Vector Direction
• A vector quantity is a quantity that is fully
described by both magnitude and
direction.
• On the other hand, a scalar quantity is a
quantity that is fully described by the
magnitude.
• The emphasis is to understand vectors in
order to understand motion in 2D.
• Examples of vector quantities are
displacement, velocity, acceleration, and
force.
• Vector diagrams depict vector by use of an
arrow drawn to scale in a specific
direction.
Vector Diagrams
• Scale clearly listed
• A vector arrow (head and tail) drawn in a
specified direction.
• The magnitude and direction of the vector
is clearly labeled.
• Ex: magnitude 20 m and direction is 30
degrees West of North
Describing Directions
• Vectors can be directed due East, due
West, due South, and due North.
• But some vectors are directed northeast
(at a 45 degree angle) and some vectors
are even directed northeast, yet more
north than east.
2 Conventions to Use
• The direction of a vector is often
expressed as an angle of rotation of the
vector about its tail from east, west, north,
or south.
• Ex: 40 degrees North of West
65 degrees East of South
• The direction of the vector is often
expressed as a counterclockwise angle of
rotation of the vector about its tail from due
East.
• Ex: 30 degrees
160 degrees
Representing Magnitude
• Magnitude of a vector is depicted by the
length of the arrow.
• The arrow is drawn a precise length to a
chosen scale.
• Ex: scale: 1 cm = 5 miles
• Vector of 20 miles would be 4 cm
You Try It!!!! (Paden)
Vector Addition
• Two vectors can be added together to
determine the result (or resultant).
• Two methods for adding vectors:
• Pythagorean theorem and trig functions
• Head to tail method using a scaled vector
diagram
Pythagorean Theorem
• Useful for adding only 2 vectors which
make a right angle to each other.
• Not applicable when adding 2 vectors
together that do not make a right angle
with each other.
• Mathematical equation which relates the
length of the sides of a right triangle to the
length of the hypotenuse of a right triangle.
a2 + b2 = c2
ex:
Eric leaves the base camp and
hikes 11 km, north and then hikes
11 km east. Determine Eric’s
resulting displacement.
• R = 15.6 km
More practice
• 10 Km, North + 5 Km, West
• 30 Km, West + 40 Km, South
• 1. R = 11.2 km
• 2. R = 50 km
Trig functions to determine
Direction
• The direction of a resultant vector can
often be determined by use of
trigonometric functions.
• SOH CAH TOA is a mnemonic which
helps remember the meaning of the 3
common trig functions - Sine, cosine, and
tangent functions
• Sine = opposite/hypotenuse
• Cosine = adjacent/hypotenuse
• Tangent = opposite/adjacent
• 11 Km, North + 11 Km, East
• Find the direction of the hiker’s
displacement.
• R= 45 degrees
Practice
• 10 Km, North + 5 Km, West
• Find the magnitude and direction of the
resultant vector.
• R = 11.2 km, 116.6 degrees
Practice
• 30 Km, West + 40 Km, South
• Find the magnitude and direction of the
resultant.
• R = 50, 233.1 degrees
Practice 3A
• An archeologist climbs the Great Pyramid
in Giza, Egypt. If the pyramid’s height is
136 m and its width is 2.30 X 102 m, what
is the magnitude and the direction of the
archeologist’s displacement while climbing
from the bottom of the pyramid to the top?
Answer
• R = 178 m
• θ = 49.8 degrees
Head to Tail Method
• The head to tail method is employed to
determine the vector sum or resultant
when two or more vectors are drawn to
scale.
• Involves drawing a vector to scale at a
beginning position. Where the head of the
first vector ends, the tail of the second
vector begins, etc.
• Once all of the vectors have been drawn
head to tail, the resultant is then drawn
from the tail of the first vector to the head
of the last vector (from start to finish).
• Then, the length and direction of the
resultant can be measured and
determined using a ruler and protractor.
Practice
• Scale 1 cm = 5 m
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•
•
•
Add the vectors:
20 m, 45 degrees
25 m, 300 degrees
15 m, 210 degrees
• R= 22 m, 310 degrees
• The order of adding vectors doesn’t
change the magnitude or direction of the
resultant.
Vector Components
• A vector resultant can be transformed into
two parts (x and y).
• For example, a vector pointed northwest
can be directed as having a northward
vector and a westward vector.
• Each part of a 2D vector is known as a
component.
Resolving Vectors
• The influence of the 2 components is
equivalent to the influence of a single 2D
vector.
Practice 3B
• Find the component velocities of a
helicopter traveling 95 km/h at an angle of
35° to the ground.
Answer
• Y = 54 km/h
• X = 78 km/h
Projectile Motion
• Objects that are thrown or launched into
the air and are subject to gravity are called
projectiles.
• Ex: throwing a ball, arrows projected
through the air
Path of a Projectile
• The path of a projectile forms a curve
called a parabola
• If a projectile has a horizontal velocity, it
will have horizontal velocity throughout the
flight.
• For our samples and problems, the
horizontal velocity of a projectile will be
considered constant.
• With air resistance, the horizontal velocity
would not be constant.
• With air resistance, an object would travel
along a shorter path, which would not be a
parabola.
• Projectile motion is free fall with an initial
horizontal velocity.
• A ball dropped straight down has no initial
velocity.
• If air resistance is disregarded, one ball
dropped and one launched horizontally will
hit the ground at the same time.
• Projectiles can be analyzed as having both
horizontal and vertical components of
motion (2D)
• In any time interval, a launched ball
undergoes the same vertical displacement
as a ball that falls straight down, thus
hitting the ground at the same time.
• The horizontal acceleration dimension (x)
is zero but the acceleration in the vertical
dimension (y) will be equal to acceleration
due to gravity (g).
• We will analyze problems in each
dimension separately and list givens for
the x and y components separately similar
to our one dimensional acceleration
problems.
Vertical Motion of a Projectile that
falls from rest
• Vy,f = -gΔt
• Vy,f2 = -2gΔy
• Δy = -1/2g(Δt)2
Horizontal Motion of a Projectile
• Vx = vx,i = constant
• Δx = vxΔt
• To find the velocity of a projectile at any
point during its flight, find the vector sum
of the components of the velocity at that
point. Use the pythagorean theorem and
tangent function to find the direction of the
velocity.
Sample Problem 3D
• The Royal Gorge Bridge in Colorado rises
321 m above the Arkansas River.
Suppose you kick a little rock horizontally
off the bridge. The rock hits the water
such that the magnitude of its horizontal
displacement is 45.0 m. Find the speed at
which the rock was kicked.
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•
•
•
•
Givens:
Δy = -321 m
Δx = 45 m
ay = g = 9.81 m/s2
Vi = ?
Formulas
• Δx = vxΔt
• Δy = -1/2g(Δt)2
• Answer = 5.56 m/s
Projectiles launched at an angle
• If a projectile is launched at an angle, then
it has an initial vertical component as well
as a horizontal component of velocity.
• We will use sine and cosine to find the
horizontal and vertical components of the
initial velocity.
• vx,i = vi(cos θ) and vy,i = vi(sin θ)
Projectiles Launched at an Angle
•
•
•
•
•
vx = vi(cos θ) = constant
Δx = vi(cos θ)Δt
vy,f = vi(sin θ) – gΔt
vy,f2 = vi2(sin θ)2 – 2gΔy
Δy = vi(sin θ)Δt – 1/2g(Δt)2
Sample 3E
• A zookeeper finds an escaped monkey hanging
from a light pole. Aiming her tranquilizer gun at
the monkey, the zookeeper kneels 10.0 m from
the light pole, which is 5.00 m high. The tip of
her gun is 1.00 m above the ground. The
monkey tries to trick the zookeeper by dropping
a banana, then continues to hold onto the light
pole. At the moment the monkey releases the
banana, the zookeeper shoots. If the
tranquilizer dart travels at 50.0 m/s, will the dart
hit the monkey, the banana, or neither one?
• Answer: the banana is 4.77 m above the
ground
• The dart is 4.76 m above the ground so
the dart hits the banana.