PHYSICS: VectorsPHYSICS: Vectors

Today’s Goals

Students will:

1. Be able to describe the difference between a vector and a scalar.

2. Be able to draw and add vector’s graphically and use Pythagorean theorem when applicable.

Day 1

Today’s Objective

By the end of today…

1.You must be able to find the magnitude and direction of a resultant mathematically.

A couple of quick definitions!A couple of quick definitions!

This should be review, but just in case…..This should be review, but just in case…..

• Scalars

• Vectors

Remember: Remember: Force, Velocity, Force, Velocity, Displacement, Acceleration are ALL Displacement, Acceleration are ALL VECTORS!!!VECTORS!!!

Representing VectorsRepresenting Vectors• Because of direction, vectors do not add the

same way scalars do.

2 + 2 does not always equal 4!!!!2 + 2 does not always equal 4!!!!

We will discuss 2 Ways to Add Vectors1. Graphically

2. Component Addition

Adding VectorsAdding Vectors• When two vectors are added together, the

results are a vector called a RESULTANT.

• In order to find the resultant, you must draw the two vectors “head to tail.”

Example: http://www.stmary.ws/highschool/physics/home/notes/forces/animations/AngleResultants.html

Another Reason why vectors are important:http://www.stmary.ws/highschool/physics/home/notes/kinematics/MotionTerms/animations/boatsRiver.swf

Find the Following ResultantsFind the Following Resultants

1. 5 N

+ 5 N = ?

2.

+6 N 3 N = ?

ComponentsComponents• Every vector can be

separated into two equivalent vectors.

• These two vectors are known as components.

• In physics, we often break vectors down into horizontal (x) and vertical (y) components.

X - component Y -

co

mp

on

ent

Calculating Components

• Use the Pythagorean Theorem to calculate the missing component for each velocity below. a2 + b2 = c2

3 m/s

4 m/s

4 m/s

- 8 m/s

?

?

5 m/s

?

8 m/s5 m/s

8.9 m/s6.24 m/s

a2 + b2 = c2

(-8)2 + 42 = c2

64 + 16 = c2

80 = c2 c = 8.9 m/s

4 m/s

?

- 8 m/s

Right Triangle Trig.Right Triangle Trig.Q: WHAT ARE TRIG FUNCTIONS?A: Sin Cos Tan

• Three functions that are useful when dealing with right triangles.

• These functions are used to determine missing sides of right triangles when other information is known.

Sine (sin)

• The Sin of any angle is equal to the ratio of the length of the “opposite side” and the length of the “hypotenuse.”

Sin (ø) =HO

O

ø

H

Cosine (cos)

• The Cosine of any angle is equal to the ratio of the length of the “adjacent side” and the length of the “hypotenuse.”

A

ø

HCos (ø) =HA

Tangent (tan)

• The Tangent of any angle is equal to the ratio of the length of the “opposite side” and the length of the “adjacent side.”

O

A

ø

Tan (ø) =AO

An easy way to remember…An easy way to remember…

• What should you do when you have dirty feet?

»SOHCAHTOA

(pronounced: Soak A Toe.. Uhhh….)

SOH CAHTOASine

OppositeHypotenuse

CosineAdjacentHypotenuse

TangentOppositeAdjacent

What is a Function?What is a Function?

• 2x = Y

• What do I get for Y as I plug in numbers for x?

x y

0 0

1 2

2 4

Sin (x) Cos (x) and Tan (x)

do the same thing!!!!

Plug in a number for the angle:

(x degrees)

and you get back a different number (usually a decimal).

Next, following SOHCAHTOA, the number you get back from the function, is equal to O/H, A/H, or O/A!

Trig. Practice Problems

#1

a.  Find x.b.  Find y.

c.  Find q.d.  Find r.

q

r25o

9

sin (30) = x/20

.5 = x/20

(.5)20 = x

x = 10

cos (30) = y/20

.866 = y/20

(.866)20 = y

y = 17.32

sin (25) = 9/q

.423 = 9/q

.423q = 9

q = 21.3

tan (25) = 9/r

.466 = 9/r

.466r = 9

r = 19.3

Trig. Problems

#1

HINT:  

Draw a diagram showing the information.

                

m

From a point on the ground 25 meters from the foot of a tree, the angle of elevation

of the top of the tree is 32º.  (Meaning you have to look up 32o to be looking at the top of the tree.) Find to the nearest meter, the height of

the tree.

                 

A ladder leans against a building. 

The foot of the ladder is 6 feet from

the building.  The ladder reaches

height of 14 feet on the building.

a.  Find the length of the ladder to the nearest foot.

Trig. Practice

#2

                 

b. What is the angle between the top of the ladder and the

wall it leans on?

Trig. Practice

#2

Trig. Practice Problem #3

3. A plane travels 25 km at an angle 35 degrees to the ground and then changes direction and travels 12.5 km at an angle of 22 degrees to the ground. What is the magnitude and direction of the total displacement?

Quiz!Quiz!• The Vectors listed below represent the velocity of an

airplane and the velocity of wind. Each Plane is trying to Go North. Draw a Vector Diagram and Calculate the magnitude and direction of the resulting velocity.

1.Wind: 50 m/s EastPlane Engines: 200 m/s North

2. Wind 30 m/s EastPlane Engines: 100 m/s North

3. Wind 120 m/s WestPlane Engines: 175 m/s North