Quick Review:
Four Kinematic Equations
Free Fall
Four Kinematic Equations
 Constant
acceleration - an object will change
its velocity by the same amount each second.
 You must have constant acceleration to use the
four kinematic equations.
 Δx = ½(vi + vf) Δt
 vf = vi + a Δt
 Δx = vi Δt + ½ a(Δt)2
 vf2 = vi2 + 2 a Δx
Four Kinematic Equations
There are always 4 variables
 To use these equations you guess and check.
 Remember to always do 4 things:

1.
2.
3.
4.

Draw a diagram
Write what you know
Write what you need
Guess and check
Let’s practice…
Free Fall
Is when an object is falling under the sole
influence of gravity
 known as “acceleration due to gravity” = g
 g = 9.81m/s2
 There are slight variations that are affected
by altitude, we will ignore this.

g
Free Fall
is independent of 3 things:
 time
it’s been falling
 mass of the object
 if it started at rest or not
 Terminal Velocity – speed when the
force of air resistance is equal and
opposite to the force of gravity.
Working Backwards
It all works backward as well.
 If a ball is thrown straight up:

It will decelerate at 9.81m/s2
 At the top of it’s path the ball “hangs” in
mid air.
 At bottom of it’s path the balls velocity is
equal to vi


See Diagram….
Part 1.
Motion of Objects Projected
Horizontally
Introduction
 Projectile
Motion:
Motion through the air without a propulsion
 Examples:
Projectile Motion
 Keep
it simple by considering motion close to the
surface of the earth for the time being
 Neglect air resistance to make it simpler
Projectiles
A
projectile has only one
force acting upon - the
force of gravity
 Examples: golf, soccer
ball, bullet, rock dropped,
javelin thrower …
Factors Influencing
Projectile Trajectory
Trajectory: the flight
path of a projectile
 Angle of projection
 Projection speed
 Relative height of
projection
Factors Influencing
Projectile Trajectory
Angle of Projection
 General shapes
 Perfectly
vertical
 Parabolic
 Perfectly
horizontal
 Implications
in
sports
 Air resistance may
cause irregularities
Factors Influencing
Projectile Trajectory
Projection speed:
 Range:
o
horizontal displacement.
 For
oblique projection angles, speed
determines height and range.
 For vertical projection angle, speed
determines height.
Factors Influencing
Projectile Trajectory
Relative Projection
Height:
 Difference
between
projection and landing
height
 Greater the relative
projection height, longer
the flight time, greater the
displacement.
Projectile Motion
 The
path (trajectory) of a projectile is a parabola
 Describe the motion of an object in TWO
dimensions
 Vertical
- vY
 Horizontal - vX
 Horizontal
(90°)
and vertical motion are independent
Projectile Motion
 Horizontal
 Motion
of a ball rolling freely along a level surface
 Horizontal velocity is ALWAYS constant
 The horizontal component of it’s velocity does not
change. vX is constant
Projectile Motion
 Vertical
 Motion
of a freely falling object
 Force due to gravity
 Vertical component of velocity changes with time
Package drop



The package follows a parabolic path and remains directly below the
plane at all times
The vertical velocity changes (faster, faster)
The horizontal velocity is constant!
Trajectory and Range
 Maximum
range
is at 45°
 Low and high
trajectory cover
the same
distance.
 30 and 60
 10 and 80
 25 and…
The path (trajectory)
of a projectile is a parabola
Parabolic motion of a projectile
y
v0
x
y
x
y
x
y
x
y
x
y
•
y-motion is accelerated
•
Acceleration is constant,
and downward
•
g = -9.81m/s2
a = g = -9.81m/s2
•
The horizontal (x)
component of velocity is
constant
•
The horizontal and
vertical motions are
independent of each
other, but they have a
common time x
Experiment
What do you think? Which ball will hit the ground first?
a)
b)
c)
The left ball will hit first
The right ball will hit first
They will hit the ground at the same time.
Projectiles
Both balls hit the ground at the same time.
Why?
As soon as both balls are released by the
launcher, they are in "freefall.
The only force acting on both objects is gravity.
Both objects accelerate at the same rate, 9.8m/s2
Both objects covering the same distance at the
same rate and therefore hit the ground at the
same time
Equations

X- Component
x  v

xi
t
Y- Component
 y  v yi  t 

2
2
2
at
2
v yf  v yi  2 a y
v yf  v yi  at

1
Note: g=
9.8 m/s^2
ANALYSIS OF MOTION
ASSUMPTIONS:
•
x-direction (horizontal):
uniform motion
•
y-direction (vertical):
accelerated motion
•
no air resistance
QUESTIONS:
•
What is the trajectory?
•
What is the total time of the motion?
•
What is the horizontal range?
•
What is the final velocity?
•
What is the initial velocity?
Example: Projectiles launched horizontally
 What is the total time of the motion?
 What is the horizontal range?
 What is the final velocity?
 What is the initial velocity?
 The Royal Gorge Bridge in Colorado rises 321 m above
the Arkansas River. Suppose you kick a rock horiaontally
off the bridge. The magnitude of the rock’s horizontal
displacement is 45m How long does it take the rock to hit
the ground? What speed did you have to initially have to
kick the rock? How fast was the rock going before hitting
the ground?

Example: Projectiles launched horizontally
 What is the total time of the motion?
 What is the horizontal range?
 What is the final velocity?
 What is the initial velocity?


People in movies often jump from buildings into pools. If
a person jumps horizontally from the 10th floor(30m) to a
pool that is 5m away from the building, how long does it
take for him to hit the water in the pool? What initial
speed must the person jump to make it? What is the final
velocity of the person before he hits the water’s surface.
Let’s try pg 99 practice D
Board Work
1. Erica kicks a soccer ball 12 m/s at horizontally from the
edge of the roof of a building which is 30.0 m high.
2. A ball thrown horizontally from the roof of a building
lands 36m from the base of the building. Just before
impact the ball had a velocity of 25m/s.
3. A boy kicked a can horizontally from a 6.5 m high rock
with a speed of 4.0 m/s.
4.A car drives straight off the edge of a cliff that is 54 m
high. The police at the scene of the accident note that
the point of impact is 130 m from the base of the cliff.

Part 2.
Motion of objects projected at an
angle
y
vi
Initial position: x = 0, y = 0
Initial velocity: vi = vi [Θ]
viy
Velocity components:
x- direction : vix = vi cos Θ
θ
y- direction : viy = vi sin Θ
x
vix
y
a =g=
- 9.81m/s2
• Motion is accelerated
• Acceleration is constant, and
downward
•
a = g = -9.81m/s2
• The horizontal (x) component of
velocity is constant
• The horizontal and vertical
motions are independent of each
other, but they have a common
time
x
ANALYSIS OF MOTION:
ASSUMPTIONS
•
x-direction (horizontal):
uniform motion
•
y-direction (vertical):
accelerated motion
•
no air resistance
QUESTIONS
•
What is the trajectory?
•
What is the total time of the motion?
•
What is the horizontal range?
•
What is the maximum height?
•
What is the final velocity?
Equations of motion:
X
Uniform motion
ax = 0
Y
Accelerated motion
ay = g = -9.81 m/s2
VELOCITY
vx = vi cos Θ
vy = vi sin Θ + a t
DISPLACEMENT
Δx = vi cos Θ t
Δy = vi sin Θ t + ½ a t2
ACCELERATION
Equations


X- Component
x  v cos  t
i
Y- Component
 y  (v i sin  ) t 

2
1
2
at
2
v yf  (v i sin  )  2 a y
2
v yf  (v i sin  )  at
Example: Projectiles launched @ an angle
Erica kicks a soccer ball 12 m/s at an angle of 40
degrees above the horizontal.
*Don’t forget to draw your chart*
What are the x and y components of the vi?
How long does it take the ball to hit the ground?
What is the max height the ball travels?
How far does she kick the ball?
Example: Projectiles launched @ an angle
An archer needs to be sure to shoot over the wall of the
castle. He raises his bow at an angle of 65° and fires
his arrow with an initial velocity of 43m/s.
*Don’t forget to draw your chart*
What are the x and y components of the vi?
How long does it take the arrow to hit the ground?
What is the max height the arrow travels?
How far does the archer shoot the arrow?
Projectile Motion – Final Equations
(0,0) – initial position, vi = vi [Θ]– initial velocity, g = -9.81m/s2
Trajectory
Total time
Horizontal range
Parabola, open down
Δt =
Δx =
2 vi sin Θ
(-g)
vi 2 sin (2 Θ)
(-g)
vi2 sin2 Θ
Max height
hmax =
2(-g)
PROJECTILE MOTION - SUMMARY
 Projectile
motion is motion with a constant
horizontal velocity combined with a constant
vertical acceleration
 The projectile moves along a parabola
The monkey and the zookeeper!!
A
golfer practices driving balls off a cliff and into
the water below. The dege of the cliff is 15m
above the water. If the golf ball is launched at
51m/s at and angle of 15°, how far does the ball
travel horizontally before hitting the water?
The monkey and the zookeeper!!
A
zookeeper finds an escaped monkey hanging
from a light pole. Aiming her tranquilizer gun at
the monkey, she kneels 10m away from the light
pole, which is 5m high. The tip of her gun is 1m
above the ground. At the same moment that
monkey drops a banana, the zookeeper shoots. If
the dart travels at 50m/s, will the dart hit the
monkey, the banana, or neither one?
PROJECTILE MOTION - SUMMARY
Review for Test 2
 Pg 109 # 2, 3, 6, 12, 13, 14, 15, 17, 18, 20, 21, 24,
25, 27, 28, 30, 31, 32, 34, 37
 Pg 69 # 18, 20, 22, 24, 26, 30, 31, 33, 35, 38, 39,
46