotenuse
adjacent cos
adjacent
opposite tan
O
H
H
A
A
O
h
eck
nother
our
f
lgebra
SohCahToa
Trigonometry
Have – hypotenuseWant – adjacent
Which function uses both?
hypotenuse
adjacent cos
55
a82 cos
55
a8829.0
6.48a
Trigonometry
Let’s find x
Have – oppositeWant – hypotenuse
Which function uses both?
hypotenuse
opposite sin
x
5 30sin
Be careful when the variable is on the bottom!
30sin
5 x
10x
30°
x5
Trigonometry
Find B
We are now looking at a different angle, so opposite, and adjacent are different!
Have – hypotenuse & adjacentWant – angle B
Which function can we use??
hypotenuse
adjacent c os
10
5 c Bos
To solve for an angle, you must use the inverse functions!
5.0 B cos
60B
B
)5.0(cos 1B
Does this number make sense?
30°
105
X
Trigonometry
Find y
Since we have both other sides, we can use the Pythagorean theory.
222 cba
222 105 y
10025 2 y
752 y
7.8y
B
30°
105
Xy
TrigonometryYou can use trig with measurements:
34 m?
345
m/s
?
12 km/hr18 km/hr
47°23°
Ɵ
10/15/13Today I will demonstrate vector additionWarm Up – Find the a and b in the triangle
below:
150 m
20°
a
b
Trig HW
Review…
What is a scalar quantity? a quantity with only magnitude
Review…
What is a vector quantity? a quantity with both magnitude and direction
Are the following quantities vectors or scalars?
Time
Acceleration
Distance
Velocity
Displacement
Speed
Scalar
Vector
Scalar
Vector
Vector
Scalar
Intro to Vectors Mini-LabPart 1 & 2a
10/17/13Today I will draw vectors, add them and find the resultant.
Warm Up – If I walk 10 m North, 5 m East, 10 m South and then 5 m West, what is my displacement?
Drawing vectors…To draw a vector you must represent both the direction and the magnitudeDirection is represented by the angle and arrow
Magnitude is represented by the length
Parts of a vector
Tip – ending point (arrow)
Tail – starting point
Drawing Vectors…Length of the vector is the
magnitudeAngle and arrow indicate
directionAngle represented by Greek letter theta
θ
Drawing Vectors - scaleWhen drawing vectors, you must use some scale.
Ex. 100 mYou clearly cannot draw 100 m on a piece of paper.
You must set up a workable scale. If 1 cm = 10 m, 10 cm = 100 m… You can draw 10 cm!
Drawing Vectors - scaleScaling rules of thumb…
Centimeters is a good scale to work in
If your numbers are too large… Divide by something and make 1 cm = that many units
If your numbers are too small… Multiply by something and make that many cm = 1 unit
*Remember to scale your unit back up at the end by doing the opposite!
Drawing Vectors - scale120 m, 90 m, 70 m
3500 km/h, 7200 km/h, 6000 km/h
0.6 mi, 0.3 mi, 0.2 mi
Adding VectorsWe can add two or more vectors together; when we do this we are finding the resultant
Two methods of adding vectors
Solving Graphically – Tip to Tail method Draw first vector Start next vector where the last
one ended (so its tail is connected to the previous vectors tip)
Draw your resultant vector Find the direction (including angle)
Solving Graphically
1st Vector
2nd Vector
Example:
Solving GraphicallyResultant - Start where the first vector starts and end where the last vector ends
Solving Graphically
1st Vector
2nd Vector
Resulta
nt
Example:
Vector Lab – Day 2Drawing your motion to scale
If you make 1 cm = 1 m, that would be OK, but quite small.
If you make 2 cm = 1 m, that would be better.
You can even make 5 cm = 1 m
Give the resultant. We will deal with direction tomorrow!
10/18/13Today I will find direction in vector addition
problems
Warm Up – Draw to scale and find the resultant:
1. You drive 27 miles North to the new Wal-mart to pick up some gift cards. You turn and drive 48 miles West to see your cousin’s baseball game. You then drive South for 62 miles for a graduation party. Find your displacement graphically.
2. What would be the best way to get home?
Vector DirectionLook at this example:
What is the direction of the resultant vector?
Is it enough to say Northeast?
No, because we could be at any angle between
North and East.
123
km/h
hypo
tenu
s
e
Opposite
(y)
50°
Adjacent (x)
Vector DirectionLook at our previous example:
123
km/h
50°
OHIO RIVER
Mrs. Nairn’s house
My house is North of the Ohio River.
What if the river were the x-axis?
OHIO RIVER
Vector DirectionLook at our previous example:
123
km/h
50°
N
W
S
EMy house would be North of the East
lineOR
North of East(N of E)
Mrs. Nairn’s house
Vector DirectionLook at our previous example:
123
km/h
50°
If we put in the directional axis….
Our resultant is 50° North of the East line. (N of E)
N
E
Vector DirectionLook at our previous example:
123
km/h
50°
It would also be appropriate to say…
N
E
Our resultant is 40° East of the North line. (E of N)
Vector DirectionWhat is the direction on the following
vectors?
30°60°
W
S
W
S
30° W of S
60° S of W
Vector DirectionWhat is the direction on the following
vectors?
80°
38°
E
S
W
N
80° S of E
38° N of W
Vector DirectionAnother way of recording direction
is to say that an angle so many degrees clock-wise or counter-clockwise from a directional line.80°
E
S 80° S of E
280° counterclockwise from East
Measuring AnglesImagine a mini-coordinate system at the tail of your vector
Place the center point of the protractor along one direction of the axis
Measure the angle making sure to go from zero
Determine the direction
50° N of E
Measuring AnglesImagine a mini-coordinate system at the tail of your vector
Place the center point of the protractor along one direction of the axis
Measure the angle making sure to go from zero
Determine the direction
40° E of N
Vector DirectionUse your ruler to measure the angles on
the Direction WS
Vector Lab – Part 3Go back and add direction into you lab!
10/21/13Today I will solve vector addition
problems graphically
Warm Up – Find x
x
127 km
25°
Solving GraphicallyA plane is flying East at 100 km/hA Westward wind is blowing at 20 km/h
When adding vectors, place them tip to tail to scale to find the resultant vector.
A good scale here might be 1cm = 10 km/h
100 km/h 20 km/h
What is the resultant vector?80 km/h East
Remember to scale your answer back up!
Solved ExamplesExample 1: Every March, the swallows return to
San Juan Capistrano, CA after the winter in the south. In the swallows fly due North and cover 200 km on the first day, 300 km on the second day, and 250 km on the third day, draw a vector diagram of their trip and find the total displacement for the three day journey.
200 km
300 km
250 km
1 cm = 100 km
750 km North
Solved ExamplesExample 2: Suppose a car pulling with a
force of 20000N was pulled back by a rope that Joe held in his teeth. Joe pulled the car with a force of 25000 N. Draw a vector diagram and find the resultant force.+20000 N
-25000 N
5000 N in the direction Joe is pulling
Solved ExamplesExample 3: If St. Louis Cardinals homerun
king, Mark McGuire, hit a baseball due West with a speed of 50.0 m/s, and the ball encountered a wind that blew it north at 5.00 m/s, what was the resultant velocity of the baseball?
50m/s
5m/sθ
The resultant is about 50.3 m/s at some angle Ɵ.
Measure the angle with the protractor
Solved ExamplesExample 4:
The Maton family begins a trip driving 700 km west. Then the family drives 600 km south, 300 km east, and 400 km north.
Measure the length and the angle with your ruler to find the answer!
HomeworkExercises 1, 2, 3, 5, 6
Draw each exercise to scale and measure for magnitude and direction
10/22/2013Today I will solve vector problems involving angles graphically.
Warm Up:A dog runs 50 m west toward a cat. His owner calls him and he runs 30 m north before deciding to chase a squirrel. He turns and runs 100 m west chasing the squirrel. Solve graphically for his resultant displacement (include direction)
Homework ReviewExercises
Drawing Problems With AnglesWe typically don’t travel directly N, S, E, W.
I drive my car 1200 km in a direction 23° S of W. What does this vector look like?
I drive my car 55 miles/hour in a direction 92° counter-clockwise from the East. Draw this vector.
ExampleYou travel 630 miles at 23° N of W. You then travel 470 miles directly South. You finish your trip by traveling 500 miles at 55° W of S.
Example: CrosswindAn airplane is flying 80 km/h north
is caught in a strong crosswind of 60 km/h blowing in a direction 20° W of S. What is the resultant velocity of the airplane?
Exercise 7Amit flies due East 5600 km. He then flies
900 km at an angle of 55.0° E of N. What is his total displacement?
Mini-Lab Part 4
10/23/2013Objective – to resolve vectors and to practice vectors.
Warm UpFind the missing side of the triangle:
54 m
?
23°
Vector Components
Components– the two vectors at right angles with one another that are added to form the resultant (the x and y vectors aka the horizontal and vertical components)
y-component
x-component
Vector ComponentsResolution or resolving – Breaking the resultant vector into its two vector components (x and y)
ResolvingA plane flies according to the vector below at 123 km/h. What are the x and y components?
123
km/h
hypo
tenu
s
e
Opposite
(y)
40°
Adjacent (x)
hypotenuse
opposite sin
hypotenuse
adjacent cos
adjacent
opposite tan
Lets look at the adjacent (x) first:
Θ = 40H = 123 km/hA = x
ResolvingA plane flies according to the vector below at 123 km/h. What are the x and y components?
123
km/h
hypo
tenu
s
e
Opposite
(y)
40°
Adjacent (x)
hypotenuse
adjacent cos Lets look at
the adjacent (x) first:
Θ = 40H = 123 km/hA = x
km/h 123
x 40 cos
km/h 123
x .7660444430
xkm/h 4.2239
ResolvingA plane flies according to the vector below at 123 km/h. What are the x and y components?
123
km/h
hypo
tenu
s
e
Opposite
(y)
40°
Adjacent (x)
hypotenuse
opposite sin
hypotenuse
adjacent cos
adjacent
opposite tan
Lets look at the adjacent (y) next:
Θ = 40H = 123 km/hO = y
ResolvingA plane flies according to the vector below at 123 km/h. What are the x and y components?
123
km/h
hypo
tenu
s
e
Opposite
(y)
40°
Adjacent (x)
hypotenuse
opposite sin Lets look at
the adjacent (y) next:
Θ = 40H = 123 km/hO = y
km/h 123
y 40sin
km/h 123
y 64278761.0
ykm/h 9.0637
Vector Components12
3 km
/h
hypo
tenu
se
Opposite
(y)
40°
Adjacent (x)
If we consider the starting point of the vector as the origin on an x-y axis…
Notice that the x-component is in the positive x direction and the y-component is in the positive y direction.
So your answers are x = +94.2 km/hy = +79.1 km/h
Vector ComponentsLook at this example:
18 km
47°
opposite
hypotenuse
adjacent
hypotenuse
opposite sin
hypotenuse
adjacent cos
adjacent
opposite tan
Lets look at the adjacent (x):
Θ = 47H = 18 kma = x
Vector ComponentsLook at this example:
18 km
47°
opposite
hypotenuse
adjacenthypotenuse
adjacent cos
Lets look at the adjacent (x):
Θ = 47H = 18 kma = x
km 18
x 47 cos
kmx 3.12
This vector is in the positive x direction so that’s OK.
Vector ComponentsLook at this example:
18 km
47°
opposite
hypotenuse
adjacenthypotenuse
opposite sin
Lets look at the opposite(y):
Θ = 47H = 18 kmo = y
km 18
y 47sin
kmy 2.13
BUT…..This vector is in the negative y direction!
kmy 2.13
Vector ComponentsWhen resolving vectors, you MUST
pay close attention to the sign on the answer. Mathematically, this serves as your direction!!
Solved ExamplesExample 5: Ralph is mowing the back yard
with a push mower that he pushes downward with a force of 20 N at an angle of 30° to the horizontal. What are the horizontal and vertical components of the force? (Lets say he is walking in a + direction)
30°
20 Ny = opp
x = adj
hyp
adjcos
N 20
x30cos
Nx 3.71
hyp
oppsin
N 20
y30sin
Ny 10
Nx 3.71 Ny 10
Do exercises 8 & 9
Resolving WS
10/24/13Today I will solve vector problems mathematically
Warm UpDraw a vector that represents
7400 miles at 31° W of S
TrigonometryIf a boat travels at 98 km/h North and a current is running at 15 km/h West, what is the resultant velocity of the boat? Find both the magnitude and direction using trigonometry.9
8
km/h
15 km/h
15 km/h
Find the magnitude of the resultant:
a2 + b2 = c2
(98 km/h)2 + (15 km/h)2 = c2
9829 km2/h2 = c2
99.1 km/h = c
99.1
km/h
TrigonometryFinding an angle
In what direction??
98
km/h
15 km/h
15 km/hθ = ??O = 15 km/hA = 98 km/h9
9.1
km/h
θ
adjacent
opposite tan
km/h 98
km/h 15 tan
70.8
final answer:
99.1 km/h at 8.7°West of North
Vector ComponentsMethod #2 – Solving Mathematically
1. Draw & resolve for the x and y components of each vector in your problem (keeping sign a priority). Does not need to be to scale or connected.
2. Add all x’s and all y’s separately3. Make a new triangle with your combined x and y (paying attention to sign). Does not need to be to scale.
4. Use trig to find the final resultant magnitude and direction
Solved ExamplesExample 1: Every March, the swallows return to
San Juan Capistrano, CA after the winter in the south. In the swallows fly due North and cover 200 km on the first day, 300 km on the second day, and 250 km on the third day, draw a vector diagram of their trip and find the total displacement for the three day journey.
200
km
300
km
250
km750 km North
x y
v1
v2
v3
+200 km+300 km+250 km+750 km
750 km
Solved ExamplesExample 2: Suppose a car pulling with a
force of 20000N was pulled back by a rope that Joe held in his teeth. Joe pulled the car with a force of 25000 N. Draw a vector diagram and find the resultant force.20000 N
25000 N
x y
v1
v2
+20000 N
-25000 N
-5000 N5000 N backwards (opposite of Joe)
-5000 N
Solved ExamplesExample 3: If St. Louis Cardinals homerun
king, Mark McGuire, hit a baseball due West with a speed of 50.0 m/s, and the ball encountered a wind that blew it north at 5.00 m/s, what was the resultant velocity of the baseball?
50m/s
5m/s
x y
v1
v2
-50 m/s
+ 5 m/s
-50 m/s +5 m/s
Solved ExamplesExample 3: If St. Louis Cardinals homerun
king, Mark McGuire, hit a baseball due West with a speed of 50.0 m/s, and the ball encountered a wind that blew it north at 5.00 m/s, what was the resultant velocity of the baseball?
50m/s
5m/s
a2 + b2 = c2
(50)2 + (5)2 = c2
c2 = 2525 m2/s2
c = 50.3 m/s
θ
The resultant is 50.3 m/s
Solved ExamplesExample 3: If St. Louis Cardinals homerun
king, Mark McGuire, hit a baseball due West with a speed of 50.0 m/s, and the ball encountered a wind that blew it north at 5.00 m/s, what was the resultant velocity of the baseball?
50m/s
5m/sθ
The resultant is 50.3 m/sAt 5.7° above the horizon
adj
opptan
sm
sm
/50
/5tan
7.5
Solved ExamplesExample 4:
The Maton family begins a trip driving 700 km west. Then the family drives 600 km south, 300 km east, and 400 km north.
1. X = -700 km y = 02. X = 0 y = -600 km3. X = 300 km y = 04. X = 0 y = 400 km
R. X = -400 km y = -200 km
400 km
200
km
a2 + b2 = c2
c = 447 kmθ
km 400
km 200 tan
26.6
Resultant is 447 km at 26.6° S of W
A car travels 120 m North, 140 m East, and then 50 m at 38° W of N. Find the resultant displacement mathematically.
You travel 630 miles at 23° N of W. You then travel 470 miles directly South. You finish your trip by traveling 500 miles at 55° W of S.
Homework – Redo exercises 6 & 7
10/25/13Today I will solve vector problems
mathematically.
Warm UpA marathoner runs a set course each day to practice.
-He starts going 10 miles in a direction 34° N of W. -He then runs 5 miles in a direction 12° S of W. -Next, he goes 6 miles in a direction 48° E of S. -He finally runs 1 mile directly south where his friend picks him up.
What is the runner’s total displacement? Solve both graphically and mathematically.
A marathoner runs a set course each day to practice. He starts going 10 miles in a direction 34° N of W. He then runs 5 miles in a direction 12° S of W. Next, he goes 6 miles in a direction 48° E of S. He finally runs 1 mile directly south where his friend picks him up and drives him home. What is the runner’s total displacement? Solve mathematically.
x y
v1
v2
v3
v4
Vector 1
34
y
x
mi
x
1034cos
10mimix 3.8-8.3
mi
y
1034sin
miy 6.5
+5.6
A marathoner runs a set course each day to practice. He starts going 10 miles in a direction 34° N of W. He then runs 5 miles in a direction 12° S of W. Next, he goes 6 miles in a direction 48° E of S. He finally runs 1 mile directly south where his friend picks him up and drives him home. What is the runner’s total displacement? Solve mathematically.
x y
v1
v2
v3
v4
Vector 2
12
y
x mi
x
512cos
5mi
mix 9.4-8.3
mi
y
512sin
miy 0.1
+5.6
-4.9
-1.0
A marathoner runs a set course each day to practice. He starts going 10 miles in a direction 34° N of W. He then runs 5 miles in a direction 12° S of W. Next, he goes 6 miles in a direction 48° E of S. He finally runs 1 mile directly south where his friend picks him up and drives him home. What is the runner’s total displacement? Solve mathematically.
x y
v1
v2
v3
v4
Vector 3
48y
x
mi
x
648sin
6mi
mix 5.4-8.3
mi
y
648cos
miy 0.4
+5.6
-4.9
-1.0
+4.5 -4.0
A marathoner runs a set course each day to practice. He starts going 10 miles in a direction 34° N of W. He then runs 5 miles in a direction 12° S of W. Next, he goes 6 miles in a direction 48° E of S. He finally runs 1 mile directly south where his friend picks him up and drives him home. What is the runner’s total displacement? Solve mathematically.
x y
v1
v2
v3
v4
-8.3 +5.6
-4.9
-1.0
+4.5 -4.0
Vector 4
-1.00
-7.1 -3.1
A marathoner runs a set course each day to practice. He starts going 10 miles in a direction 34° N of W. He then runs 5 miles in a direction 12° S of W. Next, he goes 6 miles in a direction 48° E of S. He finally runs 1 mile directly south where his friend picks him up and drives him home. What is the runner’s total displacement? Solve mathematically.
x y
v1
v2
v3
v4
-8.3 +5.6
-4.9
-1.0
+4.5 -4.0
Resultant!
-1.00
-8.7 -0.4
8.7 mi
0.4 mi
a2 + b2 = c2
(0.4mi)2 + (8.7mi)2 = c2
c = 8.7 mi
7.8
4.0tan
6.2
Resultant =8.7 mi at 2.6° S of W
HomeworkAdditional Exercises 5 & 6 – Solve BOTH
graphically and mathematically!
10/28/13Objective – to demonstrate vector
addition both graphically and mathematically
Warm Up – Draw the following vector to scale and then find the x and y components mathematically. Would these components be positive or negative?
4500 m at 28° W of S28°
y
x
m 4500
x28sin
m 4500
y28cos
mx 2113
mx 2113
my 3973
my 3973
4500
m
Vector ActivityWe are going to follow Ms. Neil’s journey
through Pittsburgh.1. Start at the starting point above the “W” in
BALDWIN.
Graphically find the Resultant1. Using your protractor and ruler, draw in
each vector. (4 cm = 1 mi)2. Draw in the FINAL RESULTANT in a different
color. Measure the displacement of the resultant with your ruler and protractor.
Vector ActivityMathematically find the resultant:
1. Draw in the x and y component of each vector (you may transpose them onto a separate sheet). They do not need to be to scale.
2. Solve for the x and y components of each.3. Add all the x’s and y’s together.4. Draw a new triangle with these as
components!5. Solve for the resultant vector including
direction!6. Compare the two methods.
10/30/13An ant is searching for food. He walks 6
inches in an eastern direction, 7 inches in a direction 56° S of E and then 10 inches in a direction 32° W of S. He finally walks another 3 inches directly east. Find his resultant mathematically and graphically.
Now, Draw the four vectors in a different order. Explain what happened.