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					2-D motion
       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.

                                           2
       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                      3
  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.

                                            4
Vectors can be drawn
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!!
    Example

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

•The magnitude of the vector
is 2.3.




                               6
 Now You Try


Direction: 47° North of West

Magnitude: 2




                           7
       Try Again




Direction: 43° East of South

Magnitude: 3

                               8
             Try Again




It is also possible to describe this
vector's direction as 47 South of East.
              Why?
                                      9
Reading directions
There are two ways of reading a vector’s
direction.
   By comparing to a cardinal direction
   By easting convention
By comparison
expressed as an angle of rotation of the
vector about its tail
Ex. 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)
Easting Convention
a counterclockwise angle of rotation of the
vector about its tail from due East.
Ex. 30 degrees, 240 degrees
Vector addition
There are two ways of adding vectors
     graphically
     Analytically
Resultant - the vector sum of two or more
vectors. It is the result of adding two or more
vectors together.
           Graphic Addition

         Head-to-Tail Method
1. 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
•Ex. 20 m, 45 deg. + 25 m, 300 deg. +
15 m, 210 deg.SCALE: 1 cm = 5 m
        Graphic Addition
•Ex. 20 m, 45 deg. + 25 m, 300 deg. +
15 m, 210 deg.SCALE: 1 cm = 5 m
         Graphic Addition
•The order of addition doesn’t matter. The
resultant will still have the same magnitude
and direction.
     Analytically Addition
         Pythagorean Theorem

This works only if the two vectors are at a
right angle.
     Analytically Addition
         Pythagorean 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
         Pythagorean 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
            Trigonometry:
       Let’s try these together
Back to Practice A and Practice B
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?
Remember: SOH CAH TOA
     Analytically Addition
            Trigonometry

This works only if the two vectors are at a
right angle.
Remember: SOH CAH TOA
     Analytically 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
            Trigonometry:
       Let’s try these together
Back to Practice A and Practice B
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?
Remember: SOH CAH TOA
     Analytically Addition
            Try this…
Remember: SOH CAH TOA
A 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
            Try this…
Remember: SOH CAH TOA
A camper travels 4.5km northeast and 4.5km
northwest. What is the camper’s total
displacement?


Answer:
    6.4km north
     Analytically Addition
             Try this…

Pg 89 Practice A #1-4
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.
                                     28
29
          A good example:




Express this vector as an ordered pair.

Answer:
    (42.7, 34.6)
                                          30
     Resolving Vectors
            Try this…
Remember: SOH CAH TOA
Find 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…
Remember: SOH CAH TOA
Find the components of the velocity of a
helicopter traveling 95km/h at an angle of
35° to the ground.


Answer:
     y = 54km/h
     x = 78km/h
     Resolving Vectors
            One more…
Remember: SOH CAH TOA
An 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…

Pg 92 Practice B #1-4
      Resolving 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.

     m o
      East
        No
    47 39
     s
      Resolving 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.

     m o
      East
        No
    47 39
     s
Let’s draw the vector
                  m
               47
                  s



     39o

                      E
         Continuing



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

Vertical
                    m
                 47
                    s



           39o

                     Horizontal
           Identifying the Sides

Vertical
                           m
                        47
                           s
           hyp
                       opp
           39o

                             Horizontal
                 adj
What Trig Function will give the
  Horizontal Component?
Vertical
                           m        d
                                   aj
                        47    o
                             cos 
                             c s
                           s        y
                                   hp
           hyp
                       opp
           39o

                             Horizontal
                 adj
Finding The Horizontal
     Component
    adj
    adj
    adj
    
    hyp
cos   co
 
 
  
cosa
cos      
        dj
        adj
   hyp
   hyp
   hyp

    m    o
 d 4  o9
 a    s
  j 7 c3
     s
          m
    aj 6
     d 3.5
          s
      Finding The Vertical
          Component
Vertical
                           m       p
                                   op
                        47    in
                             sin 
                             s 
                           s        y
                                   hp
           hyp
                       opp
           39o

                             Horizontal
                 adj
 Finding The Vertical
     Component
    opp
    opp
    opp
     sin
sin   
  
   
   
   
sin opp
sin      hy p
        opp
    hy p
    hyp
    hyp
     m    o
  p 4  i3
  o    n
   p 7 s 9
      s
           m
    p  95
        .
    op 2 6
           s
         Continuing



The two components are:
   x:36.6m/s
   y: 29.65m/s
     Resolving Vectors Practice
A 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…

Pg 92 Practice B #1-4
Analytically Addition
Analytically Addition
Analytically Addition
      Analytically 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


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
            You try…
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
     Resolving Vectors
             More practice…

Pg 94 Practice C #1-4

				
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