  Dan Fullerton

Projectile Motion

Slide Duration:

Section 1: Introduction
What is Physics?

7m 12s

Intro
0:00
Objectives
0:11
What is Physics?
0:27
Why?
0:50
0:51
Matter
1:27
Matter
1:28
Mass
1:43
Inertial Mass
1:50
Gravitational Mass
2:13
A Spacecraft's Mass
3:03
What is the Mass of the Spacecraft?
3:05
Energy
3:37
Energy
3:38
Work
3:45
Putting Energy and Work Together
3:50
Mass-Energy Equivalence
4:15
Relationship between Mass & Energy: E = mc²
4:16
Source of Energy on Earth
4:47
The Study of Everything
5:00
Physics is the Study of Everything
5:01
Mechanics
5:29
Topics Covered
5:30
Topics Not Covered
6:07
Next Steps
6:44
Three Things You'd Like to Learn About in Physics
6:45
Math Review

1h 51s

Intro
0:00
Objectives
0:10
Vectors and Scalars
1:06
Scalars
1:07
Vectors
1:27
Vector Representations
2:00
Vector Representations
2:01
2:54
2:55
Graphical Vector Subtraction
5:36
Graphical Vector Subtraction
5:37
Vector Components
7:12
Vector Components
7:13
Angle of a Vector
8:56
tan θ
9:04
sin θ
9:25
cos θ
9:46
Vector Notation
10:10
Vector Notation 1
10:11
Vector Notation 2
12:59
Example I: Magnitude of the Horizontal & Vertical Component
16:08
Example II: Magnitude of the Plane's Eastward Velocity
17:59
Example III: Magnitude of Displacement
19:33
Example IV: Total Displacement from Starting Position
21:51
Example V: Find the Angle Theta Depicted by the Diagram
26:35
Vector Notation, cont.
27:07
Unit Vector Notation
27:08
Vector Component Notation
27:25
Vector Multiplication
28:39
Dot Product
28:40
Cross Product
28:54
Dot Product
29:03
Dot Product
29:04
Defining the Dot Product
29:26
Defining the Dot Product
29:27
Calculating the Dot Product
29:42
Unit Vector Notation
29:43
Vector Component Notation
30:58
Example VI: Calculating a Dot Product
31:45
Example VI: Part 1 - Find the Dot Product of the Following Vectors
31:46
Example VI: Part 2 - What is the Angle Between A and B?
32:20
Special Dot Products
33:52
Dot Product of Perpendicular Vectors
33:53
Dot Product of Parallel Vectors
34:03
Dot Product Properties
34:51
Commutative
34:52
Associative
35:05
Derivative of A * B
35:24
Example VII: Perpendicular Vectors
35:47
Cross Product
36:42
Cross Product of Two Vectors
36:43
Direction Using the Right-hand Rule
37:32
Cross Product of Parallel Vectors
38:04
Defining the Cross Product
38:13
Defining the Cross Product
38:14
Calculating the Cross Product Unit Vector Notation
38:41
Calculating the Cross Product Unit Vector Notation
38:42
Calculating the Cross Product Matrix Notation
39:18
Calculating the Cross Product Matrix Notation
39:19
Example VII: Find the Cross Product of the Following Vectors
42:09
Cross Product Properties
45:16
Cross Product Properties
45:17
Units
46:41
Fundamental Units
46:42
Derived units
47:13
Example IX: Dimensional Analysis
47:21
Calculus
49:05
Calculus
49:06
Differential Calculus
49:49
Differentiation & Derivative
49:50
Example X: Derivatives
51:21
Integral Calculus
53:03
Integration
53:04
Integral
53:11
Integration & Derivation are Inverse Functions
53:16
Determine the Original Function
53:37
Common Integrations
54:45
Common Integrations
54:46
Example XI: Integrals
55:17
Example XII: Calculus Applications
58:32
Section 2: Kinematics
Describing Motion I

23m 47s

Intro
0:00
Objectives
0:10
Position / Displacement
0:39
Object's Position
0:40
Position Vector
0:45
Displacement
0:56
Position & Displacement are Vectors
1:05
Position & Displacement in 1 Dimension
1:11
Example I: Distance & Displacement
1:21
Average Speed
2:14
Average Speed
2:15
Average Speed is Scalar
2:27
Average Velocity
2:39
Average Velocity
2:40
Average Velocity is a Vector
2:57
Example II: Speed vs. Velocity
3:16
Example II: Deer's Average Speed
3:17
Example II: Deer's Average Velocity
3:48
Example III: Chuck the Hungry Squirrel
4:21
Example III: Chuck's Distance Traveled
4:22
Example III: Chuck's Displacement
4:43
Example III: Chuck's Average Speed
5:25
Example III: Chuck's Average Velocity
5:39
Acceleration
6:11
Acceleration: Definition & Equation
6:12
Acceleration: Units
6:19
Relationship of Acceleration to Velocity
6:52
Example IV: Acceleration Problem
7:05
The Position Vector
7:39
The Position Vector
7:40
Average Velocity
9:35
Average Velocity
9:36
Instantaneous Velocity
11:20
Instantaneous Velocity
11:21
Instantaneous Velocity is the Derivative of Position with Respect to Time
11:35
Area Under the Velocity-time Graph
12:08
Acceleration
12:36
More on Acceleration
12:37
Average Acceleration
13:11
Velocity vs. Time Graph
13:14
Graph Transformations
13:59
Graphical Analysis of Motion
14:00
Velocity and acceleration in 2D
14:35
Velocity Vector in 2D
14:39
Acceleration Vector in 2D
15:26
Polynomial Derivatives
16:10
Polynomial Derivatives
16:11
Example V: Polynomial Kinematics
16:31
Example VI: Velocity Function
17:54
Example VI: Part A - Determine the Acceleration at t=1 Second
17:55
Example VI: Part B - Determine the Displacement between t=0 and t=5 Seconds
18:33
Example VII: Tortoise and Hare
20:14
Example VIII: d-t Graphs
22:40
Describing Motion II

36m 47s

Intro
0:00
Objectives
0:09
Special Case: Constant Acceleration
0:31
Constant Acceleration & Kinematic Equations
0:32
Deriving the Kinematic Equations
1:28
V = V₀ + at
1:39
∆x = V₀t +(1/2)at²
2:03
V² = V₀² +2a∆x
4:05
Problem Solving Steps
7:02
Step 1
7:13
Step 2
7:18
Step 3
7:27
Step 4
7:30
Step 5
7:31
Example IX: Horizontal Kinematics
7:38
Example X: Vertical Kinematics
9:45
Example XI: 2 Step Problem
11:23
Example XII: Acceleration Problem
15:01
Example XIII: Particle Diagrams
15:57
Example XIV: Particle Diagrams
17:36
18:46
Free Fall
22:56
Free Fall
22:57
Air Resistance
23:24
Air Resistance
23:25
Acceleration Due to Gravity
23:48
Acceleration Due to Gravity
23:49
Objects Falling From Rest
24:18
Objects Falling From Rest
24:19
Example XVI: Falling Objects
24:55
Objects Launched Upward
26:01
Objects Launched Upward
26:02
Example XVII: Ball Thrown Upward
27:16
Example XVIII: Height of a Jump
27:48
Example XIX: Ball Thrown Downward
31:10
Example XX: Maximum Height
32:27
Example XXI: Catch-Up Problem
33:53
Example XXII: Ranking Max Height
35:52
Projectile Motion

30m 34s

Intro
0:00
Objectives
0:07
What is a Projectile?
0:28
What is a Projectile?
0:29
Path of a Projectile
0:58
Path of a Projectile
0:59
Independence of Motion
2:45
Vertical & Horizontal Motion
2:46
Example I: Horizontal Launch
3:14
Example II: Parabolic Path
7:20
Angled Projectiles
8:01
Angled Projectiles
8:02
Example III: Human Cannonball
10:05
Example IV: Motion Graphs
14:39
Graphing Projectile Motion
19:05
Horizontal Equation
19:06
Vertical Equation
19:46
Example V: Arrow Fired from Tower
21:28
Example VI: Arrow Fired from Tower
24:10
Example VII: Launch from a Height
24:40
Example VIII: Acceleration of a Projectile
29:49
Circular & Relative Motion

30m 24s

Intro
0:00
Objectives
0:08
0:32
Degrees
0:35
0:40
1:08
Example I: Part A - Convert 90 Degrees to Radians
1:09
Example I: Part B - Convert 6 Radians to Degrees
2:08
Linear vs. Angular Displacement
2:38
Linear Displacement
2:39
Angular Displacement
2:52
Linear vs. Angular Velocity
3:18
Linear Velocity
3:19
Angular Velocity
3:25
Direction of Angular Velocity
4:36
Direction of Angular Velocity
4:37
Converting Linear to Angular Velocity
5:05
Converting Linear to Angular Velocity
5:06
Example II: Earth's Angular Velocity
6:12
Linear vs. Angular Acceleration
7:26
Linear Acceleration
7:27
Angular Acceleration
7:32
Centripetal Acceleration
8:05
Expressing Position Vector in Terms of Unit Vectors
8:06
Velocity
10:00
Centripetal Acceleration
11:14
Magnitude of Centripetal Acceleration
13:24
Example III: Angular Velocity & Centripetal Acceleration
14:02
Example IV: Moon's Orbit
15:03
Reference Frames
17:44
Reference Frames
17:45
Laws of Physics
18:00
Motion at Rest vs. Motion at a Constant Velocity
18:21
Motion is Relative
19:20
Reference Frame: Sitting in a Lawn Chair
19:21
Reference Frame: Sitting on a Train
19:56
Calculating Relative Velocities
20:19
Calculating Relative Velocities
20:20
Example: Calculating Relative Velocities
20:57
Example V: Man on a Train
23:19
Example VI: Airspeed
24:56
Example VII: 2-D Relative Motion
26:12
Example VIII: Relative Velocity w/ Direction
28:32
Section 3: Dynamics
Newton's First Law & Free Body Diagrams

23m 57s

Intro
0:00
Objectives
0:11
Newton's 1st Law of Motion
0:28
Newton's 1st Law of Motion
0:29
Force
1:16
Definition of Force
1:17
Units of Force
1:20
How Much is a Newton?
1:25
Contact Forces
1:47
Field Forces
2:32
What is a Net Force?
2:53
What is a Net Force?
2:54
What Does It Mean?
4:35
What Does It Mean?
4:36
Objects at Rest
4:52
Objects at Rest
4:53
Objects in Motion
5:12
Objects in Motion
5:13
Equilibrium
6:03
Static Equilibrium
6:04
Mechanical Equilibrium
6:22
Translational Equilibrium
6:38
Inertia
6:48
Inertia
6:49
Inertial Mass
6:58
Gravitational Mass
7:11
Example I: Inertia
7:40
Example II: Inertia
8:03
Example III: Translational Equilibrium
8:25
Example IV: Net Force
9:19
Free Body Diagrams
10:34
Free Body Diagrams Overview
10:35
Falling Elephant: Free Body Diagram
10:53
Free Body Diagram Neglecting Air Resistance
10:54
Free Body Diagram Including Air Resistance
11:22
Soda on Table
11:54
Free Body Diagram for a Glass of Soda Sitting on a Table
11:55
Free Body Diagram for Box on Ramp
13:38
Free Body Diagram for Box on Ramp
13:39
Pseudo- Free Body Diagram
15:26
Example V: Translational Equilibrium
18:35
Newton's Second & Third Laws of Motion

23m 57s

Intro
0:00
Objectives
0:09
Newton's 2nd Law of Motion
0:36
Newton's 2nd Law of Motion
0:37
Applying Newton's 2nd Law
1:12
Step 1
1:13
Step 2
1:18
Step 3
1:27
Step 4
1:36
Example I: Block on a Surface
1:42
Example II: Concurrent Forces
2:42
Mass vs. Weight
4:09
Mass
4:10
Weight
4:28
Example III: Mass vs. Weight
4:45
Example IV: Translational Equilibrium
6:43
Example V: Translational Equilibrium
8:23
Example VI: Determining Acceleration
10:13
Example VII: Stopping a Baseball
12:38
Example VIII: Steel Beams
14:11
Example IX: Tension Between Blocks
17:03
Example X: Banked Curves
18:57
Example XI: Tension in Cords
24:03
Example XII: Graphical Interpretation
27:13
Example XIII: Force from Velocity
28:12
Newton's 3rd Law
29:16
Newton's 3rd Law
29:17
Examples - Newton's 3rd Law
30:01
Examples - Newton's 3rd Law
30:02
Action-Reaction Pairs
30:40
Girl Kicking Soccer Ball
30:41
Rocket Ship in Space
31:02
Gravity on You
31:23
Example XIV: Force of Gravity
32:11
Example XV: Sailboat
32:38
Example XVI: Hammer and Nail
33:18
Example XVII: Net Force
33:47
Friction

20m 41s

Intro
0:00
Objectives
0:06
Coefficient of Friction
0:21
Coefficient of Friction
0:22
Approximate Coefficients of Friction
0:44
Kinetic or Static?
1:21
Sled Sliding Down a Snowy Hill
1:22
Refrigerator at Rest that You Want to Move
1:32
Car with Tires Rolling Freely
1:49
Car Skidding Across Pavement
2:01
Example I: Car Sliding
2:21
Example II: Block on Incline
3:04
Calculating the Force of Friction
3:33
Calculating the Force of Friction
3:34
Example III: Finding the Frictional Force
4:02
Example IV: Box on Wood Surface
5:34
Example V: Static vs. Kinetic Friction
7:35
Example VI: Drag Force on Airplane
7:58
Example VII: Pulling a Sled
8:41
Example VIII: AP-C 2007 FR1
13:23
Example VIII: Part A
13:24
Example VIII: Part B
14:40
Example VIII: Part C
15:19
Example VIII: Part D
17:08
Example VIII: Part E
18:24
Retarding & Drag Forces

32m 10s

Intro
0:00
Objectives
0:07
Retarding Forces
0:41
Retarding Forces
0:42
The Skydiver
1:30
Drag Forces on a Free-falling Object
1:31
Velocity as a Function of Time
5:31
Velocity as a Function of Time
5:32
Velocity as a Function of Time, cont.
12:27
Acceleration
12:28
Velocity as a Function of Time, cont.
15:16
Graph: Acceleration vs. Time
16:06
Graph: Velocity vs. Time
16:40
Graph: Displacement vs. Time
17:04
Example I: AP-C 2005 FR1
17:43
Example I: Part A
17:44
Example I: Part B
19:17
Example I: Part C
20:17
Example I: Part D
21:09
Example I: Part E
22:42
Example II: AP-C 2013 FR2
24:26
Example II: Part A
24:27
Example II: Part B
25:25
Example II: Part C
26:22
Example II: Part D
27:04
Example II: Part E
30:50
Ramps & Inclines

20m 31s

Intro
0:00
Objectives
0:06
Drawing Free Body Diagrams for Ramps
0:32
Step 1: Choose the Object & Draw It as a Dot or Box
0:33
Step 2: Draw and Label all the External Forces
0:39
Step 3: Sketch a Coordinate System
0:42
Example: Object on a Ramp
0:52
Pseudo-Free Body Diagrams
2:06
Pseudo-Free Body Diagrams
2:07
Redraw Diagram with All Forces Parallel to Axes
2:18
Box on a Ramp
4:08
Free Body Diagram for Box on a Ramp
4:09
Pseudo-Free Body Diagram for Box on a Ramp
4:54
Example I: Box at Rest
6:13
Example II: Box Held By Force
6:35
Example III: Truck on a Hill
8:46
Example IV: Force Up a Ramp
9:29
Example V: Acceleration Down a Ramp
12:01
Example VI: Able of Repose
13:59
Example VII: Sledding
17:03
Atwood Machines

24m 58s

Intro
0:00
Objectives
0:07
What is an Atwood Machine?
0:25
What is an Atwood Machine?
0:26
Properties of Atwood Machines
1:03
Ideal Pulleys are Frictionless and Massless
1:04
Tension is Constant
1:14
Setup for Atwood Machines
1:26
Setup for Atwood Machines
1:27
Solving Atwood Machine Problems
1:52
Solving Atwood Machine Problems
1:53
Alternate Solution
5:24
Analyze the System as a Whole
5:25
Example I: Basic Atwood Machine
7:31
Example II: Moving Masses
9:59
Example III: Masses and Pulley on a Table
13:32
Example IV: Mass and Pulley on a Ramp
15:47
Example V: Ranking Atwood Machines
19:50
Section 4: Work, Energy, & Power
Work

37m 34s

Intro
0:00
Objectives
0:07
What is Work?
0:36
What is Work?
0:37
Units of Work
1:09
Work in One Dimension
1:31
Work in One Dimension
1:32
Examples of Work
2:19
Stuntman in a Jet Pack
2:20
A Girl Struggles to Push Her Stalled Car
2:50
A Child in a Ghost Costume Carries a Bag of Halloween Candy Across the Yard
3:24
Example I: Moving a Refrigerator
4:03
Example II: Liberating a Car
4:53
Example III: Lifting Box
5:30
Example IV: Pulling a Wagon
6:13
Example V: Ranking Work on Carts
7:13
Non-Constant Forces
12:21
Non-Constant Forces
12:22
Force vs. Displacement Graphs
13:49
Force vs. Displacement Graphs
13:50
Hooke's Law
14:41
Hooke's Law
14:42
Determining the Spring Constant
15:38
Slope of the Graph Gives the Spring Constant, k
15:39
Work Done in Compressing the Spring
16:34
Find the Work Done in Compressing the String
16:35
Example VI: Finding Spring Constant
17:21
Example VII: Calculating Spring Constant
19:48
Example VIII: Hooke's Law
20:30
Example IX: Non-Linear Spring
22:18
Work in Multiple Dimensions
23:52
Work in Multiple Dimensions
23:53
Work-Energy Theorem
25:25
Work-Energy Theorem
25:26
Example X: Work-Energy Theorem
28:35
Example XI: Work Done on Moving Carts
30:46
Example XII: Velocity from an F-d Graph
35:01
Energy & Conservative Forces

28m 4s

Intro
0:00
Objectives
0:08
Energy Transformations
0:31
Energy Transformations
0:32
Work-Energy Theorem
0:57
Kinetic Energy
1:12
Kinetic Energy: Definition
1:13
Kinetic Energy: Equation
1:55
Example I: Frog-O-Cycle
2:07
Potential Energy
2:46
Types of Potential Energy
2:47
A Potential Energy Requires an Interaction between Objects
3:29
Internal energy
3:50
Internal Energy
3:51
Types of Energy
4:37
Types of Potential & Kinetic Energy
4:38
Gravitational Potential Energy
5:42
Gravitational Potential Energy
5:43
Example II: Potential Energy
7:27
Example III: Kinetic and Potential Energy
8:16
Example IV: Pendulum
9:09
Conservative Forces
11:37
Conservative Forces Overview
11:38
Type of Conservative Forces
12:42
Types of Non-conservative Forces
13:02
Work Done by Conservative Forces
13:28
Work Done by Conservative Forces
13:29
Newton's Law of Universal Gravitation
14:18
Gravitational Force of Attraction between Any Two Objects with Mass
14:19
Gravitational Potential Energy
15:27
Gravitational Potential Energy
15:28
Elastic Potential Energy
17:36
Elastic Potential Energy
17:37
Force from Potential Energy
18:51
Force from Potential Energy
18:52
Gravitational Force from the Gravitational Potential Energy
20:46
Gravitational Force from the Gravitational Potential Energy
20:47
Hooke's Law from Potential Energy
22:04
Hooke's Law from Potential Energy
22:05
Summary
23:16
Summary
23:17
Example V: Kinetic Energy of a Mass
24:40
Example VI: Force from Potential Energy
25:48
Example VII: Work on a Spinning Disc
26:54
Conservation of Energy

54m 56s

Intro
0:00
Objectives
0:09
Conservation of Mechanical Energy
0:32
Consider a Single Conservative Force Doing Work on a Closed System
0:33
Non-Conservative Forces
1:40
Non-Conservative Forces
1:41
Work Done by a Non-conservative Force
1:47
Formula: Total Energy
1:54
Formula: Total Mechanical Energy
2:04
Example I: Falling Mass
2:15
Example II: Law of Conservation of Energy
4:07
Example III: The Pendulum
6:34
Example IV: Cart Compressing a Spring
10:12
Example V: Cart Compressing a Spring
11:12
Example V: Part A - Potential Energy Stored in the Compressed Spring
11:13
Example V: Part B - Maximum Vertical Height
12:01
Example VI: Car Skidding to a Stop
13:05
Example VII: Block on Ramp
14:22
Example VIII: Energy Transfers
16:15
Example IX: Roller Coaster
20:04
Example X: Bungee Jumper
23:32
Example X: Part A - Speed of the Jumper at a Height of 15 Meters Above the Ground
24:48
Example X: Part B - Speed of the Jumper at a Height of 30 Meters Above the Ground
26:53
Example X: Part C - How Close Does the Jumper Get to the Ground?
28:28
Example XI: AP-C 2002 FR3
30:28
Example XI: Part A
30:59
Example XI: Part B
31:54
Example XI: Part C
32:50
Example XI: Part D & E
33:52
Example XII: AP-C 2007 FR3
35:24
Example XII: Part A
35:52
Example XII: Part B
36:27
Example XII: Part C
37:48
Example XII: Part D
39:32
Example XIII: AP-C 2010 FR1
41:07
Example XIII: Part A
41:34
Example XIII: Part B
43:05
Example XIII: Part C
45:24
Example XIII: Part D
47:18
Example XIV: AP-C 2013 FR1
48:25
Example XIV: Part A
48:50
Example XIV: Part B
49:31
Example XIV: Part C
51:27
Example XIV: Part D
52:46
Example XIV: Part E
53:25
Power

16m 44s

Intro
0:00
Objectives
0:06
Defining Power
0:20
Definition of Power
0:21
Units of Power
0:27
Average Power
0:43
Instantaneous Power
1:03
Instantaneous Power
1:04
Example I: Horizontal Box
2:07
Example II: Accelerating Truck
4:48
Example III: Motors Delivering Power
6:00
Example IV: Power Up a Ramp
7:00
Example V: Power from Position Function
8:51
Example VI: Motorcycle Stopping
10:48
Example VII: AP-C 2003 FR1
11:52
Example VII: Part A
11:53
Example VII: Part B
12:50
Example VII: Part C
14:36
Example VII: Part D
15:52
Section 5: Momentum
Momentum & Impulse

13m 9s

Intro
0:00
Objectives
0:07
Momentum
0:39
Definition of Momentum
0:40
Total Momentum
1:00
Formula for Momentum
1:05
Units of Momentum
1:11
Example I: Changing Momentum
1:18
Impulse
2:27
Impulse
2:28
Example II: Impulse
2:41
Relationship Between Force and ∆p (Impulse)
3:36
Relationship Between Force and ∆p (Impulse)
3:37
Example III: Force from Momentum
4:37
Impulse-Momentum Theorem
5:14
Impulse-Momentum Theorem
5:15
Example IV: Impulse-Momentum
6:26
Example V: Water Gun & Horizontal Force
7:56
Impulse from F-t Graphs
8:53
Impulse from F-t Graphs
8:54
Example VI: Non-constant Forces
9:16
Example VII: F-t Graph
10:01
Example VIII: Impulse from Force
11:19
Conservation of Linear Momentum

46m 30s

Intro
0:00
Objectives
0:08
Conservation of Linear Momentum
0:28
In an Isolated System
0:29
In Any Closed System
0:37
Direct Outcome of Newton's 3rd Law of Motion
0:47
Collisions and Explosions
1:07
Collisions and Explosions
1:08
The Law of Conservation of Linear Momentum
1:25
Solving Momentum Problems
1:35
Solving Momentum Problems
1:36
Types of Collisions
2:08
Elastic Collision
2:09
Inelastic Collision
2:34
Example I: Traffic Collision
3:00
Example II: Collision of Two Moving Objects
6:55
Example III: Recoil Velocity
9:47
Example IV: Atomic Collision
12:12
Example V: Collision in Multiple Dimensions
18:11
Example VI: AP-C 2001 FR1
25:16
Example VI: Part A
25:33
Example VI: Part B
26:44
Example VI: Part C
28:17
Example VI: Part D
28:58
Example VII: AP-C 2002 FR1
30:10
Example VII: Part A
30:20
Example VII: Part B
32:14
Example VII: Part C
34:25
Example VII: Part D
36:17
Example VIII: AP-C 2014 FR1
38:55
Example VIII: Part A
39:28
Example VIII: Part B
41:00
Example VIII: Part C
42:57
Example VIII: Part D
44:20
Center of Mass

28m 26s

Intro
0:00
Objectives
0:07
Center of Mass
0:45
Center of Mass
0:46
Finding Center of Mass by Inspection
1:25
For Uniform Density Objects
1:26
For Objects with Multiple Parts
1:36
For Irregular Objects
1:44
Example I: Center of Mass by Inspection
2:06
Calculating Center of Mass for Systems of Particles
2:25
Calculating Center of Mass for Systems of Particles
2:26
Example II: Center of Mass (1D)
3:15
Example III: Center of Mass of Continuous System
4:29
Example IV: Center of Mass (2D)
6:00
Finding Center of Mass by Integration
7:38
Finding Center of Mass by Integration
7:39
Example V: Center of Mass of a Uniform Rod
8:10
Example VI: Center of Mass of a Non-Uniform Rod
11:40
Center of Mass Relationships
14:44
Center of Mass Relationships
14:45
Center of Gravity
17:36
Center of Gravity
17:37
Uniform Gravitational Field vs. Non-uniform Gravitational Field
17:53
Example VII: AP-C 2004 FR1
18:26
Example VII: Part A
18:45
Example VII: Part B
19:38
Example VII: Part C
21:03
Example VII: Part D
22:04
Example VII: Part E
24:52
Section 6: Uniform Circular Motion
Uniform Circular Motion

21m 36s

Intro
0:00
Objectives
0:08
Uniform Circular Motion
0:42
Distance Around the Circle for Objects Traveling in a Circular Path at Constant Speed
0:51
Average Speed for Objects Traveling in a Circular Path at Constant Speed
1:15
Frequency
1:42
Definition of Frequency
1:43
Symbol of Frequency
1:46
Units of Frequency
1:49
Period
2:04
Period
2:05
Frequency and Period
2:19
Frequency and Period
2:20
Example I: Race Car
2:32
Example II: Toy Train
3:22
Example III: Round-A-Bout
4:07
Example III: Part A - Period of the Motion
4:08
Example III: Part B- Frequency of the Motion
4:43
Example III: Part C- Speed at Which Alan Revolves
4:58
Uniform Circular Motion
5:28
Is an Object Undergoing Uniform Circular Motion Accelerating?
5:29
Direction of Centripetal Acceleration
6:21
Direction of Centripetal Acceleration
6:22
Magnitude of Centripetal Acceleration
8:23
Magnitude of Centripetal Acceleration
8:24
Example IV: Car on a Track
8:39
Centripetal Force
10:14
Centripetal Force
10:15
Calculating Centripetal Force
11:47
Calculating Centripetal Force
11:48
Example V: Acceleration
12:41
Example VI: Direction of Centripetal Acceleration
13:44
Example VII: Loss of Centripetal Force
14:03
Example VIII: Bucket in Horizontal Circle
14:44
Example IX: Bucket in Vertical Circle
15:24
Example X: Demon Drop
17:38
Example X: Question 1
18:02
Example X: Question 2
18:25
Example X: Question 3
19:22
Example X: Question 4
20:13
Section 7: Rotational Motion
Rotational Kinematics

32m 52s

Intro
0:00
Objectives
0:07
0:35
Once Around a Circle: In Degrees
0:36
Once Around a Circle: In Radians
0:48
0:51
1:08
Example I: Convert 90° to Radians
1:09
Example I: Convert 6 Radians to Degree
1:23
Linear vs. Angular Displacement
1:43
Linear Displacement
1:44
Angular Displacement
1:51
Linear vs. Angular Velocity
2:04
Linear Velocity
2:05
Angular Velocity
2:10
Direction of Angular Velocity
2:28
Direction of Angular Velocity
2:29
Converting Linear to Angular Velocity
2:58
Converting Linear to Angular Velocity
2:59
Example II: Angular Velocity of Earth
3:51
Linear vs. Angular Acceleration
4:35
Linear Acceleration
4:36
Angular Acceleration
4:42
Example III: Angular Acceleration
5:09
Kinematic Variable Parallels
6:30
Kinematic Variable Parallels: Translational & Angular
6:31
Variable Translations
7:00
Variable Translations: Translational & Angular
7:01
Kinematic Equation Parallels
7:38
Kinematic Equation Parallels: Translational & Rotational
7:39
Example IV: Deriving Centripetal Acceleration
8:29
Example V: Angular Velocity
13:24
Example V: Part A
13:25
Example V: Part B
14:15
Example VI: Wheel in Motion
14:39
Example VII: AP-C 2003 FR3
16:23
Example VII: Part A
16:38
Example VII: Part B
17:34
Example VII: Part C
24:02
Example VIII: AP-C 2014 FR2
25:35
Example VIII: Part A
25:47
Example VIII: Part B
26:28
Example VIII: Part C
27:48
Example VIII: Part D
28:26
Example VIII: Part E
29:16
Moment of Inertia

24m

Intro
0:00
Objectives
0:07
Types of Inertia
0:34
Inertial Mass
0:35
Moment of Inertia
0:44
Kinetic Energy of a Rotating Disc
1:25
Kinetic Energy of a Rotating Disc
1:26
Calculating Moment of Inertia (I)
5:32
Calculating Moment of Inertia (I)
5:33
Moment of Inertia for Common Objects
5:49
Moment of Inertia for Common Objects
5:50
Example I: Point Masses
6:46
Example II: Uniform Rod
9:09
Example III: Solid Cylinder
13:07
Parallel Axis Theorem (PAT)
17:33
Parallel Axis Theorem (PAT)
17:34
Example IV: Calculating I Using the Parallel Axis Theorem
18:39
Example V: Hollow Sphere
20:18
Example VI: Long Thin Rod
20:55
Example VII: Ranking Moment of Inertia
21:50
Example VIII: Adjusting Moment of Inertia
22:39
Torque

26m 9s

Intro
0:00
Objectives
0:06
Torque
0:18
Definition of Torque
0:19
Torque & Rotation
0:26
Lever Arm ( r )
0:30
Example: Wrench
0:39
Direction of the Torque Vector
1:45
Direction of the Torque Vector
1:46
Finding Direction Using the Right-hand Rule
1:53
Newton's 2nd Law: Translational vs. Rotational
2:20
Newton's 2nd Law: Translational vs. Rotational
2:21
Equilibrium
3:17
Static Equilibrium
3:18
Dynamic Equilibrium
3:30
Example I: See-Saw Problem
3:46
Example II: Beam Problem
7:12
Example III: Pulley with Mass
10:34
Example IV: Net Torque
13:46
Example V: Ranking Torque
15:29
Example VI: Ranking Angular Acceleration
16:25
Example VII: Café Sign
17:19
Example VIII: AP-C 2008 FR2
19:44
Example VIII: Part A
20:12
Example VIII: Part B
21:08
Example VIII: Part C
22:36
Example VIII: Part D
24:37
Rotational Dynamics

56m 58s

Intro
0:00
Objectives
0:08
Conservation of Energy
0:48
Translational Kinetic Energy
0:49
Rotational Kinetic Energy
0:54
Total Kinetic Energy
1:03
Example I: Disc Rolling Down an Incline
1:10
Rotational Dynamics
4:25
Rotational Dynamics
4:26
Example II: Strings with Massive Pulleys
4:37
Example III: Rolling without Slipping
9:13
Example IV: Rolling with Slipping
13:45
Example V: Amusement Park Swing
22:49
Example VI: AP-C 2002 FR2
26:27
Example VI: Part A
26:48
Example VI: Part B
27:30
Example VI: Part C
29:51
Example VI: Part D
30:50
Example VII: AP-C 2006 FR3
31:39
Example VII: Part A
31:49
Example VII: Part B
36:20
Example VII: Part C
37:14
Example VII: Part D
38:48
Example VIII: AP-C 2010 FR2
39:40
Example VIII: Part A
39:46
Example VIII: Part B
40:44
Example VIII: Part C
44:31
Example VIII: Part D
46:44
Example IX: AP-C 2013 FR3
48:27
Example IX: Part A
48:47
Example IX: Part B
50:33
Example IX: Part C
53:28
Example IX: Part D
54:15
Example IX: Part E
56:20
Angular Momentum

33m 2s

Intro
0:00
Objectives
0:09
Linear Momentum
0:44
Definition of Linear Momentum
0:45
Total Angular Momentum
0:52
p = mv
0:59
Angular Momentum
1:08
Definition of Angular Momentum
1:09
Total Angular Momentum
1:21
A Mass with Velocity v Moving at Some Position r
1:29
Calculating Angular Momentum
1:44
Calculating Angular Momentum
1:45
Spin Angular Momentum
4:17
Spin Angular Momentum
4:18
Example I: Object in Circular Orbit
4:51
Example II: Angular Momentum of a Point Particle
6:34
Angular Momentum and Net Torque
9:03
Angular Momentum and Net Torque
9:04
Conservation of Angular Momentum
11:53
Conservation of Angular Momentum
11:54
Example III: Ice Skater Problem
12:20
Example IV: Combining Spinning Discs
13:52
Example V: Catching While Rotating
15:13
Example VI: Changes in Angular Momentum
16:47
Example VII: AP-C 2005 FR3
17:37
Example VII: Part A
18:12
Example VII: Part B
18:32
Example VII: Part C
19:53
Example VII: Part D
21:52
Example VIII: AP-C 2014 FR3
24:23
Example VIII: Part A
24:31
Example VIII: Part B
25:33
Example VIII: Part C
26:58
Example VIII: Part D
28:24
Example VIII: Part E
30:42
Section 8: Oscillations
Oscillations

1h 1m 12s

Intro
0:00
Objectives
0:08
Simple Harmonic Motion
0:45
Simple Harmonic Motion
0:46
Circular Motion vs. Simple Harmonic Motion (SHM)
1:39
Circular Motion vs. Simple Harmonic Motion (SHM)
1:40
Position, Velocity, & Acceleration
4:55
Position
4:56
Velocity
5:12
Acceleration
5:49
Frequency and Period
6:37
Frequency
6:42
Period
6:49
Angular Frequency
7:05
Angular Frequency
7:06
Example I: Oscillating System
7:37
Example I: Determine the Object's Angular Frequency
7:38
Example I: What is the Object's Position at Time t = 10s?
8:16
Example I: At What Time is the Object at x = 0.1m?
9:10
Mass on a Spring
10:17
Mass on a Spring
10:18
Example II: Analysis of Spring-Block System
11:34
Example III: Spring-Block ranking
12:53
General Form of Simple Harmonic Motion
14:41
General Form of Simple Harmonic Motion
14:42
Graphing Simple Harmonic Motion (SHM)
15:22
Graphing Simple Harmonic Motion (SHM)
15:23
Energy of Simple Harmonic Motion (SHM)
15:49
Energy of Simple Harmonic Motion (SHM)
15:50
Horizontal Spring Oscillator
19:24
Horizontal Spring Oscillator
19:25
Vertical Spring Oscillator
20:58
Vertical Spring Oscillator
20:59
Springs in Series
23:30
Springs in Series
23:31
Springs in Parallel
26:08
Springs in Parallel
26:09
The Pendulum
26:59
The Pendulum
27:00
Energy and the Simple Pendulum
27:46
Energy and the Simple Pendulum
27:47
Frequency and Period of a Pendulum
30:16
Frequency and Period of a Pendulum
30:17
Example IV: Deriving Period of a Simple Pendulum
31:42
Example V: Deriving Period of a Physical Pendulum
35:20
Example VI: Summary of Spring-Block System
38:16
Example VII: Harmonic Oscillator Analysis
44:14
Example VII: Spring Constant
44:24
Example VII: Total Energy
44:45
Example VII: Speed at the Equilibrium Position
45:05
Example VII: Speed at x = 0.30 Meters
45:37
Example VII: Speed at x = -0.40 Meter
46:46
Example VII: Acceleration at the Equilibrium Position
47:21
Example VII: Magnitude of Acceleration at x = 0.50 Meters
47:35
Example VII: Net Force at the Equilibrium Position
48:04
Example VII: Net Force at x = 0.25 Meter
48:20
Example VII: Where does Kinetic Energy = Potential Energy?
48:33
Example VIII: Ranking Spring Systems
49:35
Example IX: Vertical Spring Block Oscillator
51:45
Example X: Ranking Period of Pendulum
53:50
Example XI: AP-C 2009 FR2
54:50
Example XI: Part A
54:58
Example XI: Part B
57:57
Example XI: Part C
59:11
Example XII: AP-C 2010 FR3
1:00:18
Example XII: Part A
1:00:49
Example XII: Part B
1:02:47
Example XII: Part C
1:04:30
Example XII: Part D
1:05:53
Example XII: Part E
1:08:13
Section 9: Gravity & Orbits
Gravity & Orbits

34m 59s

Intro
0:00
Objectives
0:07
Newton's Law of Universal Gravitation
0:45
Newton's Law of Universal Gravitation
0:46
Example I: Gravitational Force Between Earth and Sun
2:24
Example II: Two Satellites
3:39
Gravitational Field Strength
4:23
Gravitational Field Strength
4:24
Example III: Weight on Another Planet
6:22
Example IV: Gravitational Field of a Hollow Shell
7:31
Example V: Gravitational Field Inside a Solid Sphere
8:33
Velocity in Circular Orbit
12:05
Velocity in Circular Orbit
12:06
Period and Frequency for Circular Orbits
13:56
Period and Frequency for Circular Orbits
13:57
Mechanical Energy for Circular Orbits
16:11
Mechanical Energy for Circular Orbits
16:12
Escape Velocity
17:48
Escape Velocity
17:49
Kepler's 1st Law of Planetary Motion
19:41
Keller's 1st Law of Planetary Motion
19:42
Kepler's 2nd Law of Planetary Motion
20:05
Keller's 2nd Law of Planetary Motion
20:06
Kepler's 3rd Law of Planetary Motion
20:57
Ratio of the Squares of the Periods of Two Planets
20:58
Ratio of the Squares of the Periods to the Cubes of Their Semi-major Axes
21:41
Total Mechanical Energy for an Elliptical Orbit
21:57
Total Mechanical Energy for an Elliptical Orbit
21:58
Velocity and Radius for an Elliptical Orbit
22:35
Velocity and Radius for an Elliptical Orbit
22:36
Example VI: Rocket Launched Vertically
24:26
Example VII: AP-C 2007 FR2
28:16
Example VII: Part A
28:35
Example VII: Part B
29:51
Example VII: Part C
31:14
Example VII: Part D
32:23
Example VII: Part E
33:16
Section 10: Sample AP Exam
1998 AP Practice Exam: Multiple Choice

28m 11s

Intro
0:00
Problem 1
0:30
Problem 2
0:51
Problem 3
1:25
Problem 4
2:00
Problem 5
3:05
Problem 6
4:19
Problem 7
4:48
Problem 8
5:18
Problem 9
5:38
Problem 10
6:26
Problem 11
7:21
Problem 12
8:08
Problem 13
8:35
Problem 14
9:20
Problem 15
10:09
Problem 16
10:25
Problem 17
11:30
Problem 18
12:27
Problem 19
13:00
Problem 20
14:40
Problem 21
15:44
Problem 22
16:42
Problem 23
17:35
Problem 24
17:54
Problem 25
18:32
Problem 26
19:08
Problem 27
20:56
Problem 28
22:19
Problem 29
22:36
Problem 30
23:18
Problem 31
24:06
Problem 32
24:40
1998 AP Practice Exam: Free Response Questions (FRQ)

28m 11s

Intro
0:00
Question 1
0:15
Part A: I
0:16
Part A: II
0:46
Part A: III
1:13
Part B
1:40
Part C
2:49
Part D: I
4:46
Part D: II
5:15
Question 2
5:46
Part A: I
6:13
Part A: II
7:05
Part B: I
7:48
Part B: II
8:42
Part B: III
9:03
Part B: IV
9:26
Part B: V
11:32
Question 3
13:30
Part A: I
13:50
Part A: II
14:16
Part A: III
14:38
Part A: IV
14:56
Part A: V
15:36
Part B
16:11
Part C
17:00
Part D: I
19:56
Part D: II
21:08
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• ## Related Books 1 answer Last reply by: Professor Dan FullertonWed Oct 23, 2019 12:53 PMPost by Cyrus Seyrafi on October 23, 2019For Example I: Horizontal Launch, why do we make 9.8 m/s/s positive? Would it mess with the results if we made it negative? 1 answer Last reply by: Professor Dan FullertonMon Sep 3, 2018 8:37 AMPost by Magic Fu on September 2, 2018Hi, Professor Fullerton. I have a homework problem I don't know how to solve.A rifle that shoots bullets at 460m/s is to be aimed at a target 45.7m away. If the center of the target is level with the rifle, how high above the target must be the rifle barrel be pointed so that the bullet hits dead center?about onlyI try to use the range equation but only able to find the angle. 1 answer Last reply by: Professor Dan FullertonFri Oct 20, 2017 1:56 PMPost by Kevin Wiggins on October 20, 2017for the vertical velocity in a projectile, being as though its symmetric, wouldnt the initial velocity and the final velocity be the same right before the object hits the ground? 1 answer Last reply by: Professor Dan FullertonThu Oct 20, 2016 3:28 PMPost by Sarmad Khokhar on October 20, 2016Your calculation in Example V11 is wrong as 260 into 30.7 is 7982 not 7970.Thanks 1 answer Last reply by: Professor Dan FullertonThu Mar 3, 2016 5:50 AMPost by Joy Ojukwu on March 3, 2016y= 15, so why did you set it up with two yy= 15m +.557x my question is, y is 15m, I taught it will be 15m = .557x +.......... 1 answer Last reply by: Professor Dan FullertonThu Jan 14, 2016 7:27 AMPost by Jerica Cui on January 13, 2016Hello Professor.Can you explain a little bit further why an object will travel max horizontally when the launch angle is 45 degree? (why at this specific angle?) 0 answersPost by Professor Dan Fullerton on January 5, 2016Hi Sohan.  Yes, either 9.8 or 10 is acceptable and will get you full credit. 0 answersPost by Sohan Mugi on January 4, 2016Hello Professor Fullerton. I just had a quick question about the calculations for each of these problems. For the acceleration in the previous kinematics part 2 lecture, you have stated that it is possible to use 10m/s^2 instead of 9.8m/s^2. However, in this lecture, you seem to have used only 9.8m/s^2. I have used 10m/s^2 when I was doing these problems, and when I came to the second to last problem with the pirate cannonball, I got an answer that was about 100 meters less than the answers you have put. However, when I did the same thing again with the 9.8m/s^2, I got it to be the answer you put. Do you think that it might be better off to use the 9.8 instead of the 10 or does it makes no difference if they realize that you use 10m/s^2 in your work for the AP Physics C:Mechanics Exam FRQ and possibly still see that I did all the other calculations right and give the full credit?

### Projectile Motion

• The horizontal acceleration of a projectile in flight is zero.
• The vertical acceleration of a projectile in flight is the acceleration due to gravity on the surface of Earth.
• You may use the kinematic equations to separately analyze the horizontal and vertical components of a projectile’s motion.

### Projectile Motion

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Objectives 0:07
• What is a Projectile? 0:28
• What is a Projectile?
• Path of a Projectile 0:58
• Path of a Projectile
• Independence of Motion 2:45
• Vertical & Horizontal Motion
• Example I: Horizontal Launch 3:14
• Example II: Parabolic Path 7:20
• Angled Projectiles 8:01
• Angled Projectiles
• Example III: Human Cannonball 10:05
• Example IV: Motion Graphs 14:39
• Graphing Projectile Motion 19:05
• Horizontal Equation
• Vertical Equation
• Example V: Arrow Fired from Tower 21:28
• Example VI: Arrow Fired from Tower 24:10
• Example VII: Launch from a Height 24:40
• Example VIII: Acceleration of a Projectile 29:49

### Transcription: Projectile Motion

Hello, everyone, and welcome back to www.educator.com.0000

In this lesson we are going to talk about projectile motion.0003

Our objectives for the lesson include analyzing situations of projectile motion in uniform gravitational fields.0008

Writing down expressions for both the horizontal and vertical components of velocity and positions as functions of time0014

and using those expressions and analyzing the motion of a projectile that was projected with some arbitrary initial velocity.0020

What is a projectile?0028

A projectile is an object that is acted upon only by gravity.0030

In reality we know that air resistance plays a role but as we began our study of projectile motion we are going to neglect air resistance.0035

When we talk about projectiles typically what we are talking about is something that is launched at an angle0042

and follows some sort of path something like that.0048

Although technically something going straight up and down as a projectile that it is not really nearly as interesting.0051

Let us look at the path of the projectile.0058

Projectiles launched an angle move in parabolic arcs.0060

Let us draw a baseline here.0066

If we have some nice flat level ground and we launch some object to some initial velocity let us call it Vi for V initial.0070

At some angle θ i with respect to the ground it is going to follow a parabolic arc.0081

It is going to come up and come back down in a symmetric path assuming we neglect air resistance.0087

A couple of interesting things to point out here.0096

However, the long it takes to go up is the exact same amount of time it takes to go down assuming it returns to level ground.0099

Its initial speed right when it leaves the ground is its speed right before it hits the ground.0107

Its initial angle is going to be the same angle it makes with the ground over here so we have this symmetry of motion.0113

We could look at the maximum height that our projectile attains which is halfway through its path.0121

There is our maximum height.0132

We could also talk about the range of our projectile.0134

How far it travels horizontally?0137

Our range would be that distance.0140

Projectiles follow parabolic paths and assuming they are coming down to the same level they left at it will follow it will have a symmetric path.0153

As we study these, one thing that will help us simplify this tremendously is looking at the independence of motion.0165

When projectiles are launched at an angle they have some piece of vertical motion and some piece of horizontal motion.0171

We can treat those 2 separately.0178

The vertical motion is acted upon by the acceleration due to gravity by the force of gravity.0180

Horizontal motion there is no acceleration because there is no air resistance.0185

As we split those up we can greatly simplify our problem solving process.0189

Let us take a look with an example.0195

Fred throws a baseball 42 m/s horizontally from a height of 2m.0197

How far will the ball travel before it reaches the ground?0203

Let us start off with a little diagram.0207

He throws it from a height of 2m.0210

It is the same as if we tossed a ball off a cliff of a height of 2m, not that much of a cliff.0213

Here is our ball and it is traveling horizontally at 42 m/s.0219

Our height here is 2m.0226

Let us take a look at what we know.0231

Horizontally we know the velocity the speed is 42 m/s and that is going to remain the same because there is no acceleration until it is acted upon by some other force.0235

Once it hits the ground at which point or into a different studying that projectile motion any longer.0245

We will have some change in horizontal position Δ x and it will take some amount of time to do this.0251

We could also look at the vertical components of its motion.0259

Vertically its initial velocity is 0.0263

It is moving horizontally but up and down vertically initially it is not moving.0268

Let us call down our positive y direction since that is the direction it moves first and the vertical.0272

We will have some final velocity we do not know that yet.0278

It will travel some vertical distance Δ y which will be 2m, +2m because we call down the positive y direction.0282

It will accelerate in the y direction at 9.8 m/s² g+ because we call down positive.0291

We do not yet know the time.0300

As we look at these what is the same between horizontal and vertical motion the amount of time the ball is in the air.0302

The time is the one thing that will be the same between our 2 tables of information.0309

If we can find the time it is in the air vertically we could then use that to figure out how far it went horizontally.0313

Let us do that.0321

Let us see if we can find a way with what we know to solve for the time vertically.0322

I think I would start with our kinematic equation Δ y = V initial t + 1/2 ay t².0328

It is nice here since V initial is 0 that whole term become 0 and we can simplify this to Δ y = 1/2 at² ay t²0342

or t² is going to be equal to 2 Δ y / ay which will be 2 × 2m / 9.8 m/s².0356

If I want just t I take the square root of that so if I want t I will take the square root of that and find that t comes out to about 0.639s0371

that is how long it takes for the ball to hit the ground to travel the 2m vertically.0384

That is the same as the amount of time it is traveling horizontally so that 0.639 s over there.0389

Since we have a constant velocity horizontally we can go right to our equation to find how far it travels.0397

Δ x = the velocity in the x direction × time over which it is traveling that speed or 0.639 s × velocity 42 m/s.0404

I come up with a horizontal distance that travels of 26.8m.0419

A little bit more involved but really it is just breaking up a problem into horizontal and vertical components to keep things nice and simple mathematically.0432

Let us take a look at another example.0441

The diagram represents the path of a stunt car that is driven off a cliff neglecting friction.0443

Compared to the horizontal component of the car's velocity over here at a, the horizontal component of the car's velocity at b0449

is we need to realize here the horizontal component of velocity is not going to change because acceleration due to gravity is down.0457

It is not affecting the horizontal motion of the car.0466

Therefore, we can say the horizontal component of the velocity is the same.0469

If it asks about the vertical component of the velocity well of course it is beating up the longer it falls.0473

For objects launched at an angle in order to figure out the components of velocity we typically0482

have to break that velocity up to x and y components using their trigonometry.0488

We will use those initial velocity components in our horizontal and vertical motion tables.0493

An object will travel the maximum horizontal distance on level ground with the launch angle of 45° as long as you are neglecting air resistance.0499

That comes up quite a bit.0507

If you want an object to go the furthest possible distance what angle do you want is 45°.0508

Let us take a look at what this would look like if we have something that is launched with a velocity at an angle.0515

There is our ground and let us say we have some initial velocity vector right there.0523

We will call that V initial.0530

It is launched at some angle θ with respect to the ground.0534

In order to find its components we will look at the x component first.0537

That piece we call the initial x and the component vertically V initial in the y.0546

From our trigonometry remember V initial y is just going to be V initial sin θ and V initial x the x component of that velocity vector0558

is going to be V initial cos θ because we are at the adjacent side of that angle.0567

Sin θ because we are the opposite side here.0575

For range, if we want the maximum possible range on our projectile.0579

If we want to go as far as possible, there is our range.0586

The trick is a launch angle of 45° assuming it is on level ground and you are neglecting air resistance of course.0595

Let us take an example of the projectile launched in an angle.0606

Herman, the human cannonball is launched from level ground at an angle of 30° above the horizontal with the initial velocity of 26 m/s.0610

Interesting career path for Herman.0618

How far does he travel horizontally before reuniting with the ground?0621

Now let us start by taking a look at his initial velocity vector.0626

I am going to make some axis here.0633

These initially launched an angle of 30° above the ground, above the horizontal and that initial velocity is 26 m/s.0638

The x component the horizontal component of that velocity we can find the initial x is just going to be 26 m/s cos 30° or 22.5 m/s.0654

The y component the initial y will be 26 m/s sin 30° or about 13 exactly 13 m/s sin 30 is ½.0677

How far did he travel horizontally before reuniting with the ground?0699

Again, I think the most important thing for us to know here is how long he is in the air.0703

We will start by answering that with a vertical analysis.0707

Vertically we have V initial, V final, some horizontal displacement, ay and t.0712

Herman's path is going to take him up and down like that.0722

Remember our trick for simplifying this is let us look at either the way up or the way down that by cutting this in half.0726

Let us say that we want to look at the way up.0732

Time to go up, time to go down, and V initial then if we are going up we are going to call positive y up.0740

V initial is 13 m/s because we said up was positive, V final at its highest point it stops.0751

We do not know the Δ y.0759

We know the acceleration is -9.8 m/s² negative because we call up positive and the acceleration is down due to gravity.0760

We are trying to find the time and we can use V = V initial + at.0769

We arrange that to find the time is V - V initial / a is 0 -13 m/s / -9.8 m/s².0780

Our time is going to be about 1.33s but note that is the time just to go up.0794

If it takes 1.33s to go up how long does it take to go up and down?0800

Twice that so our total time is twice the time it takes to go up or 2.6 and if we do not round here I get closer to 2.65s.0806

We know the time in the air and we can figure out the range how far Herman travels horizontally.0820

Let us do that.0826

Horizontally we do not know Δ x yet but we know the velocity in the x direction was our 22.5 m/s.0828

We found that out initially by breaking at vector in the components.0841

And the total time Herman's traveling horizontally is 2.65s.0845

We just found that by analyzing the vertical motion.0850

Δ x = the velocity in the x × the time which is 22.5 m/s × 2.65s for 59.6m.0857

Let us take a look at some motion graphs then.0877

An arrow was launched from level ground with initial velocity V0 at an angle of θ above the horizontal.0881

Sketch a graph of the displacement velocity and acceleration of the arrow was functions of time and neglect air resistance.0887

Let us do this for both x and the y so we are going to make some graphs here.0894

Make another graph for velocity and another graph over here for our acceleration.0905

This those a little bit but we will get the idea across I think and our x axis.0917

We will have a look at the x up here and look at the y down here.0944

Take a look at graphs of position vs. time, velocity in the x vs. time.0948

Acceleration in the x vs. Time and for the y we will have Δ y for time and we will have the velocity0957

and the y compared the time and will have acceleration in the y in time.0965

Let us start off on the right hand side with acceleration usually those are an easy place to start.0973

In the x direction we know there is no acceleration so that graph is nice and flat at 0 no acceleration.0979

In the y direction the entire time our acceleration is 9.8 m/s² down.0988

Assuming we are calling up positive let us just draw our line down here at -9.8 m/s² for acceleration.0995

Now let us move backwards and take a look at the velocity.1004

In the x direction the velocity must be constant.1008

We have some constant velocity and that makes sense because if we take the slope of velocity which we will get our acceleration.1012

The slope of a straight line is 0 so our acceleration is 0.1020

Down here for the y we have a negative acceleration which means our velocity must have a negative slope.1025

For something launched from level ground with initial velocity of V 0 at an angle it is going to start off that is maximum velocity.1031

It is going to slow down stop and then start going faster and faster in the negative direction.1039

There is our graph of velocity vs. time in the y direction.1050

Note that the slope is negative in constant and our acceleration is negative in constant.1054

Now for the displacement vs. time.1060

We know that we are changing our position horizontally at a constant rate.1063

We would see something that looks kind of like this.1070

Assuming we are starting from the point we call 0.1072

We are going to have a 0 nice straight linear line slightly redundant.1075

The slope of this is positive in constant.1082

I will look positive in constant velocity.1086

All of these match up when we look at our slopes.1088

When we look at displacement in the y direction what we are going to have change in values here1092

because we have a constant but we have a change in value here so we have changing slopes.1098

Right in the middle the value of our velocity is 0, right in the middle we must have a slope of 0.1103

I would draw something kind of like this for our vertical displacement.1108

We start off and 0 we reach a maximum here halfway through the slope here is 0 corresponding to a velocity of 0.1117

Then we start coming back the other way steeper and steeper negative slope more and more negative velocity.1125

That is how I would graph our displacement velocity and acceleration for each of these components for an object that is launched from the ground.1132

Graphing projectile motion.1145

Contract a vector component as a function of time or plot their path as a y = function of x.1147

We have been doing is function of time but if we wanted to do this is as a function of x solve for the time in the horizontal1152

and then we can use that to eliminate time in the vertical equation.1159

What does that mean?1163

Let us look for an analysis of something launched an angle horizontally Δ x is our initial x velocity × time.1164

And therefore time would be our horizontal displacement divided by our horizontal velocity.1177

When we move to our vertical analysis Δ y = Vy initial t + 1/2 ay t².1187

We can now substitute in what we know for time from our horizontal equation recognizing that t = Δ x / our initial x velocity.1200

Δ y = V initial y × our time Δ x / V x initial + ½ ay × Δ x² / x initial².1213

Or just cleaning this up a little bit we could write this as our y position.1237

Δ y assuming we are starting at 0 equals if we do this as a function of x we are going to have Vy initial / Vx initial ×1245

our x coordinate + we will have ay / 2 Vx initial² × x².1258

Y = some constant × x + another constant x² that is the form of a parabola, a parabolic motion.1274

Let us take a look at an example here.1288

We got an air fighter from a tower, if an arrow was fired from a 15m high tower with an initial velocity 50 m/s and angle of 30° above the horizontal,1290

find the height of the projectile above the ground as a function of its horizontal displacement.1300

Since our object goes up first let us call up our positive y direction.1307

We can take a look and figure out our initial x velocity, Vx initial is going to be V cos θ, V cos 30° or 50 m/s cos 30° which will be 43.3 m/s.1312

Doing the same thing for the y the initial velocity in the y will be V sin θ, V sin 30° or 50 sin 30 sig 30 is half so that is just going to be 25 m/s.1330

Since we called up the positive direction our acceleration in the y is -g or -9.8 m/s².1346

We can go back to our form for the motion of a projectile in parabolic form.1355

We had y = V initial / Vx initial x + ay / 2 Vx initial² x².1363

Substituting in our values that we just determined here y = 25 / 43.3 x + -9.8 / 2 × 43.3² x².1379

Let us set our origin to 0, 15 because that is where we are starting.1403

I get that y = our 15m we are starting with + 25 / 43.3, .577x - 9.8 / the quantity of 2 × √43.3 it is going to be 0.00261 x².1415

And there is our formula for the parabola that shows our x and y motion.1438

Let us actually check this and let us make a graph of that.1444

I plotted a nice pretty one in a software package that shows this formula y is a function of x.1448

Notice that we start here at 15 m are going up, coming back down and we are going to hit the ground right around there.1456

It should be pretty reasonable to say 1234 looks like we are about 244 m or so that is how far1465

that arrow would travel before it struck the ground when launch from that 50m high tower.1472

Alright speaking of launching things from a height.1480

Black Bear the pirate fires a cannonball from the deck of his ship an angle of 30° above the horizontal with an initial velocity of 300 m/s.1483

How far does the cannonball travel before contacting the ocean waters if the ship's deck is 8m above the waterline?1494

A quick diagram here if there is our ground we are starting from 8m above and doing something like that.1503

We have got 8m as our initial height.1512

We will let us say let us take a look at our horizontal analysis.1519

Horizontally our initial velocity in the x direction is going to be 300 cos 30 or about 260 m/s.1524

Our final velocity is the same.1537

There is no acceleration horizontally so that is 260 m/s.1539

Our acceleration in the x is 0.1543

We do not know how far it goes horizontally that would be nice to know, that is what we are after.1546

We do not know time yet.1550

How are we going to figure out time?1553

Now I would go to the vertical.1556

Let us take a look at our vertical analysis to figure out some more information.1559

Vertically let us call up the positive direction since a cannonball travels up vertically first.1564

V initial in the y direction is going to be 300 m/s sin 30° or 150 m/s.1572

Let us say we have the final y we do not know yet.1584

Δ y while the entire trip if it starts here and it ends up 8m below its total displacement is going to be -8m.1592

-8 because we call up positive it is going to travel much further vertical distance but its displacement will be -8.1601

Acceleration in the y will be -9.8 m/s² negative because we called up the positive direction.1609

Let us see what we can find out here.1618

As I do this it looks like we could solve for Vy first if we try to find t initially we are going to come up with a quadratic.1621

Let us go with the finding final velocity in the y first.1631

Vy² = V initial y² + 2ay Δ y our kinematic equation complies that Vy² is going to be equal to our initial velocity1635

in the vertical 150² + 2 × -9.8 × -8.1648

Vy² = 22,657 m² / s² or if I take the square root of that Vy = + or - about 150.5 m/s.1658

We are going to take some common sense here.1676

Go here when it gets this point is it traveling up or down.1679

It must be going down vertically because we called up positive so we must be negative value Vy is -150.5 m/s.1683

Find the velocity in the y -150.5 m/s.1700

From here we can figure out the time it took to do that.1707

Using another kinematic equation let us go with if ay is Δ Vy / t then t is going to be Δ Vy / a.1712

That is going to be our -150.5 m/s -150 m/s our initial velocity divided by acceleration -9.8 m/s² or about 30.7s.1725

T= 30.7s and that is the same amount of time that the cannonball is going to be traveling horizontally.1743

I can fill in my time here horizontally as 30.7s.1751

We can take a look now at the horizontal displacement of the cannonball.1757

Δ x is going to be the x velocity × time or 260 m/s the horizontal velocity our time of 30.7s or about 7,970 m.1762

It is quite the path that is traveling.1784

Taking a look at one last question here.1790

Kevin kicks a football across level ground, if the ball follows the path shown what is the direction of the balls acceleration when it gets to point b?1793

Fairly common question but a trick question.1802

The entire time the ball is in the air with the only force acting on it is gravity.1806

Gravity pulls down so the acceleration on the ball is a very pretty down, straight down there is the direction of our acceleration.1812

Any point in the path even when vertically it stops at the highest point the acceleration is always straight down.1824

Thank you for watching www.educator.com, we will see you again soon.1830

Make it a great day everyone.1833

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