  Dan Fullerton

Describing Motion II

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 Sep 20, 2017 12:54 PMPost by Silvia Basu on September 20, 2017  I tried to do example IX before watching the way you have solved it. I used the 2nd formula dx=v0t+1/2at^2. I found the time and then solved fro the speed. x/t.  You solved for v final, thus the velocity. Why? The question was asking for the speed. Did I misinterpreted the question?   Thank you. 3 answers Last reply by: Professor Dan FullertonMon Jan 4, 2016 6:52 AMPost by Sohan Mugi on January 3, 2016Hello Mr.Fullerton! I have one question.So how would you know how to determine change in velocity and acceleration from a velocity-time graph? 1 answer Last reply by: Professor Dan FullertonMon Jan 4, 2016 6:15 AMPost by Yuhuan Ye on January 3, 2016Hi Mr.Fullerton,Will we have a formula sheet that has all the equations(which includes those kinematic equations) when we take the exam?Your videos are very helpful. Thank you! 1 answer Last reply by: Professor Dan FullertonThu Dec 4, 2014 12:33 PMPost by Brian Bartley on December 4, 2014Are we not able to download the lecture slides? 1 answer Last reply by: Professor Dan FullertonSun Sep 21, 2014 8:04 PMPost by Nick Cadogan on September 21, 2014Do we need to know how to derive the kinematics equations as shown at 6:50 for the ap physics c exam?

### Describing Motion II

• An object’s position is its location at a given point in time.
• The vector from the origin to the object’s position is the position vector, r.
• The change in an object’s position is called displacement.
• Velocity is the time rate of change of displacement: v=dx/dt.
• Acceleration is the time rate of change of velocity: a=dv/dt.
• The slope of the position-time graph is the velocity. The slope of the velocity-time graph is the acceleration.
• The area under the acceleration-time graph gives you change in velocity. The area under the velocity-time graph gives you change in position.
• For cases of constant acceleration, you can utilize the kinematic equations to solve for unknown quantities.
• Objects under the force of gravity only are said to be in free fall.
• The acceleration due to gravity on the surface of Earth is 9.8 meters per second per second toward the center of the Earth.

### Describing Motion II

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 0:09
• Special Case: Constant Acceleration 0:31
• Constant Acceleration & Kinematic Equations
• Deriving the Kinematic Equations 1:28
• V = V₀ + at
• ∆x = V₀t +(1/2)at²
• V² = V₀² +2a∆x
• Problem Solving Steps 7:02
• Step 1
• Step 2
• Step 3
• Step 4
• Step 5
• 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
• Example XV: Quadratic Solution 18:46
• Free Fall 22:56
• Free Fall
• Air Resistance 23:24
• Air Resistance
• Acceleration Due to Gravity 23:48
• Acceleration Due to Gravity
• Objects Falling From Rest 24:18
• Objects Falling From Rest
• Example XVI: Falling Objects 24:55
• Objects Launched Upward 26:01
• Objects Launched Upward
• 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

### Transcription: Describing Motion II

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

In this lesson we are going to continue to talk about describing motion and kinematics.0003

Our goals understand the relationship among position, velocity, and acceleration for the motion of the particle.0010

Use the kinematic equations to solve problems of motion with constant acceleration0016

and write appropriate differential equations and solve for velocity in cases in which acceleration is specified function of velocity and time.0021

Let us start talking about constant acceleration.0030

For cases where we have a constant acceleration, where velocity is always changing by a constant rate or not changing at all,0034

we can derive a set of kinematic equations that will allow us to solve for unknown quantities.0040

These pre key equations are the final velocity = the initial velocity + acceleration × time.0047

The final position is the initial position + the initial velocity × time + half the acceleration ×0054

the square of the time that has elapsed or the square of the final velocity = the square of the initial velocity +0061

2 × the acceleration × the change in position.0068

When you know any of these 3 kinematic quantities with constant acceleration you can use these formulas to solve for the other 2.0072

The trick is always going to be finding and understanding what 3 of these are before you go on to get the rest of them.0080

How did we come up with these equations?0088

Let us take a minute and see if we can derive them.0089

Here we are showing a velocity time curve for something that is steadily increasing its speed.0092

It is accelerating at a constant rate.0098

To find our kinematic equations I'm going to start with the definition of acceleration a = Δ V / T which is our final velocity - our initial velocity ÷ time.0101

Or I can rearrange that to say that the final velocity = initial velocity + acceleration × time.0113

There is our first kinematic equation.0119

For our second, we are going to take a look at this in a little more detail and pull out my ruler here.0126

I'm going to look at this where we have this as a function of time.0132

We are going to call this for some final value time T and I'm going to break up our shape here into a triangle and0138

a rectangle that is going to come in handy in just a minute.0146

As we look at our graph, if this is our initial velocity, this is our final velocity, then this distance here is V – V initial.0152

If we want to take a look at the change in position Δ x that is the area under the velocity time curve.0165

Δ x is the area just ½ base × height for our triangle + length × width our rectangle down here0174

which is going to be ½ , our base is T and our height of our triangle is V - V initial + our height V knot × T.0187

But we already know from equation 1 that V =V initial + at.0203

Therefore V - V initial must equal at.0209

This implies then that Δ x = ½ T × V – V initial, we are just going to replace with at + V knot T which implies then that Δ x must equal V knot t + ½ at².0214

There is equation 2.0243

For our 3rd equation, we are going to come back up to the formula we have there in green, Δ x = ½ T × V – V knot + V knot t we have right here.0247

We are just going to manipulate it a little bit differently.0264

We are going to say then that ½ Vt – ½ V knot t + V knot t which implies then that Δ x = ½ Vt + ½ V knot t0265

which implies then that Δ x / t = V + V knot / 2 which is another formula not exactly whenever kinematic equations but useful.0295

That left hand side is V average and V average is V final + V initial ÷ 2 for something that is accelerating that has a constant acceleration.0309

We are going to use that here in a little bit, I believe.0319

Let us keep going with derivation now.0323

V average = Δ x / t which implies then that Δ x = V average × t which we just said is V + V knot / 2 × t.0325

Another tricky little substitution here if V = V initial + at, we can solve that for t to say that t = V - V initial / a.0350

We can then substitute in to say that Δ x = we have our V + V knot / 2.0363

For our t, we have V – V knot / a which implies then if we multiply through here Δ x is going to be V² - V initial² / 2a0372

which implies then that V² - V initial² = 2a Δ x or solving for V² V final² = V initial² + 2a Δ x.0381

There is that 3rd kinematic equation.0412

It is fairly easy to derive these.0418

As we start dealing with a bunch of these kinematic problems, there are couple problem solving steps0423

when you can use kinematic equations for constant acceleration problems that will help things out.0429

First is to label your analysis for horizontal and vertical motion.0434

Choose a direction to call positive and stick with that throughout the problem.0439

Typically it helps if you call the direction the object is moving initially as the positive direction.0443

Create a motion analysis table.0448

Fill in your givens and once you know 3 items in your table you can always solve for the unknowns using your kinematic equations.0450

Let us take a look at an example here.0460

A race car starting from rest V initial = 0 accelerates uniformly at a rate of 4.9 m / s² that must be your acceleration.0462

What is the car's speed?0473

We are looking for V final after it has traveled 200m Δ x.0475

Let us pick a direction to call positive and we will call to the right positive in this case.0482

Horizontal motion problem and we will list our items of interest, V initial, V Δ x, a, and t.0487

We will fill in what we know, V initial is 0, we are trying to find V.0500

The final Δ x is 200 m, acceleration is 4.9 m / s², and we do not know time.0505

What I am going to do now is I'm going to look at what I have and what I'm trying to find0515

and see if I can find a kinematic equation that has all the items that I want in there.0521

I do not care about t, what I am looking for a formula that has those things in it.0527

There is one that would be V² = V initial² + 2a Δ x.0532

V² = 0² + 2 × 4.9 m / s² × Δ x 200m or V² = 1960 m² / s².0542

I do not want V², I want V.0560

V is going to be equal to the square root of that which is + or - 44.3 m / s.0563

Common sense tells me of course I'm looking for the positive so I can get rid of the negative there.0571

My answer is 44.3 m / s.0576

Let us take a look at a vertical problem.0584

An astronaut is standing on the platform on the Moon drops a hammer.0587

If the hammer fall 6 m vertically in 2.7 s what is its acceleration?0591

This is a vertical problem and since the hammer goes down first that is called down our +y direction.0597

Then the list of our information of interest V knot, V, Δ y, a, and t.0603

We know V initial is 0, we know Δ y is 6m and there time is 2.7 s.0613

What we are trying to find? Acceleration.0621

Our 4 items that would be nice to have all a nice formula are right there.0625

We can do that using the formula Δ y = V initial t + ½ at².0631

V initial is 0 that term becomes 0.0641

This becomes a little simpler Δ y = 1/2 at² which is ½ .0645

Actually we are trying to solve for a so this implies then that a = 2 Δ y / t² which is 2 × 6 m / 2.7 s² or about 1.65 m / s or 1.65 m / s².0653

How about a problem that is a little bit more involved, a 2 step problem?0683

We have a car traveling on the straight road at 50 m / s and it accelerates uniformly to speed of 21 m / s in 12 s.0688

Find the total distance traveled by the car in that 12 s interval.0695

We will call to the right our positive direction and list our items of interest V knot = 15 m / s, V final is 21 m / s, Δ x is what we are trying to find.0700

a we do not know and time is 12 s, that is what I'm after.0718

However, it does not look to me like we have a very straightforward path to getting that as it currently stands.0730

What formula has those 4 things in it I do not see it.0738

What we could do know is we could remember the average velocity is V initial + V final / 2.0743

That is going to be 15 m / s + 21 m / s / 2 or 18 m / s.0749

Average velocity is distance traveled ÷ time, therefore the distance traveled is going to be V average × time or 18 m / s × 12 s which is 216 m.0757

One way you can go about solving that.0773

We do not actually have to do it that way especially if you do not remember that equation.0775

We could also go and we can find acceleration first.0780

We know enough to find acceleration even though we do not need it and then plug that back into another kinematic equation.0784

Let us try that a = change in velocity / time which is final velocity - initial velocity / t or 21 m / s - 15 m / s ÷ 12 s.0789

It is just going to be 0.5 m / s², we know a = 0.5 m / s².0806

We can choose any kinematic equation we want now because we know the 4 other items as long as it has Δ x in it.0814

Let us choose Δ x = V initial t + ½ at² which will be 15 m / s × 12 s + ½ × the acceleration 0.5 m / s² × 12 s² is going to give us 216 m again.0821

Or we could have picked even a different kinematic equation, V final² = V initial² + 2a Δ x.0851

Solving for Δ x that is going to be V final² - V initial² / 2a which will be (21 m / s² -15 m / s² ) / 2 × 0.5 m / s² or 216 m.0860

A lot of different path to a solution.0881

Something we are going to see coming up again and again and again in physics.0883

There may not be any correct single path, there may be multiple paths to an answer.0886

Some may be considerably easier than others or more straightforward but oftentimes0891

there more than one solution to a problem that will lead you to a correct answer.0895

Taking a look at acceleration.0903

How long must a 5 kg kitty cat accelerate at 3 m / s² in order to change its velocity by 9 m / s?0905

That looks like a horizontal problem again, V initial, V Δ x, a, and t.0914

What do we know?0923

We want to have a change in velocity of 9 m / s.0924

Δ V = 9 m / s and acceleration is 3 m / s².0928

We can do this quite simply if we wanted to recognizing that acceleration is change in velocity /time.0935

Therefore time = Δ V / a or 9 m/s / 3m/s² which is 3s.0942

Let us take a look at some particle diagrams.0959

A spark timer is a little device that shots at regular intervals and what you do is0962

you have a paper go through it and it makes dots on the paper.0967

Those dots correspond to that time interval so you can examine the dots to help you see what is going on with the motion of an object.0972

A spark timer is used record the position of a lab card accelerating uniformly from rest.0980

We probably tied this piece of paper onto a lab card and that card go through it and had the card move0985

as it went through the timer it makes a dot and set time intervals that gives you a feel for what is going on.0990

Each 0.1s the timer marks a dot on a recording tape to indicate the position of the cart.0995

If that is a 0s and that means this must been 0.1 s, that dot must have been at 0.2 s, 0.3 s, 0.4 s, you get the idea.1002

Find its placement of the card at time t = 0.3 s.1016

I will do that and just go to 0.3 s to recognize from my ruler that must be 9 cm, Δ x = 9 cm.1021

Find the average speed of a cart from 0 to 0.3 s.1031

Average speed is displacement ÷ time so that would be 0.09 m or 9 cm ÷ 2.3 s to get there which is 0.3 m / s.1035

Let us go a little further with this problem.1055

Find the acceleration of the cart.1058

To find the acceleration of the cart we could use Δ x = V knot t + ½ at² realizing that the cart began at rest.1062

It says it is accelerating from rest V knot 0.1078

Therefore we can rearrange this for a to say that a is going to be 2 Δ x / t² or 2 × it went 0.09 m here at 0.3 s ÷ 0.3 s²1082

going to give us an acceleration of about 2 m/s².1104

On the blank diagram draw at least 4 dots indicating a cart moving at constant velocity.1110

I will probably have something like that constant velocity they should be evenly spaced.1117

When you do these problems there are times when you can come into quadratic situations.1126

Arnie, the aardvark accelerates at the constant 2 m/s² from an initial velocity of 1 m/s.1130

How long does it take Arnie to cross a distance of 50 m?1137

Horizontal problem will start with our known information V initial as 1 m/s.1141

We do not know V final, displacement is 50m, acceleration is 2 m/s².1146

We do not know the time and we are trying to find that.1157

As I go to solve this I look for my items of interest, we have got those 4.1161

We are worried about that one time.1170

We have got a formula that has those in there.1173

Δ x = V initial t + ½ at² which implies that Δ x 50 = V initial is 1.1177

T + ½ a and ½ × 2 is going to be 1.1190

T² which implies then that t² + t - 50 = 0.1196

It is a quadratic equation and we can use the quadratic formula.1206

Let us do that for practice t = -b + or - the square root of b² – 4ac / 2a which is going to be -b is going to be -1 + or –1210

the square root of b² 1² - 4 × a which is 1 × c which is - 50 / 2a which is 1.1225

That is going to be equal to -1 + 14.18 / 2 or -1 -14.18 / 2 which implies that t = 6.59 s or -7.59 s.1248

Here we got to use a little bit of common sense and know which one of those does not really fit here?1271

Of course that is the negative time, the correct answer must be 6.59 s.1276

That is a lot of work.1281

Typically when you see a quadratic formula pop up in a kinematic equation like this you can get around it by being a little bit clever.1284

An alternate solution, imagine instead we want to solve for the final velocity first.1293

We could do something like final velocity² = the initial velocity² + 2a Δ x which is going to be equal to 1² + 2 × acceleration 2 × displacement 50,1298

which implies that V² is going to be equal to 201 m / s² or V final is about 14.18 m / s.1313

Maybe a little bit familiar.1327

Once we have that we could go to another kinematic equation V = V initial + at1329

therefore t = V - V initial / a which is 14.18 m / s - 1 m/s ÷ 2 m/s².1336

All looks familiar perhaps which gives us 6.59 s same answer but we did not have to go to use the quadratic formula.1351

And really is just another way of getting to the same thing.1362

If you do see you got a quadratic and you are not a big fan of using a quadratic formula solve for something else first1366

and then you can go use a different kinematic equation.1372

Free fall when the only force acting on an object is the force of gravity the objects weight we refer to the motion of that object as freefall.1378

That we are going to ignore the force of friction or air resistance.1386

If it drops something it is in free fall, that includes objects that have a non 0 initial velocity.1391

If you throw something down as long as the only force acting on it is the force of gravity we will call that freefall.1397

If we drop a ball and a sheet of paper of course it is obvious there not going to fall at the same rate.1406

If you can remove all the air from the room however you would find that they do follow at the same rate.1411

We are going to analyze the motion of objects by neglecting air resistance that type of friction at least for the time being.1416

Eventually we are going to pull air resistance back into the mix here in this course.1422

Near the surface of earth objects accelerate down at the rate of 9.8 m/s /s we call1429

this acceleration the acceleration due to gravity and gets a special symbol g.1436

For the purposes of the AP exam we want to make the math a little quicker you can round 9.8 up to 10.1441

More accurately g is the gravitational field strength.1448

As we move further away from earth of course the acceleration due to gravity decreases.1452

Let us take a look at objects falling from rest.1458

That object falls from rest that means its initial velocity is 0 since the objects initial motion is down1461

it is typically useful to call down the positive y direction and acceleration is +g.1467

If we called down the y direction and the object accelerates down that would be a = 9.8 m/s².1474

G is always 9.8 m/s² whether the acceleration is +g or -g depends on your reference.1485

Let us take a look at falling objects.1496

How far will a brick starting from rest fall freely in 3s neglect air resistance.1498

Since it is going down first we will call down our y direction.1504

We will set up our table of information V initial = 0 from rest.1509

V final we do not know, Δ y we do not know.1513

Acceleration we do know it is 9.8 m/s² and let us round that to 10 m/s² and that is positive because we call down positive and time is 3s.1517

Using our kinematic equations to find how far Δ y = V initial t + ½ ay t²1531

and since V initial is 0 that term becomes 0, Δ y = 1/2 a 10 × t² 3².1541

9 × 10, 90 × ½ 45m1552

Alright how about objects launched upward?1562

In this case you must examine the motion of the object on the way up and the way down.1565

Since the objects initial motion is up typically it is useful to call that the positive direction.1570

If we call up the + y direction the acceleration and the problem would be -g or -10 m/s²1575

because it is pointing in the opposite direction of what we called positive.1584

It is nice when things coming up and back down is it the highest point when it gets to its highest point,1588

that peak for a split second its vertical motion stop since velocity there is 0 before it starts accelerating again on the way back down.1594

There is some symmetry of motion there.1601

We throw something up and we are neglecting air resistance it takes the exact same amount of time1604

to go up as it does become down to that same level.1609

The initial velocity you threw it up with is the same as the initial velocity it comes down with.1612

Theoretically if you neglect air resistance and you shoot a bullet straight up, the velocity of the bullets as it comes out of the gun1617

is going to be the exact same as the velocity when it is coming down just in the opposite direction.1625

In reality it does not work that way because of air resistance but I think you get the idea there.1631

A ball thrown upward, a ball thrown vertically upward reaches a maximum height of 30m above the surface of the Earth,1638

find the speed of the ball and its maximum height.1644

Let us call up the positive y direction and we are going to have to recognize as the object goes up and comes back down at its highest point the vertical velocity is 0.1648

Therefore the speed of the ball at its maximum height 0 m/s.1660

A basketball player jumps straight up to grab a rebound, if she is in the air for 0.8 s which is pretty good jump how high does she jumped?1669

Let us call up our +y direction and realize that she comes up and back down and all of that happens in 0.8 s1679

which means she hits her highest point at 0.4 s half of that.1691

If we wanted to we could analyze her motion on the way up or the way down that breaking this in half is going to make it a lot easier1697

because if you want to know how high she jumps we want to know what is happening halfway through that motion.1704

Let us look at the way up first.1709

Up at y direction we know now V initial, we do not know we know final velocity is going to be 0, Δ y is what we are trying to find.1713

A must be -10 because we call the positive in the time to go up is 0.4 s.1727

Let us see do we have an equation that will give us Δ y write off?1735

I do not see anything easy.1739

Let us solve for initial velocity first.1741

If a V = V knot + at and we could say that V initial = V - at which is 0 - a -10 × 0.4 s is 4 m/s.1745

Our initial velocity must have been 4 m/s.1763

We can use an equation with Δ y we want to find how high she goes.1767

Her highest point Δ y is V initial t + ½ at² which is going to be V initial is 4 m/s ×1772

our time 0.4 s + ½ × acceleration -10 × time squared 0.4²1782

Complies that Δ y is about 0.8 m that is a pretty serious vertical leap.1793

That was one way to do it.1801

We could have also looked at what was going on by analyzing our motion on the way down instead of the path up.1803

If we want to look at the way down what we know she starts at initial velocity 0 so it is called down the positive y.1809

V initial before looking at just this window where she is coming down the initial is 0.1818

We do not know V final, we do not know Δ y.1823

Acceleration now is 10 m/s² because she is accelerating down and we called down y direction and t is 0.4 s.1827

Δ y = V initial t + ½ at² which is ½ × 10 m/s² × 0.4² or 0.8 m.1838

Noticed that analyzing on the way down we only have these one formula so it was a little bit faster.1856

Either one will work but another case where there are multiple ways to solve the problem1861

but one is typically a little more streamlined than the other.1865

How about a ball thrown downward?1870

We threw a ball straight down there with the speed of 0.5 m/s from a height of 4m.1872

What is the speed of the ball 0.7 s after it is released?1877

All right it is called down the y direction because that is the direction the ball moves initially.1882

V initial is .5 m/s we do not know V and we want to figure that out.1888

Δ y is a little tricky I will put 4 m in there but it does not say the ball travels 4m it says1895

it was thrown from a height of 4m and does not say the hits the ground.1902

Therefore we do not know how far it is gone when it is 0.7 s.1906

From now we do not know the Δ y, A is 10 m / s² and our time is 0.7 s.1911

Now V = V initial + at which is 0.5 m/s + a 10 m/s² × 0.7 s which is going to be 7.5 m/s.1921

There is our final velocity.1942

Alright a quarter kilogram baseball is thrown upward with the speed of 30 m/s.1949

Neglecting friction the maximum height reached by the baseball is what?1955

Let us take a look and we are going to call up the y direction and as we do that the maximum height all we know V initial is 30 m/s.1960

Its highest point, its top, just by looking at just the way up we can call its final velocity 0 m/s.1971

Δ y is what we are trying to find, acceleration is -10 m/s² because we called up our positive direction and time we do not know either.1979

I'm looking for something that has those quantities in it.1993

V² = V initial² + 2a Δ y which implies that Δ y = V² - V initial² / 2a.1999

Now we can substitute in our values Δ y = 0² - 30² / 2 × -10 – 900 /-20 is going to give us 45m.2012

Another type of popular problem is called catch up problems or something like this.2033

Rush, the crime fighting super hero, can run at the maximum speed of 30 m/s while Evil Eddie the criminal mastermind can run 5 m/s.2037

If evil Eddie is 500m ahead of Rush, how much time does Evil Eddie have to devise an escape plan?2047

Let us see here, we can start by looking at their positions.2056

The position of Rush is going to be given by the speed × time.2062

Displacement is velocity × time so 30 m/s × time.2068

The position of Evil Eddie is he starts 500m ahead of Rush but adds at the rate of 5 m/s 500 + 5t.2072

How much time does Evil Eddie have?2083

He has until Rush catches him, when their positions are the same.2085

If we have let xr Rush’s position = Evil Eddie’s position and solve for time that should tell us how long Evil Eddie has to figure out how to get out of the situation.2089

That implies then that 30t = 500 + 5t or 25t = 500 therefore t = 500 / 25 or 20s.2101

Evil Eddie better do a lot of thinking in that 20s.2118

How far must Rush run to capture Evil Eddie?2121

The distance Rush covers is going to be his position, since he started at 0 that is going to be 30t which is 30 × 20 s or 600m.2127

Rush is going to run 600 m in the same time Evil Eddie runs is 100m because he started at 500.2141

Alright, one more sample here.2149

3 A model rocket of varying mass are launched vertically upward from the ground with varying initial velocities,2154

from highest to lowest rank the maximum height reached by each rocket neglecting air resistance.2161

It looks like they have different masses, from lightest to heaviest.2167

This one has the highest initial velocity, the lowest, and medium.2173

The key here is there are no mass dependents.2177

The only thing that is going to matter is that initial velocity.2180

The one with the highest initial velocity with the biggest initial velocity is going to go the highest.2183

That will be one first which is going to then go to the rocket 3 the next highest2189

and rocket 2 will have the lowest maximum height because there is no mass dependents.2197

All right thank you for watching www.educator.com and make it a great day everyone.2204

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