  Dan Fullerton

Retarding & Drag Forces

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 4 answers Last reply by: Professor Dan FullertonWed Oct 19, 2016 7:41 AMPost by Brad Greer on October 10, 2016In many of the downloadable lecture slides, several identical slides in a row appear (e.g. in Mech-10 (Retarding and Drag forces), slide Velocity-as-a-Function-of-Time_332 duplicates slide 747 and slide 916, and all have only a title. Also, in some cases, the slide is unreadable because it is distorted or partially off the page, e.g. slide 748 in this lecture. Is there any way to get better copies of the slides, that contain all the information from the lecture? 1 answer Last reply by: Professor Dan FullertonThu Aug 4, 2016 1:26 PMPost by Peter Ke on August 1, 2016For Example 2, at 28:18 I understand why you put -k next to the dv but how did you get -1/k on the outside of the integral? 1 answerLast reply by: Yuhuan YeSun Jan 3, 2016 9:58 PMPost by Yuhuan Ye on January 3, 2016Hi Mr.Fullerton,I have a question on "velocity as a function of time", just right before you integrate the equation, where did the "d" in "b/m*dt" go? 1 answer Last reply by: Professor Dan FullertonWed Oct 15, 2014 2:14 PMPost by Scott Beck on October 15, 2014Is the Ap C Physics exam different from Ap physics 1 and 2? Could this course also prepare for the algebra based ap physics exams excluding optics and modern physics? 3 answersLast reply by: Scott BeckWed Oct 15, 2014 1:25 PMPost by Scott Beck on October 15, 2014Why isn't the equation for net force in y direction Vo - mg + kv=ma?   Since the ball was launched upward, and since the equation would be written Vo - mg -Fdrag but since Fdrag is -kv, to negatives make a positive turning it into Vo-mg+kv=ma??  Why isn't Vo included in the equation from the free body diagram?

### Retarding & Drag Forces

• When the frictional force is a function of an object’s velocity, the frictional force is known as a drag, or retarding, force.
• Typically we assume the drag force takes the form F=bv or F=cv^2, where b and c are constants.
• Once an object reaches its maximum velocity, when the net force on the object is zero, we say the object has reached its terminal velocity.
• To find velocity as a function of time, write your Newton’s 2nd Law equation in the form of a differential equation. Standard bodies falling through the air with air resistance can typically be solved using the method of separation of variables.

### Retarding & Drag Forces

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
• Retarding Forces 0:41
• Retarding Forces
• The Skydiver 1:30
• Drag Forces on a Free-falling Object
• Velocity as a Function of Time 5:31
• Velocity as a Function of Time
• Velocity as a Function of Time, cont. 12:27
• Acceleration
• Velocity as a Function of Time, cont. 15:16
• Graph: Acceleration vs. Time
• Graph: Velocity vs. Time
• Graph: Displacement vs. Time
• Example I: AP-C 2005 FR1 17:43
• Example I: Part A
• Example I: Part B
• Example I: Part C
• Example I: Part D
• Example I: Part E
• Example II: AP-C 2013 FR2 24:26
• Example II: Part A
• Example II: Part B
• Example II: Part C
• Example II: Part D
• Example II: Part E

### Transcription: Retarding & Drag Forces

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

I'm Dan Fullerton and in this lesson we are going to talk about retarding and drag forces.0003

You have got it we are actually going to get to air resistance.0008

Our objectives are going to be to find the terminal velocity of an object.0011

To describe the motion of a particle under the effect of a retarding a drag force in terms of its displacement, velocity, and acceleration.0015

Use Newton’s second law to write a differential equations for the velocity of the object.0024

Derive the equation for velocity for Newton’s second law using separation of variables.0029

Determine the acceleration for an object falling under the influence of drag forces.0035

Let us start off by talking about these retarding a drag forces.0040

Sometimes the frictional force is a function of how fast an object is going.0044

Its velocity and air resistance is a great example of this.0048

These forces are called drag or retarding forces.0052

Now, as we go to our analysis of drag this is going to be our first lesson in a while that is going to get pretty heavy into the map and calculus.0056

Probably, seeing some things if you are taking calculus concurrently where you are seeing it in physics before you might in calculus.0064

That is ok, it is not uncommon.0071

Do your best to stick with it and then come back to this lesson later on once you are a little bit more solid0073

on the calculus and some of the integration we are going to be doing.0079

That is not uncommon at all.0082

If you leave this lesson a little fuzzy that is ok.0084

Alright, let us start by talking about a skydiver.0090

Assume we drop our dear friend Alex from an airplane.0094

Typically, the drag force on a freefalling object takes the form where the force of drag is some constant × velocity or constant × square of velocity.0097

Or sometimes even something in between that where b and c are constants.0104

For the purposes of this problem, let us assume that the drag force is to equal to a constant times the velocity.0111

Let us draw a free body diagram for our dear friend Alex who are about to push from an airplane.0118

We have the weight of Alex pulling down and the drag force backup.0123

We will call down the y direction since that is the way Alex was going to start his motion and finishes motion.0132

As I analyze this, the net force in the y direction using Newton’s second is going to be mg - the drag force0139

and that is all equal to mass times the acceleration in the y direction.0150

We just said that our drag force is equal to bv.0155

We can write that mg - bv = Ma.0162

Let us take a look at what happens when we first push Alex from the plane.0173

Initially at time T = 0 Alex’s velocity is 0, therefore, the drag force is 0 mg = Ma and a =g.0178

The acceleration = acceleration due to gravity.0194

Therefore, we can write at that point acceleration is equal to g.0197

As T increases as we head toward infinity eventually Alex reaches a maximum or terminal velocity.0205

Terminal velocity we will abbreviate that Vt and at that point acceleration is equal to 0.0228

At that point, all the forces must be balanced if there is no acceleration.0242

The drag force must equal mg at that point.0246

In Newton’s second law equation we have Mg – bv = ma with the condition of terminal velocity, acceleration is 0 and V = Vt.0251

Our equation becomes Mg – b Vt = 0.0268

Therefore, we can solve for terminal velocity and say terminal velocity is just going to be Mg divided by the constant b.0275

Our formula for terminal velocity based on that constant in our f drag = bv equation and we also know initially T =0.0285

Velocity 0 is equal to the acceleration due to gravity g.0294

I think that will set us up to get into our math using Newton’s second law of equation.0299

If we go back to our Newton’s second law equation which has velocity and velocities derivative acceleration in it,0306

we are really have 2 different forms of the same variable in the equation.0312

The variable itself and its derivative and that is called a differential equation.0318

We are going to solve that using a strategy known as separation of variables.0322

Let us start on the next page to give ourselves all kind of room here.0329

We will begin with our Newton’s second law of equation mg - bv = ma.0334

Like we said, acceleration is the derivative of velocity a = dv dt0343

We have mg - bv = m dv dt.0349

Our differential equation with V and its derivative in the same equation.0357

You can take entire courses on solving differential equations.0361

We are only to have to deal with a couple types, couple simple ones here in the course.0364

This one we can solve with that separation of variable strategy.0368

Let us walk through and see how we do this.0372

If mg – bv= m dv dt I can divide the whole thing by b so we would have mg / b - V = m/b dv dt.0374

Mg/b we just defined in the previous page that is what we called terminal velocity.0391

We can maybe pretty this up just a little bit by saying that VT = mg / b0397

Therefore our left hand side becomes Vt - V= m/ b dv dt.0403

Which implies then I will try to get all the variables V’s and derivative of these on the same side.0416

I have dv / Vt – V = b / m dt.0423

It is just an algebraic rearrangement.0432

I have dv / Vt – V I rather have V there than have V – Vt.0436

I am going to multiply both sides by -1 to get the left hand side as dv / V – Vt = - b / m T.0441

We have got all of V on one side and their variable in terms of T on the right hand side.0455

What we are going to do is we are going to integrate both sides.0461

As I integrate the left hand side I am going to integrate dv / V – VT.0465

My variable of integration is that velocity.0472

I am going to integrate from some initial velocity v= 0 to some final velocity V.0475

I have to integrate do the same thing to the right hand side so that will be the integral of - b / dt.0481

My variable of integration is T so we are going to integrate from T = 0 to some final value t.0495

How do I integrate those?0504

The left hand side fits the form du / u where if I said that u = v - Vt the differential of u would just be dv.0506

In the formula maybe you have learned or you have not got in there yet in the rule of d/u is the natural log of u + that constant of integration.0523

Since, we have the limits here we do not have to worry about our constant integration.0533

That means that our left hand side is going to become the natural log of u V - Vt evaluated from 0 to V.0537

In the right, inside -b / m that is a constant that can come out of the integral sign we are just integrating dt from 0 to T0549

That is just going to be T and this becomes - b / MT.0557

The left hand side here becomes log of V - VT substituting in V - VT - log of 0 – Vt plugging in 0 for V0568

That is - Vt = - b / Mt.0583

We are going to use the identity of the difference of two logs is the one of the quotient.0590

We can use a log a – log b = log a/b to say that the left hand side is log of V - Vt /- VT.0598

In the right hand side still - b / MT.0613

A little bit more manipulation to do here.0619

Implies that the log of V – Vt /- Vt.0623

Let us see if we can rearrange them a little bit and take that negative to the top.0631

We can write that as the log of c VT - V / dt = -b/MT which implies that the log of b1- V/VT = -b/MT.0636

If I want to get rid of that nasty log I can use this as a power we raised e2 on both sides to state then that 1 - V / VT0660

Even the natural log is whether you happen to have there = eb/MT.0672

Alright, next up.0682

We are trying to get V all by itself so I could take and rearrange this add V / VT to that side and subtract that from the other and come up with V/ VT = 1 – eb/MT,0684

which implies then that V = VT × 1 – e-b/MT.0703

Or we said VT was mg / b so if we wanted to we can put that back in those terms as well that V =MG / b × (1-e-b/MT).0716

Those are equivalent statements.0734

We found velocity as a function of time.0738

Alright, now knowing VT we can solve for the acceleration and I'm going to give ourselves more room for that.0743

If we wanted to do that let us take a look here.0748

A = dv dt which would be the derivative of what we just found from our velocity which is VT - VT e-b/MT.0753

I will expand that out so that is going to be equal to the derivative with respect to T of the derivative of the constant is going to be 0.0766

We will just have – VT e-b/MT.0779

We can pull our constant out -VT derivative with respect to T of e-b/MT0785

which implies then that a = - VT × derivative eu is eu du.0796

We have our eu e-b/MT × du is going to be another – b/m0806

which implies then that a is going to be equal to - Vt was mg /b replace that there we have - b/m and we still have e-b/MT0815

which implies then that a = we can do some simplifications.0835

We got some m there and there.0838

B there and there our negative signs.0840

I'm just going to get that a = g × e-b/MT.0843

You are going to see forms of the solutions like this for differential equations quite regularly where0854

you have a constant times e-b/MT raised to some power times T.0860

Or a constant ×1 – enegative power.0866

It will come up again and again.0870

These are all of the form where you have 1 – e-T / some time constant.0872

Here time would be m/b or something times e-T / time itself.0879

It is going to come up here.0886

We are going to see it in the electricity and magnetism course.0887

When we talk about capacitors and get into inductors that form keeps coming up again and again.0890

It is really nice that once you start to get a feel for that form you can almost guess the answer to these problems0896

before you go all the way through the math to actually prove it.0901

Or look at the initial and final conditions to help draw graphs of these before you actually go solve them.0906

Let us take a look at how these would look graphically.0913

I'm going to draw a couple graphs here and we are going to draw the acceleration time graph, velocity time graph,0917

and there is our position or displacement time graph.0927

Let us see what all these are going to look like.0932

We will start up here in acceleration time graph.0951

We will do a velocity time graph and also a y displacement time graph.0955

We said initially as far as acceleration goes to moment Alex left the airplane when velocity was 0 the acceleration of Alex was g.0966

We are going to start at a maximum value and over time as Alex goes faster and faster the acceleration0974

is going to decrease and decrease until reaching terminal velocity when there is no more acceleration.0981

We start at g and we have to decay down here to 0 when it is an exponential decay so we can radiate like that as we approach that asymptote.0987

That should be right on the line there.0996

As far as velocity goes, we know when we first push Alex out of the plane the velocity is 01000

We know that point and after a long time we know we eventually get to some value of terminal velocity which is mg / b.1005

We will write that in there as an asymptote and we will have an exponential increase there following that same basic shape.1014

For displacement, displacement begins at 0 and increases as speed increases1024

until reaching a constant rate of increase when the velocity reaches VT.1029

As far as displacement goes we are going to start at 0.1035

Increase and increase until we get to some point where it is just going to be linear as1038

we have a constant velocity when we are running it just terminal velocity.1046

There would be graphs of acceleration, velocity, and displacement as functions of time that correspond to those calculations we just did.1051

Let us take a look and see how this would look in a free response problem.1063

You can download the problem yourself if you want 2005 free response 1 from the mechanics exam.1067

A link to it there where you can google it and will take a few minutes look it over and give it a try and come back here and let us see how you did.1073

2005 mechanics 1 looking at part A, we have a ball that is thrown vertically upward at some initial speed.1083

It has some air resistance given by - kv the positive direction we are going to call up.1092

Is the magnitude of the acceleration of the ball increases or decrease or remain the same as it moves upward?1098

We are calling up the positive y direction and if I were to draw a free body diagram of the ball1105

as it moves up we have its weight down and we have that drag force kv down.1110

Net force in the y direction is just going to be - mg - kv is equal to MA y which implies that Ay was just going to be -g - kv/M.1119

If we look at that as V goes down, as it slows down, as it gets higher and higher, V gets smaller.1134

It looks to me like our acceleration, the magnitude of our acceleration in the y direction must decrease based on our formula.1142

There is A must decrease.1152

Let us take a look at part B.1157

Write but do not solve the differential equation for the instantaneous speed of the ball in terms of time as it moves upward.1162

From our free body diagram again Ay = -g - kv/ m which implies then since our A is the derivative of velocity with respect to time.1170

We can write it as dv dt = - g –kv /m.1185

That is a differential equation that would probably give us credit but let us clean it up just a little bit.1194

I'm going to write that in a manner that will be a lot more useful that is going to be M dv dt = - mg – kv1200

but either one of those you have your different equation down.1215

For C, find the terminal speed of the ball as it moves downward.1219

For C, looking at terminal speed.1224

At terminal velocity we know the net force =01227

which implies then that our free body diagram is going to have something like this kv,1235

Let us be careful it is M according to this problem Mg which implies that terminal velocity kv terminal must equal Mg or a terminal velocity Mg / k.1244

Let us move on to check out part D.1269

For part D, we are asked does it take longer for the ball to rise to its maximum height1273

or to fall from its maximum height back to the height from which it was thrown?1278

A tricky question.1283

Let us see.1284

On the way up, friction brings the ball to a stop quickly.1286

This helps bring it to a stop more quickly than if they were no fiction.1300

On the way, down the friction slows the ball down so it has more time in the air on the way down.1308

That means that average velocity on the way up has to be greater than1323

the average velocity on the way down because it happen more quickly.1328

And because the distance is constant distance traveled on the way up and the way down1334

the time to go down is greater than the time to go up because T = d/V.1341

I would say then that it takes longer to fall.1349

Some sort of explanation like that to go along with your answer.1360

Part E, on the graph, sketch a graph of velocity vs. Time for the upward and downward parts of the ball's flight.1365

We are going to need a graph here.1377

Here is our velocity, there is our time, and we are looking at what happens for the upward and downward parts.1387

It starts at some initial velocity V0.1402

It is going to cross the axis here to have a velocity of 0 at its highest point.1407

It is going to take less time than it does on the rest of the trip because it is going to take longer to fall.1412

On the way down, we are going to have some value of terminal velocity.1418

We will draw an asymptote in here for our V terminal.1427

I would think that our graph would probably look something like this where it is approaching terminal velocity.1434

Something like that final velocity.1446

Something like that is your approach of final velocity at time Tf.1451

I think that covers that one.1454

Let us take a look at one more free response problem.1462

Let us go to the 2013 APC mechanics exam.1466

It will take a few minutes to look it over and print it out, give it a try, and come back here and hit play.1474

We will see how it worked for you.1478

In this problem, we have a box of mass M at rest in the constant applied force being applied1483

there is a frictional drag force proportional to kv where V is the speed of the box,1489

k is some positive constant, and we are given a dot to draw on label the forces.1495

Draw our free body diagram actually.1499

Let us draw our free body diagram first.1501

There is our box.1504

We know we have the normal force, we have the weight of the box, the force of gravity.1507

We have some applied force which we are calling Fa.1513

We must have our frictional force our drag force kv.1518

For part B, it asks us to write but do not solve the differential equation that can be used to determine the speed of the box and that sounds familiar.1528

We have done that sort of thing.1536

Let us take a look net force in the x direction is going to be the applied force - kv assuming we are calling to the right positive.1539

All that must be equal to Ma but as a differential equation A = dv dt therefore Fa – kv = M dv dt.1550

That will work.1571

There is our different equation.1572

We have velocity and its derivative in the same equation.1574

Moving on to part C, determine the magnitude of the terminal velocity of the box.1583

At terminal velocity acceleration is 0, f net is 0 therefore we know that the applied force in the x direction must equal kv in magnitude.1591

Therefore, the applied force = kv terminal or solving for V terminal that is just going to be our applied force divided by k.1606

Part D, use the differential equation to derive the equation for the speed.1624

Alright, we have to do some math and let us give ourselves some room.1630

Starting with our equation f - kv = M dv dt, we are going to do the separation of variables again.1636

This implies then that dv / f - kv = dt / M.1646

Let us see how it would integrate that.1657

The integral of the left hand side dv / F - kv must equal the integral of dt / M.1659

We are going to integrate here from our velocity V = 0 to some final value V in the right hand side from T = 0 to some final value T.1668

If we are going to fit this into the form du / u we would need -dv in the top and we would also need –k.1678

You need –k on the top.1686

If we are going to put –k on the top to fit that form we have to multiply by -1/k so that we maintain the same value.1689

We cannot just arbitrary throw things in there.1696

In the right hand side it looks ok to integrate.1699

This implies then, that we have -1/k integral of du/U is going to be the natural log of our U which was f-kv evaluated from 0 to V.1702

The right hand side is just going to be t/M.1709

Expanding out our left hand side we have if we do this log I am going to take a moment and I am going to put our –k over the right hand side.1726

If I multiply both sides by –k let us put –k there.1738

It can go away and can make this a little bit simpler to see.1743

Our left hand side becomes the log of f –k and I plug in V for my variable v - the log of f-0 for our V.1747

That is going to be – the log of f = - kt/M.1758

Which implies that the difference of the logs is the log of the quotient so we have on the left hand side our log of f-kv/f = -kt/M.1768

If I raise both sides to the e, the left hand side becomes f-kv/f = e-k/Mt.1784

A little bit more rearrangement here.1798

F – kv / f let us multiply both sides by f to get f –Kv= f e-k/Mt.1801

We will get V all by itself.1813

Let us get kv= f-fe-k/Mt.1816

We can factor out that f ÷ k so the velocity is going to be f/k × (1- e-k/Mt).1824

There is part D.1847

Finally for part E, on the axis sketch a graph as the speed as a function of time and label the asymptotes things like that.1851

We are getting pretty good at graph and these sorts of things by now.1863

Let us give it a shot.1864

We have V on the y axis, time on our x, and we know it is going to start at some velocity 0.1877

We can also plug that in for T in our formula.1886

If T is 0, e⁰ is 1.1890

1-1 Is 0 so the velocity would be 0.1892

We will start here at 0 and as T gets big that whole term goes to 0 so we have f/k as our asymptote.1895

Let us mark that here f/k.1906

The shape of our graph is something like that.1913

Retarding forces and drag forces, air resistance.1920

Hopefully that gets you a good start.1923

Thank you so much for watching here at www.educator.com.1924

I look forward to seeing you soon and make it a great day everyone.1927

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