2-d motion. 2 scalars and vectors a scalar is a single number that represents a magnitude –ex....

2-D motion

2

Scalars and Vectors

• A scalar is a single number that represents a magnitude– Ex. distance, mass, speed,

temperature, etc.

• A vector is a set of numbers that describe both a magnitude and direction– Ex. velocity (the magnitude of velocity

is speed), force, momentum, etc.

3

Scalars and Vectors

•Notation: – a vector-valued variable will be

Bold, – a scalar-valued variable will be in

italics. – if hand written vectors can be

denoted by an arrow over the value.

a

4

Characteristics of Vectors

A Vector is something that has two and only two defining characteristics:

1. Magnitude: the 'size' or 'quantity'

2. Direction: the vector is directed from one place to another.

A point at the beginning and an arrow at the end. The length of the arrow corresponds to the magnitude of the vector.•The direction the arrow points is the vector direction.Vectors are drawn to scale!!

Vectors can be drawn

6

Example

•The direction of the vector is 55° North of East

•The magnitude of the vector is 2.3.

7

Now You Try

Direction:

Magnitude:

47° North of West

2

8

Try Again

Direction:

Magnitude:

43° East of South

3

9

Try Again

It is also possible to describe this vector's direction as 47 South of East.

Why?

There are two ways of reading a vector’s direction.

By comparing to a cardinal directionBy easting convention

Reading directions

expressed as an angle of rotation of the vector about its tailEx. 40 degrees North of West

(a vector pointing West has been rotated 40 degrees towards the northerly direction)

Ex. 65 degrees East of South(a vector pointing South has been rotated 65 degrees towards the easterly direction)

By comparison

a counterclockwise angle of rotation of the vector about its tail from due East.Ex. 30 degrees, 240 degrees

Easting Convention

There are two ways of adding vectorsgraphicallyAnalytically

Resultant - the vector sum of two or more vectors. It is the result of adding two or more vectors together.

Vector addition

Graphic Addition Graphic Addition

Head-to-Tail MethodHead-to-Tail Method1. Draw the first vector with the proper length and orientation.

2. Draw the second vector with the proper length and orientation originating from the head of the first vector.

3. The resultant vector is the vector originating at the tail of the first vector and terminating at the head of the second vector.

4. Measure the length and orientation angle of the resultant.

Graphic Addition Graphic Addition •Ex. 20 m, 45 deg. + 25 m, 300 deg. + 15 m, 210 deg.SCALE: 1 cm = 5 m

Graphic Addition Graphic Addition •Ex. 20 m, 45 deg. + 25 m, 300 deg. + 15 m, 210 deg.SCALE: 1 cm = 5 m

Graphic Addition Graphic Addition •The order of addition doesn’t matter. The resultant will still have the same magnitude and direction.

Analytically Addition Addition Pythagorean TheoremPythagorean Theorem

This works only if the two vectors are at a right angle.

Analytically Addition Addition Pythagorean TheoremPythagorean Theorem

Ex. Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.

Analytically Addition Addition Pythagorean TheoremPythagorean Theorem

Practice A: 10km North plus 5 km West. What is the resultant vector?

Practice B: 30km West plus 40km South. What is the resultant vector?

Analytically Addition Addition Trigonometry:

Let’s try these togetherLet’s try these togetherBack to Practice A and Practice BPractice A: 10km North plus 5 km West. What is the resultant vector?Practice B: 30km West plus 40km South. What is the resultant vector? Remember: SOH CAH TOA

Analytically Addition Addition Trigonometry

This works only if the two vectors are at a right angle.Remember: SOH CAH TOA

Analytically Addition Addition Trigonometry

Ex. Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.Remember: SOH CAH TOA

Analytically Addition Addition Trigonometry:

Let’s try these togetherLet’s try these togetherBack to Practice A and Practice BPractice A: 10km North plus 5 km West. What is the resultant vector?Practice B: 30km West plus 40km South. What is the resultant vector? Remember: SOH CAH TOA

Analytically Addition Addition

Try this…Try this…Remember: SOH CAH TOAA plane travels from Houston, Texas to Washington D.C., which is 1540km east and 1160Km north of Houston. What is the total displacement of the plane?

Answer:1930 km at 37° north of east

Analytically Addition Addition

Try this…Try this…Remember: SOH CAH TOAA camper travels 4.5km northeast and 4.5km northwest. What is the camper’s total displacement?

Answer:6.4km north

Analytically Addition Addition

Try this…Try this…

Pg 89 Practice A #1-4

28

Resolving Vectors/Expressing Vectors as Ordered Pairs

How can we express this vector as an ordered pair?

Use Trigonometry

These ordered pairs are called the components of the vector.

29

30

A good example:

Express this vector as an ordered pair.

Answer:(42.7, 34.6)

Resolving Vectors

Try this…Try this…Remember: SOH CAH TOAFind the components of the velocity of a helicopter traveling 95km/h at an angle of 35° to the ground.

Answer:6.4km north

Resolving Vectors

Try this…Try this…Remember: SOH CAH TOAFind the components of the velocity of a helicopter traveling 95km/h at an angle of 35° to the ground.

Answer:y = 54km/hx = 78km/h

Resolving Vectors

One more…One more…Remember: SOH CAH TOAAn arrow is shot from a bow at an angle of 25° above the horizontal with an initial speed of 45m/s. Find the horizontal and vertical components of the arrow’s initial velocity.

Answer:41m/s, 19m/s

Resolving Vectors

Try this…Try this…

Pg 92 Practice B #1-4

Resolving VectorsResolving Vectors

What if the vectors aren’t at right angles?Resolving a vector is breaking it down into its x and y components.First, we need a vector.

North39East

s

m47 o

Resolving VectorsResolving Vectors

What if the vectors aren’t at right angles?Resolving a vector is breaking it down into its x and y components.First, we need a vector.

North39East

s

m47 o

39o

E

Let’s draw the vector

s

m47

Continuing

Next we will draw in the component vectors which we are looking for.

Horizontal

Drawing the Components

Vertical

39o

s

m47

Identifying the Sides

Vertical

opphyp

adj

39o

s

m47

Horizontal

What Trig Function will give the Horizontal Component?

opphyp

adj

39o

s

m47

Vertical

Horizontal

coscos adj

hyp

Finding The Horizontal Component

adjm

so 47 39cos

adjm

s36 5.

hyp

adjcos adj

hyp

adjcos coshypdja

hyp

adjcos

opphyp

adj

Finding The Vertical Component

39o

s

m47

Vertical

Horizontal

sinsin opp

hyp

oppm

so 47 39sin

Finding The Vertical Component

oppm

s 29 65.

hyp

oppsin opp

hyp

oppsin sinhypopp

hyp

oppsin

Continuing

The two components are:x:36.6m/sy: 29.65m/s

Resolving Vectors PracticeResolving Vectors PracticeA plane takes off at a 35° ascent with a velocity of 195 km/h. What are the horizontal and vertical components of the velocity?

A child slides down a hill that forms an angle of 37° with the horizontal for a distance of 24.0 m. What are the horizontal and vertical components?

How fast must a car travel to stay beneath an airplane that is moving at 105 km/h at an angle of 33° to the ground (What is the horizontal component?) What is the vertical component of the plane’s velocity?

Resolving Vectors

More practice… More practice…

Pg 92 Practice B #1-4

Analytically Addition Addition

Analytically Addition Addition

Analytically Addition Addition

Analytically Addition Addition

What if the vectors aren’t at right angles?There are four steps.1.We have to resolve the vectors into their components.2.Add all the x components.3.Add all the y components. 4.Find the magnitude and direction of the resultant.

Analytically Addition Addition

A hiker walks 27km from her camp at 35° south of east. The next day, she walks 41km at 65° north of east and discovers a forest ranger’s tower. Find the magnitude and direction of her resultant displacement between the base camp and the tower.

Answer:45km at 29° north of east

Analytically Addition Addition

A camper walks 4.5km at 45° north of east then 4.5 km due south. Find the camper’s total displacement, including direction.

Answer:3.4km at 22° south of east

You try…You try…

Resolving Vectors

More practice… More practice…

Pg 94 Practice C #1-4