10/11/2013

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


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


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


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


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


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


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


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.


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



50m/s

5m/s

x y

v1

v2

-50 m/s

+ 5 m/s

-50 m/s +5 m/s



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



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


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


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


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


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.

10/11/2013

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