Transcript Chapter 4 - Cloudfront.net

Vectors
Chapter 4
Cross Wind of 60 km/hr
Memphis
Plane traveling
at 80 km/hr
Houston
Flying Airplanes Requires
an Understanding of Vectors
Steering a Boat Requires an
Understanding of Vectors
Campsite
Across the
Shore
You can’t just steer your boat
directly towards the campsite
when the current is flowing east
Boat Dock
Vectors
• A vector has both magnitude and
direction. (20 m East, 40 m/s North)
• Vectors are represented by arrows.
• An arrow will show the vector’s:
magnitude: Length of the vector
direction: Angle of the vector measured
from east.
• A resultant vector:
One vector that acts as the sum of two
or more vectors.
Also called the Displacement Vector
• Direction of a vector is measured
by its angle from East.
•
•
•
•
East is 0 degrees
North is 90 degrees
West is 180 degrees
South is 270 degrees
• Sometimes direction is given as
so many degrees _______ of
________. ( North of East, etc.)
• Substitute “from” for “of”.
Vector
Direction
Vectors://www.physicsclassroom.com/mmedia/vectors/vd.cfm
2 Methods of Vector Addition
1. Graphical method –
align vectors tip to tail
and then draw the
resultant from the tail
of the first to the tip of
the last.
2. Analytical- Algebraic
method
Both give displacement or
final position from the point of
origin.
Head and Tail Winds: Tip to Tail
Tail Wind
Head Wind
20 km/hr
100 km/hr
120 km/hr
80 km/hr
100 km/hr
20 km/hr
A tail wind increases velocity and a head wind decreases velocity
Pythagorean Theorem
http://screencast.com/t/ZDg3NzM3Nm
http://www.physicsclassroom.com/Class/vectors/u3l1b.cfm#vamethods
A man walks 11 km N, then turns and walks 11 km E. What is his
final displacement from his starting point?
The final displacement is found by solving for the length of R using the
pythagorean theorem.
This only works for RIGHT TRIANGLES- must have a 90 degree angle!!
http://www.physicsclassroom.com/Class/vectors/u3l1b.cfm#vamethods
Algebraic Method using
Trig Functions
• Sine θ = opposite/hypotenuse
• Cosine θ = adjacent/hypotenuse
• Tangent θ = opposite/adjacent
θ
Trigonometric Functions
• Sin θ = opposite/hypotenuse
• Cos θ = adjacent/hypotenuse
• Tan θ = opposite/adjacent
Soh
Cah
Toa
a2 + b2 = c2
sin-1, cos-1, and tan-1
functions give θ
With any 2 values, you can
find all sides and all angles
θ
Keys to Successful Graphical Addition
•Clearly define
your scale factor
•Always add tip to
tail
Sample Problems
• Using a treasure map, a pirate walks
35 m east then 15 m north. What
single line could the pirate have taken
instead?
• A pilot sets a plane’s controls at 250
km/hr to the north. If the wind blows
at 75 km/hr toward the southeast,
what is the plane’s resultant velocity?
Sample Problems
• Using a treasure map, a pirate walks
35 m east then 15 m north. What
single line could the pirate have taken
instead?
• A2 + b2 = c2
• C= 38.1 m
• A pilot sets a plane’s controls at 250
km/hr to the north. If the wind blows
at 75 km/hr toward the southeast,
what is the plane’s resultant velocity?
• a2 + b2 = c2
Writing Vectors in Polar Coordinates
• A polar vector has both a magnitude [R] and a
direction [degrees from east]
• A Polar vector is written in polar coordinates [R,θ]
• Example: [50 m/s, 40°]
• Polar vectors must be changed to rectangular form
(x,y) in order to be added.
θ = 40°
Changing from Polar to Rectangular
• Since we always measure our angles from east (0
degrees) when in polar format, we can use the
following formulas:
• x = R cos θ
• y = R sin θ
Example Problems
• Convert the following polar coordinates to
rectangular coordinates:
• 54 m/s, θ = 60 degrees
• 4.5 N, θ = 235 degrees
x = R cos θ
y = R sin θ
Changing from Rectangular to Polar
• A Rectangular coordinate shows the x and y
components of a vector.
• Example: [38.3 m/s, 32.1 m/s]
• For converting from rectangular to polar, we
use the following formulas:
• x2 + y2 = R2
• θ = tan-1 (y/x)
Example Problems
• Convert the following rectangular coordinates
to polar coordinates:
• (36 m/s, 22 m/s)
• (-60 m, 35 m)
x2 + y2 = R2
θ = tan-1 (y/x)
Warm Ups
• A ladder is used to climb a building that is 9 meters
tall. The ladder makes an angle of 65 degrees with
the ground. How long is the ladder?
• A skateboard ramp is 3.3 meters long and makes an
angle of 35 degrees with the ground. How high is
the skateboarder when she leaves the ramp?
x = R cos θ
y = R sin θ
Adding Polar Vectors
• Convert each polar vector to rectangular coordinates.
x = R cos θ
y = R sin θ
• Add all the x coordinates to get a single x
coordinate. Add all the y coordinates to get a single
y coordinate.
• Convert back to polar coordinates.
x2 + y2 = R2 θ = tan-1 (y/x)
Warm Ups
• Convert to rectangular coordinates [15, 120°]
• Convert to polar coordinates (4.2,-6.8)
• Add [30, 115°] and [18, 255°]
• An airplane flying due south at 230 km/hr
experiences a crosswind from the east of 35 km/hr.
What is the magnitude and direction of the resultant
velocity?
Quick Review
• How do you know if a pair of coordinates is
polar or rectangular?
• Which formulas do you use to convert from
polar to rectangular?
• Which formulas do you use to convert from
rectangular to polar?
• What are the three steps for adding polar
vectors?
Warm Ups
• Add the following vectors graphically to find the
magnitude and direction. Then, add them
analytically to find magnitude and direction.
• 3 km to the East. 6 km at 220° from East. 4 km at
150° from East.
• A mover is loading a refrigerator into a truck with a
ramp that is 8 feet long at an angle of 25 degrees
with the ground. How far off the ground is the back
of the truck? How far away from the truck is the
ramp positioned?
Trigonometric Functions
• Sine θ = opposite/hypotenuse
• Cosine θ = adjacent/hypotenuse
• Tangent θ = opposite/adjacent
Angle 1
Sin θ = 6/10 = .60
Cos θ = 8/10 = .80
Tan θ = 6/8 = 0.75
Angle 2
Sin θ = 8/10 = .80
Cos θ = 6/10 = .60
Tan θ = 8/6 = 1.33
Use sin-1 to get θ
2 Methods of Vector Addition
2. Analytical Method –
use Pythagorean
Theorem and SOH CAH
TOA
• Sin = Opp / Hyp
• Cos = Adj / Hyp
• Tan = Opp / Adj