Chapter 4 Acceleration

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Chapter 7 Projectile motion
Two
Kinematics in Dimensions
2
Projectile Motion

• Projectile motion is motion in two directions
• Motion in the x-direction is independent of the
  y-direction
• Motion in the y-direction is independent of the
  x-direction

3
What is a Projectile?

• A projectile is any object that is placed into
  free flight and is being affected by gravity.
• The path of a projectile is called the
  trajectory.

Trajectory
4
More on Trajectories

• The trajectory of a projectile forms the shape of
  a parabola.

Parabola
5
Different Trajectories

• Depending on initial location and angle,
  projectiles can form different paths, all of
  which possess some parabolic shape.

This path is called a half parabola.
6
The Half Parabola

• An object is launched horizontally and allowed to
  fall.
• There is no vertical velocity at the start, v1y
  is zero.
• The time of flight is equal to the time it would
  take to drop the object from rest.

7
Different Trajectories

• Depending on initial location and angle,
  projectiles can form different paths, all of
  which still possess some parabolic shape.

This path is called a partial parabola.
8
Different Trajectories

• Depending on initial location and angle,
  projectiles can form different paths, all of
  which still possess some parabolic shape.

This path is called a full parabola.
Full Parabola
9
The Full Parabola

• The key to the full parabola is symmetry.
• Try to identify some points of symmetry.

Throw
10
Quick Comparison of Paths

• Half Parabola
• Launched Horizontally from some height
• Full Parabola
• Launched at an angle from ground level
• Symmetrical trajectory
• Partial Parabola
• Launched at an angle from some height

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Half Parabola Timing

• The time of flight of a half parabolic path is
  equal to that of simply dropping the object from
  the same height.

Horizontal velocity (vx) has no affect on flight
time because it is not affected by gravity.
12
Jill drops the yellow ball off of a cliff. What
happens to the ball? Does it have constant
velocity?
13
Now Jill drops the yellow ball and throws the red
ball horizontally. Which ball will hit the
ground first?
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Half Parabola Summary

• Objects must be dropped from some height d1y.
• The vertical reference point is usually the
  ground or floor.
• The time of projectiles flight is identical for
  that of simply dropping the object the same
  distance (straight down).
• Horizontal velocity remains constant in all
  projectile problems (v1x v2x).

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• A stone is thrown horizontally from the top of a
  78.4m high cliff at 5m/s.
• a) How long does it take to reach the bottom?

Var X Y Want
a
v1
v2
d1
d2
t
16

• A stone is thrown horizontally from the top of a
  78.4m high cliff at 5m/s.
• a) How long does it take to reach the bottom?

17

• A stone is thrown horizontally from the top of a
  78.4m high cliff at 5m/s.
• b) How far from the base does it land?

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• A stone is thrown horizontally from the top of a
  78.4m high cliff at 5m/s.
• c) What are the final vy and vx ?

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Sample Partial

• A cannon nestled in the side of a cliff (d1y
  65m) fires a cannon ball at 26 . How long
  until the ball splashes into the sea?

Fire
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Sample Problem

• A toy car is raced off a table (1.1m high) onto
  the floor below.
• How long did it take for the car to crash on the
  floor?

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Object launched horizontallyMulti media studio
What do you notice about the horizontal velocity
in each of the following animations?

• Horizontally launched projectiles
• The plane and the package
• The truck and the ball

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Projectile Motion Type
Not all object are launch horizontally
Objects can be launched at an angle
23

• Recall the trajectory of the golf ball when hit
  with a 3 iron.
• What would the trajectory of a 9 iron look like?
• The loft of the club changed the launch angle.

24

• Object 1 was launched at 60o
• Object 2 was launched at 30o

25

• Object 1 was launched from a 25m high cliff at 0o
• Object 2 was launched at 60o

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Projectile motion generator
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Analytical Vector Addition

• You will need to understand the basic trig
  functions.

SOH
CAH
hyp
opp
TOA
adj
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Initial Velocity Breakdown

• When an object is launched at some angle, its
  initial velocity (v1) can be broken down into two
  components.
• Horizontal Component (Vx)
• Vertical Component (Vy)
• What shape is formed?
• Consider also the launch angle (q).

Please Note horizontal and vertical components
are independent of one another. The only
commonality is time.
Right Triangle
q
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Initial Velocity Breakdown (Cont.)

• Consider the breakdown from the previous slide
  again.
• There are trigonometric relationships between the
  sides and angles of a right triangle.

q
30
Practicing Trig Functions

• Consider the triangle below.
• Solve for the unknown values.

Searching for x
Searching for y
31
Sample Velocity Breakdown

• A dart gun is fired at an angle of 30 with a
  muzzle velocity of 40m/s.
• Calculate the components of the velocity?

Horizontal Component (x)
Vertical Component (y)
q
Make sure your calculator is in Degree mode!
32
Object launched at an angleMulti media studio

• Parabolic motion of projectiles
• Non-Horizontally launched projectiles
• Maximum Range
• Monkey and the Zoo Keeper

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• Given an initial velocity of 40m/s and an angle
  of 25
• find v1x v1y

Searching for y
Searching for x
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Sample Full Parabola Problem

• A golf ball is struck at an angle of q 36 with
  the horizontal at a velocity of 45m/s.
• What are the components of the velocity (v1x and
  v1y)?

Horizontal Component
Strike
Vertical Component
q
35
Practice Problems

• Homework
• WS 7b 1-2
• WS 7c 1-3

36
X Y are Independent
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Problem Solving Strategies

• Solve for the horizontal component Vxi
• Use trig functions
• Solve for the vertical component Vyi
• Use trig functions
• Solve each direction (x y) separately
• Symmetry can be used when the launching landing
  places are the same height.

38
A football player kicks a ball at 27m/s at an
angle of 30.

• Find the hang time
• find the horizontal distance the ball travels.
• The maximum height of the ball.

Horizontal Horizontal Horizontal Vertical Vertical Vertical
V K W V K W
a g -9.8
V1x v1y
v2x v2y
dx dx dy dy
t t t
39
Problem Solving Strategies

• Step 1 Solve for the horizontal and vertical
  components (V1x V1y )

V27m/s
V1y ?m/s
V1x ?m/s
Searching for y
Searching for x
40
Problem Solving Strategies

• Symmetry can be used when the launching landing
  places are the same height.

Vy
15.0m/s
12.5m/s
10.0m/s
7.50m/s
5.00m/s
2.50m/s
0.00m/s
41
A football player kicks a ball at 27m/s at an
angle of 30.

• Find the hang time
• find the horizontal distance the ball travels.
• The maximum height of the ball.

Horizontal Horizontal Horizontal Vertical Vertical Vertical
V K W V K W
a g -9.8
v1x 23.4 V1y 13.5
v2x 23.4 v2y
dx dx dy dy
t t t
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A football player kicks a ball at 27m/s at an
angle of 30.

• Find the hang time

Symmetry
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A football player kicks a ball at 27m/s at an
angle of 30.
b) Find the horizontal distance
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A football player kicks a ball at 27m/s at an
angle of 30.
c) Find the maximum height
What is true about the vertical velocity at the
maximum height?
Vy
Vy0m/s
15.0m/s
12.5m/s
10.0m/s
7.50m/s
5.00m/s
2.50m/s
0.00m/s
45

• A football player kicks a ball at 27m/s at an
  angle of 30. Find the max height

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An arrow is shot at 44m/s at an angle of 60

• Find the maximum height of the arrow.
• Find the horizontal distance the arrows travels.
• Find the hang time

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Problem Solving Strategies

• Step 1 Solve for the horizontal and vertical
  components (V1x V1y )

V44m/s
V1y ?m/s
V1x ?m/s
Searching for y
Searching for x
48
An arrow is shot at 44m/s at an angle of 60

• Find the hang time
• find the horizontal distance the arrows travels.
• c) The maximum height of the arrow.

Horizontal Horizontal Horizontal Vertical Vertical Vertical
V K W V K W
a g -9.8
v1x 22 v1y 38.1
v2x v2y
dx dx dy dy
t t t
49
A football player kicks a ball at 44m/s at an
angle of 60.

• Find the hang time

Symmetry
50
A football player kicks a ball at 44m/s at an
angle of 60.
b) Find the horizontal distance
51

• A football player kicks a ball at 44m/s at an
  angle of 60. Find the max height

Recall, vy0 at dy max
52
Practice Problems

• WS 7c
• 4
• WS 7d
• 1-3

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Concept Questions

• A stone is thrown horizontally from a cliff.
• How would the x distance change if the stone was
  thrown twice as fast?

• dx distance would double

54
Concept Questions

• How would the v2y change if the stone was thrown
  twice as fast?
• Vx and Vy are independent
• Since it was thrown horizontally, it would not
  change.

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Concept Questions

• How would the v2y change if the cliff was twice
  as high?

56
The Partial Parabola

• Recall, this path has elevation and launch angle.
• The trajectory again has an apex.
• This is mathematically the most complex path.

Fire
57
Sample Partial Parabola Problem

• A cannon nestled in the side of a cliff (d1y
  20m) fires a cannonball at 130m/s at a 30 angle.

Find v1x v1y Find y max Find t1 to apex Find t2
apex to ground (half parabola) Find total
time Find x max
Fire
58
Sample Partial Parabola Problem

• A cannon nestled in the side of a cliff (d1y
  20m) fires a cannonball at 130m/s at a 40 angle.

Fire
59
Partial Parabola
Variable x y
a
v1
v2
d1
d2
t

• You are trying to win a prize by throwing an
  apple into a basket on top of a pedestal.
• The apple leaves your hand 1.00 m beneath the top
  of the pedestal.
• The apple flies 3.09 m horizontally before
  landing in the bottom of the basket.
• The apples maximum height was 1.26 m.
• What was the apples initial velocity (magnitude
  and direction)?

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Partial Parabola

• Find V1y knowing Dy max 1.26m

Dy (max) 1.26m
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Partial Parabola

• Find the hang time knowing v1y and ending height
  d2y1m

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Partial Parabola

• Partial parabolas can be represented by two half
  parabolas
• Total hang time is the time of the ½ parabola on
  the way up plus the time of the ½ parabola on the
  way down.

Down
Up
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The Partial Parabola

• If you look at this path carefully, you can see
  two half parabolas, which simplifies things
  considerably.
• You still must consider the launch angle and the
  components of the velocity when trying to solve.

1
2
64
Practice Problems

• WS 7c
• 1-6
• 8-9

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Mortar Problems
V x y
a
v1
v2
d1
d2
t

• A mortar crew fires a projectile at an enemy
  ammunitions storage facility that is protected by
  a wall located on top of a 200.0 m high cliff.
• The ammunition is located a horizontal distance
  of 314.68 m from the mortars position.
• The projectile passes directly over the wall at
  its maximum height of 215.24 m.
• What was the projectiles initial velocity
  (magnitude and direction)?

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Mortar Problems
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EndChapter 7 Projectile motion
Kinematics in Dimensions
Two
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Simple Harmonic Motion

• Simple Harmonic Motion Motion caused by a linear
  restoring force that has a period independent of
  amplitude.
• Period The time required to repeat one complete
  cycle
• Amplitude Maximum displacement from equilibrium.

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1
11
2
10
3
9
4
8
5
7
6
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Simple Harmonic Motion
Periodic Motion is a form of Simple Harmonic
Motion
Restoring force
Restoring force
Equilibrium Position
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Conservation of Energy
PE
KE
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Var Known Want
a 9.8m/s2
vyi
vyf
d1 44m
d2 0m
t t

• A stone is thrown horizontally from the top of a
  44m high cliff at 15m/s.
• a) How long does it take to reach the bottom?

73

• A stone is thrown horizontally from the top of a
  44m high cliff at 15m/s.
• a) How long does it take to reach the bottom?

74

• A stone is thrown horizontally from the top of a
  44m high cliff at 15m/s.
• b) How far from the base does it land?

75
Acceleration due to Gravity
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Free Fall

• If you drop a book and a piece of paper which
  will hit the floor first?

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Acceleration due to Gravity
Simply replace a with g.
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Acceleration due to Gravity
Gravity Time Instant speed Average speed Distance
10m/s2 0s 0m/s 0m/s 0m
10m/s2 1s 10m/s 5m/s 5m
10m/s2 2s 20m/s 10m/s 20m
10m/s2 3s 30m/s 15m/s 45m
10m/s2 4s 40m/s 20m/s 80m
10m/s2 5s 50m/s 25m/s ½ gt2
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Graph shapes

• What geometric shape would that object have on a
  position versus time graph after experiencing
  acceleration?

80

• Given an initial velocity of 30m/s and an angle
  of 35
• find v1x v1y

Searching for y
Searching for x
V130m/s
v1y
35o
v1x