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Vector Objectives
Add and subtract displacement vectors to describe
changes in position.
2. Calculate the x and y components of a displacement,
velocity, and force vector.
1.
Write a velocity vector in polar and x-y coordinates.
4. Calculate the range of a projectile given the initial velocity
vector.
5. Use force vectors to solve two-dimensional equilibrium
problems with up to three forces.
6. Calculate the acceleration on an inclined plane when
given the angle of incline.
3.
Chapter 7 Vocabulary Terms
 vector
 displacement
 projectile
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

 trajectory
 Cartesian
coordinates
 range
scalar
magnitude
x-component
y-component
 cosine
 parabola
 Pythagorean
theorem
resultant
position
resolution
right triangle
 sine
 dynamics
 tangent
 velocity vector
 equilibrium
 inclined plane
 normal force
 polar coordinates
 scale component
7.1 Vectors and Direction
Key Question:
How do we accurately
communicate length
and distance?
*Students read Section 7.1 AFTER Investigation 7.1
Vectors and Direction
 A scalar is a quantity that
can be completely
described by one value: the
magnitude.
 You can think of magnitude
as size or amount,
including units.
Vectors and Direction
 A vector is a quantity that
includes both magnitude
and direction.
 Vectors require more than
one number.
 The information “1
kilometer, 40 degrees east
of north” is an example of a
vector.
Vectors and Direction
 In drawing a vector as an
arrow you must choose a
scale.
 If you walk five meters
east, your displacement
can be represented by a 5
cm arrow pointing to the
east.
Vectors and Direction
 Suppose you walk 5 meters east,
turn, go 8 meters north, then
turn and go 3 meters west.
 Your position is now 8 meters
north and 2 meters east of
where you started.
 The diagonal vector that
connects the starting position
with the final position is called
the resultant.
Vectors and Direction
 The resultant is the sum of two
or more vectors added together.
 You could have walked a shorter
distance by going 2 m east and
8 m north, and still ended up in
the same place.
 The resultant shows the most
direct line between the starting
position and the final position.
Calculate a resultant vector
 An ant walks 2 meters West, 3 meters North,
and 6 meters East.
 What is the displacement of the ant?
Finding Vector Components
Graphically
 Draw a
displacement
vector as an arrow
of appropriate
length at the
specified angle.
 Mark the angle and
use a ruler to draw
the arrow.
Finding the Magnitude of a Vector
 When you know the x- and y- components of a vector, and
the vectors form a right triangle, you can find the
magnitude using the Pythagorean theorem.
7.1 Adding Vectors
 Writing vectors in components make it easy to add them.
Subtracting Vectors
Calculate vector magnitude
 A mail-delivery robot
needs to get from where it
is to the mail bin on the
map.
 Find a sequence of two
displacement vectors that
will allow the robot to
avoid hitting the desk in
the middle.
Projectile Motion and the Velocity
Vector
 Any object that is
moving through the air
affected only by gravity
is called a projectile.
 The path a projectile
follows is called its
trajectory.
Projectile Motion and the Velocity
Vector
 The trajectory of a
thrown basketball
follows a special type of
arch-shaped curve called
a parabola.
 The distance a projectile
travels horizontally is
called its range.
Projectile Motion and the Velocity
Vector
 The velocity vector (v) is a way
to precisely describe the speed
and direction of motion.
 There are two ways to
represent velocity.
 Both tell how fast and in what
direction the ball travels.
Calculate magnitude
Draw the velocity vector v
= (5, 5) m/sec and
calculate the magnitude
of the velocity (the
speed), using the
Pythagorean theorem.
Components of the Velocity Vector
 Suppose a car is driving 20
meters per second.
 The direction of the vector
is 127 degrees.
 The polar representation
of the velocity is v = (20
m/sec, 127°).
Calculate velocity
 A soccer ball is kicked at a speed of 10 m/s and an angle
of 30 degrees.
 Find the horizontal and vertical components of the ball’s
initial velocity.
Adding Velocity Components
 Sometimes the total velocity of an object is a combination of
velocities.
 One example is the motion of a boat on a river.
 The boat moves with a certain velocity relative to the
water.
 The water is also moving with another velocity relative to
the land.
Adding Velocity Components
Calculate velocity components
 An airplane is moving at a velocity of 100 m/s in a direction 30
degrees NE relative to the air.
 The wind is blowing 40 m/s in a direction 45 degrees SE relative
to the ground.
 Find the resultant velocity of the airplane relative to the
ground.
Projectile Motion
Vx
 When we drop a ball
from a height we know
that its speed increases
as it falls.
 The increase in speed is
due to the acceleration
gravity, g = 9.8 m/sec2.
Vy
y
x
Horizontal Speed
 The ball’s horizontal velocity
remains constant while it
falls because gravity does not
exert any horizontal force.
 Since there is no force, the
horizontal acceleration is
zero (ax = 0).
 The ball will keep moving to
the right at 5 m/sec.
Horizontal Speed
 The horizontal distance a projectile moves can be
calculated according to the formula:
Vertical Speed
 The vertical speed (v y) of the
ball will increase by 9.8 m/sec
after each second.
 After one second has passed,
v y of the ball will be 9.8 m/sec.
 After the 2nd second has
passed, v y will be 19.6 m/sec
and so on.
Calculate using projectile motion
 A stunt driver steers a car off
a cliff at a speed of 20
meters per second.
 He lands in the lake below
two seconds later.
 Find the height of the cliff
and the horizontal distance
the car travels.
Projectiles Launched at an Angle
 A soccer ball kicked
off the ground is
also a projectile, but
it starts with an
initial velocity that
has both vertical and
horizontal
components.
*The launch angle determines how the initial velocity
divides between vertical (y) and horizontal (x) directions.
Steep Angle
 A ball launched at
a steep angle will
have a large
vertical velocity
component and a
small horizontal
velocity.
Shallow Angle
 A ball launched at
a low angle will
have a large
horizontal velocity
component and a
small vertical one.
Projectiles Launched at an Angle
The initial velocity components of an object launched at a velocity vo
and angle θ are found by breaking the velocity into x and y
components.
Range of a Projectile
 The range, or horizontal distance, traveled by a
projectile depends on the launch speed and the launch
angle.
Range of a Projectile
 The range of a projectile is calculated from the
horizontal velocity and the time of flight.
Range of a Projectile
 A projectile travels farthest when launched at 45
degrees.
Range of a Projectile
 The vertical velocity is responsible for giving the
projectile its "hang" time.
"Hang Time"
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You can easily calculate your own hang time.
Run toward a doorway and jump as high as you can, touching the wall or door frame.
Have someone watch to see exactly how high you reach.
Measure this distance with a meter stick.
The vertical distance formula can be rearranged to solve for time:
Projectile Motion and the Velocity
Vector
Key Question:
Can you predict the landing spot of a projectile?
*Students read Section 7.2 BEFORE Investigation 7.2
Marble’s Path
Vx
t=?
Vy
y
x=?
In order to solve “x” we must know
“t”
Y = vot – ½ g t2
vot = 0 (zero)
Y = ½ g t2
2y = g t2
t2 = 2y
g
t = 2y
g
Forces in Two Dimensions
 Force is also represented in x-y components.
Force Vectors
 If an object is in
equilibrium, all of the
forces acting on it are
balanced and the net force
is zero.
 If the forces act in two
dimensions, then all of the
forces in the x-direction
and y-direction balance
separately.
Equilibrium and Forces
 It is much more difficult
for a gymnast to hold his
arms out at a 45-degree
angle.
 To see why, consider that
each arm must still
support 350 newtons
vertically to balance the
force of gravity.
Forces in Two Dimensions
 Use the y-component to find the total force in the
gymnast’s left arm.
Forces in Two Dimensions
 The force in the right arm must also be 495 newtons
because it also has a vertical component of 350 N.
Forces in Two Dimensions
 When the gymnast’s arms
are at an angle, only part of
the force from each arm is
vertical.
 The total force must be
larger because the vertical
component of force in each
arm must still equal half his
weight.
Forces and Inclined Planes
 An inclined plane is a straight surface, usually with
a slope.
 Consider a block sliding
down a ramp.
 There are three forces that
act on the block:
 gravity (weight).
 friction
 the reaction force
acting on the block.
Forces and Inclined Planes
 When discussing forces, the word “normal” means
“perpendicular to.”
 The normal force acting
on the block is the
reaction force from the
weight of the block
pressing against the
ramp.
Forces and Inclined Planes
 The normal force on
the block is equal
and opposite to the
component of the
block’s weight
perpendicular to the
ramp (Fy).
Forces and Inclined Planes
 The force parallel to
the surface (Fx) is
given by
Fx = mg sinθ.
Acceleration on a Ramp
 Newton’s second law can be used to calculate the
acceleration once you know the components of all the
forces on an incline.
 According to the second law:
Acceleration
(m/sec2)
a=F
m
Force (kg . m/sec2)
Mass (kg)
Acceleration on a Ramp
 Since the block can only accelerate along the ramp, the force that
matters is the net force in the x direction, parallel to the ramp.
 If we ignore friction, and substitute Newtons' 2nd Law, the net
force is:
Fx = m g sin θ
a= F
m
Acceleration on a Ramp
 To account for friction, the horizontal component of
acceleration is reduced by combining equations:
Fx = mg sin θ - m mg cos θ
Acceleration on a Ramp
 For a smooth surface, the coefficient of friction (μ) is
usually in the range 0.1 - 0.3.
 The resulting equation for acceleration is:
Calculate acceleration on a ramp
 A skier with a mass of 50 kg is on a hill making an angle of
20 degrees.
 The friction force is 30 N.
 What is the skier’s acceleration?
Vectors and Direction
Key Question:
How do forces balance
in two dimensions?