Transcript Analytically Addition

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.
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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
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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.
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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.
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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?
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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.
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A good example:
Express this vector as an ordered pair.
Answer:
(42.7, 34.6)
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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
47
East
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No
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
47
East
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No
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
adj
m
47
cos

cos
s
hyp
hyp
opp
39o
Horizontal
adj
Finding The Horizontal
Component
adj
adj
adj
cos



a
dj

hyp

co

cos

 

adj

cos


hyp
hyp
hyp
m
o
a
d
j
4
7 c
o
s
3
9
s
m
a
d
j3
6
.5
s
Finding The Vertical
Component
Vertical
opp
m
47
sin

sin
s
hyp
hyp
opp
39o
Horizontal
adj
Finding The Vertical
Component
opp
opp
opp
sin



opp

hyp

sin

sin



opp

sin



hyp
hyp
hyp
m
o
o
p
p

4
7 s
i
n
3
9
s
m
o
p
p2
9
.6
5
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