  Dan Fullerton

Math Review

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 2 answers Last reply by: Professor Dan FullertonTue Mar 21, 2017 6:12 AMPost by Woong Ryeol Yoo on March 20, 2017Hi Mr. Fullerton.I was wondering to what extent of knowlege of trig identities need for Ap Physics C mech exam. Do I have to know like coscos-sinsin=cos and other complex forumulas?Or is knowing only sin, cos, tan, sin sq + cos sq = 1 enough for the to exam? 1 answer Last reply by: Professor Dan FullertonWed Aug 17, 2016 1:08 PMPost by Cathy Zhao on August 16, 2016I still feel confused about the right hand rule. Can you explain it a little bit more? Thanks 2 answersLast reply by: Shikha BansalFri May 27, 2016 3:46 PMPost by Shikha Bansal on May 24, 2016Hi Mr.FullertonI have finished ap physics 1 as well as algebra 2 this year in school, and I was hoping to study physics C on here over the summer as I would love to go in more depth on physics. However, I do not know much precal currrently. Should I learn precal first or is this math review enough to more or less get me ready for the math in this course?Thanks! 1 answer Last reply by: Professor Dan FullertonWed Apr 1, 2015 9:50 AMPost by Luvivia Chang on March 31, 2015Hello Professor Dan FullertonCan we take the derivative of a vector? Because in the part of dot product properties, you write "d(vector A)/dt (dot)times vector B" while in the properties of cross product, you just write "dA/dt (cross)times vector B" .Is there any difference between the two?Thank you. 3 answers Last reply by: Professor Dan FullertonTue Dec 23, 2014 6:53 AMPost by John Powell on December 20, 2014Should dot product property given as "associative" be distributive? 1 answerLast reply by: Daniel FullertonSun Dec 7, 2014 3:25 PMPost by Shaina M on December 6, 2014On lSlide with examples of derivatives the last one is wrong. Instead of 12x it's supposed to be 12x^5.

### Math Review

• Matter is anything that has mass and takes up space. It is the amount of "stuff" making up an object. Mass is measured in kilograms.
• Energy is the ability or capacity to do work. Work is the process of moving an object. Therefore, energy is the ability or capacity to move an object.
• Mass-Energy Equivalence states that the mass of an object is really a measure of its energy.
• The source of all energy on Earth is the conversion of mass into energy.
• Physics is the study of matter and energy. This course will focus on mechanics, fluids, thermal physics, electricity and magnetism, waves and optics, and selected topics in modern physics.
• Intro 0:00
• Objectives 0:10
• Vectors and Scalars 1:06
• Scalars
• Vectors
• Vector Representations 2:00
• Vector Representations
• Graphical Vector Subtraction 5:36
• Graphical Vector Subtraction
• Vector Components 7:12
• Vector Components
• Angle of a Vector 8:56
• tan θ
• sin θ
• cos θ
• Vector Notation 10:10
• Vector Notation 1
• Vector Notation 2
• 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
• Vector Component Notation
• Vector Multiplication 28:39
• Dot Product
• Cross Product
• Dot Product 29:03
• Dot Product
• Defining the Dot Product 29:26
• Defining the Dot Product
• Calculating the Dot Product 29:42
• Unit Vector Notation
• Vector Component Notation
• Example VI: Calculating a Dot Product 31:45
• Example VI: Part 1 - Find the Dot Product of the Following Vectors
• Example VI: Part 2 - What is the Angle Between A and B?
• Special Dot Products 33:52
• Dot Product of Perpendicular Vectors
• Dot Product of Parallel Vectors
• Dot Product Properties 34:51
• Commutative
• Associative
• Derivative of A * B
• Example VII: Perpendicular Vectors 35:47
• Cross Product 36:42
• Cross Product of Two Vectors
• Direction Using the Right-hand Rule
• Cross Product of Parallel Vectors
• Defining the Cross Product 38:13
• Defining the Cross Product
• Calculating the Cross Product Unit Vector Notation 38:41
• Calculating the Cross Product Unit Vector Notation
• Calculating the Cross Product Matrix Notation 39:18
• Calculating the Cross Product Matrix Notation
• Example VII: Find the Cross Product of the Following Vectors 42:09
• Cross Product Properties 45:16
• Cross Product Properties
• Units 46:41
• Fundamental Units
• Derived units
• Example IX: Dimensional Analysis 47:21
• Calculus 49:05
• Calculus
• Differential Calculus 49:49
• Differentiation & Derivative
• Example X: Derivatives 51:21
• Integral Calculus 53:03
• Integration
• Integral
• Integration & Derivation are Inverse Functions
• Determine the Original Function
• Common Integrations 54:45
• Common Integrations
• Example XI: Integrals 55:17
• Example XII: Calculus Applications 58:32

### Transcription: Math Review

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

I am Dan Fullerton and in this lesson we are going to talk about some of the math skills that we are going to need to be successful in this course.0003

To begin with, our objectives are going to be explaining how vectors and scalars are used to describe physical quantities.0009

Calculating the dot and cross products of vectors, a little bit of vector multiplication.0017

Utilizing dimensional analysis to evaluate the units of a quantity.0022

Calculating the derivative, a basic functions, calculating the integral of basic functions and0027

explaining the meaning of the derivative and integral in terms of graphical analysis.0033

I know at this point a lot of folk start to get worried about the math involved in physics.0037

This is a calculus based math, a calculus base physics course.0042

However that does not mean it is a math course.0047

The math is used as the language of physics.0050

To help explain things much more efficiently than you could in words.0053

It is not really about math, it is about the physics applying those math principles and putting them to good use.0057

Let us start by talking about vectors and scalars.0064

Scalars are physical quantities that have a magnitude only.0068

They do not need to be described at the direction.0072

Things like temperature, mass, time, all of those are scalar quantities.0075

We could say when time move forward and backward but we are talking North, South, East, West directions.0082

If it has a direction we call it a vector quantity, things like velocity.0088

You are driving 55mph down the highway west.0093

Force, I pushed Susie forward.0098

Acceleration, I accelerated to the east.0103

They have a direction as well as a magnitude.0107

Vectors are typically represented by arrows where the direction is given by the direction of the arrow and the longer the arrow is the larger the magnitude,0110

if we want to show vectors graphically.0118

Let us take a look at a couple vector representations.0121

Let us assume that we have some vector in blue here.0124

Let us call it A and we have another vector here in red - let us call it B.0127

They both have the same direction but B has a larger magnitude than A, that is pretty straightforward.0134

Now one of the rules of vectors though is you are allowed to move it in space as long as you do not change it's direction or it's size you can slide it anywhere you want.0141

We could take this A vector if we wanted to.0149

Let us say we slide it down here and I am going to redraw our A vector.0152

I think it is roughly that length.0157

We will call that our new A vector and make that one go away.0160

Perfectly reasonable thing to do as a long as you keep the same magnitude and directions, vectors are free to move around in space.0165

We can also add two vectors.0175

We have vector A here in red and we have vector B here in blue.0178

We wanted to add A + B to get the vector C.0186

The way we do that is by aligning these two vectors up tip to tail.0196

They are not connected but what if instead of having it just like this I am going to redraw this over here and then try my best to do about the same length.0201

Here is a vector B but I'm also going to move vector A the same direction, same magnitude, and I'm going to slide over here so that vector A is now aligned tip to tail.0210

Its tip is touching the tail of vector B.0216

Once you have the vectors lined up tip to tail in order to find the addition of those two,0229

what we call the resultant, we draw a line from the starting point of our first vector to the ending point of our last vector.0235

That would be vector C, the resultant of A + B or what happens when you add vectors A and B.0245

It does not matter which order you do this or how many vectors you do with it.0253

As long as you add them all up to tip to tail it will work for 100 vectors as easily as it will for two.0257

Let us demonstrate that for a second by taking our B vector I am going to redraw again0262

and going to draw it down here, roughly the same length and same direction.0271

But now instead of having A point to its end, I'm going to have that point it to the end of A.0276

A was right about there so we will try drawing A here again tip to tail but with a different vector in front.0281

Once again to find the resultant I go from the starting point of my first vector to the endpoint of my last vector.0292

There is C notice regardless of how I did it which order I have roughly the same magnitude and same direction.0303

You can tell they are a little bit off on the drawing because I'm doing it by hand0311

and very carefully with the protractor so we are not here all day.0314

But that also shows the order of addition for vectors does not matter.0318

We could just as easily have written B + A = C vector.0322

Let us take a look at graphical vector subtraction.0336

We will have A in red again.0339

We will put B over here in blue and now if we want to know what A – B.0342

The trick to doing that is realizing that subtraction is just addition of a negative that is the same as saying A + -B.0352

We are going to have A + -B we are going to call that vector D.0365

We want to add A + -B because we know how to add vectors.0370

We got b here but how do we get the negative?0374

It as easy as you might think.0377

If that direction B, if that is vector B, all we have to do to get –B is switch its direction.0378

There is –B.0387

To add A + -B all I do is I line them up tip to tail again and I am going to take A and slide A over0389

so it is right about there roughly the same length and direction.0397

Line them up tip to tail so if this is A+ -B is the same A – B.0404

Draw a line from the starting point of my first vector to the ending point of my last.0413

I suppose we made D in purple.0420

Let us do that, there is D in purple.0421

There would be vector D, graphical vector subtraction.0425

Sometimes dealing with these graphically can get little bit tedious.0432

If you have a vector at an angle lot of times what you might want to do is break it down into components that are parallel perpendicular to the primary axis you are dealing with.0436

Let us say that we have a vector A here at some angle θ with the horizontal.0445

It can be considerably more efficient to break into a component that is along the x axis.0451

Let us do the y axis first just to make it easier to draw.0459

We call this the y component of vector A and we will have some x component of vector A.0462

Noticed that A axis a vector + y as a vector gives you the total A.0480

You could replace vector A with the equivalent set of vectors Ax and Ay where Ax is in the x direction.0487

Ay is along the y axis so we could break that up into components.0494

If you want to know how big those components are to find their magnitudes, the size of them, their length, Ay in trigonometry that is the side that is opposite our angle.0500

That is going to be equal to the magnitude of A our total vector × sin angle θ.0511

In a similar fashion, to find this component, the adjacent side of our right triangle Ax is going to be equal to A cos θ where A is the magnitude of this vector.0518

How about finding the angle then?0533

Here is a right triangle, we have an adjacent and opposite side and the hypotenuse.0536

If we know 2 or 3 sides we can find the angle.0542

The tan of θ is the opposite side over the adjacent side.0545

If we know those two sides then the angle between them, θ is going to be the inverse tan0553

of the opposite side over the adjacent side not the angle between the angle of the triangle.0559

If you know the opposite and the hypotenuse you can use the sin θ.0567

Sin θ that is opposite over hypotenuse therefore θ will be the inverse sin of the opposite over the hypotenuse.0573

If you know the adjacent side and hypotenuse, cos θ is adjacent over hypotenuse.0586

When you know those two you can find θ is the inverse cos of the adjacent divided by the hypotenuse.0595

If you know any two of the three sides you can go find the angles.0604

It is so wonderful.0609

As we talk about all these vectors and in this course we are going to be dealing with vectors in three dimensions in the x, y, and the z planes.0611

Also if you are using a textbook, a lot of times they are different notational styles in different textbooks.0622

Probably we are talking about those for a little bit to have some consistency throughout the course.0627

First thing I am going to do is I'm going to draw a three dimensional axis up here.0632

We will give ourselves y, x and z.0639

There is a three dimensional axis to begin with.0655

If we have some vector let us call it A, it can have components in the x, y, and z directions.0658

We could write that as in this bracket notation is its x value, its y value, and that z value.0665

That is a fairly common way of writing these and one of my favorites.0674

If you take a moment you define what we call some unit vectors, there are some other ways we can deal with this two.0677

That is our x, let us call a vector of the unit length 1, a vector that has a length of 1 in the x direction0684

that special vector we are going to call I ̂.0694

The hat means is the unit vector, its length and its magnitude is always 1.0699

In the x direction we call it I ̂.0703

In the y direction, we are going to do the same basic thing.0706

Make a unit vector of length 1 and we are going to call it j ̂.0710

In the z direction I am sure you have not guess by now.0716

A unit vector of length 1 we are going to call k ̂.0720

We could also write our A vector now whatever happens to be as having components of x coordinate × I ̂.0726

That is the unit of vector 1 in the x direction + y value × j ̂ + z value × k ̂.0740

That will get a little funky until you get used to it.0760

That is pretty common in a bunch of textbooks to see these unit vectors along with the vector in front of them.0763

These two types of notation really mean the exact same thing.0770

Let us see how that can be useful down here.0777

Let us try it again with a fairly simple vector addition problem.0779

I am going to draw my axis again, give ourselves y, x, and the z axis.0784

What we are going to do is we are going to define a couple of vectors.0805

The first one I am going to define is called P.0808

Where P goes from the origin to some point that is 4/x, 3 in the y, 1 in the z.0812

It might be somewhere about there.0824

We will draw our vector from the origin there and that is our point P 4, 3, 1.0828

Let us take a second vector and we will call it q.0836

We are going to go 2 points in the x, 0 in the y, 4 in the z.0842

We call this the vector q which leads us to q 2, 0, 4.0849

If we want to add this two to get r well when you are in three dimensional space0859

it is starting get pretty tough to see what is going on if you we want to line this up tip to tail0864

and come up with a reasonable solution that you can actually make sense of graphically.0869

What we are going to do is we are going to say that our r vector is equal to P + q which implies then our P vector is equal to 4, 3, 1, if we use bracket notation.0874

Our q vector is equal to 2, 0, 4.0899

If we want to add these up to find r all we have to do is in its bracket notation is to add up the individual components.0907

4 + 2= 6, 3 + 0= 3, 1 + 4= 5 so 6, 3, 5 will give us the vector to our resultant.0917

Let us draw that in here.0928

We go 1, 2, 3, 4 about 6 on the x, 3 in the y, 5 on the z.0930

As I draw it is going to come out somewhere around there depending on your perspective we will call that r.0939

What if you just did that graphically which you can get by lining these up?0949

It is going to be a really tough to see what is going on.0953

We get in the three dimensions especially analytically looking at these vectors sure makes a lot more sense and it makes life a whole lot easier.0956

A little bit more with vector components.0967

A soccer player kicks the ball with an initial velocity of 10 m/s if an angle of 30° above the horizontal.0970

Find the magnitude of the horizontal component in vertical component of the balls velocity.0977

I like to start with graphs wherever possible to help me visualize the problem.0983

And this look like it is in two dimensions.0989

We got a vertical and horizontal component to our problem.0991

We call this our x and y and it has an initial velocity of 10 m/s in angle of 30° above the horizontal.0996

Let us see if we can eyeball roughly 30°.1006

The magnitude of our vector is 10 m/s at some angle 30°.1010

We want the horizontal component and vertical component.1016

Let us draw these in first.1020

Our vertical component will be that piece and our horizontal component will be that piece.1022

The x component of V, our initial velocity V = 10 m/s.1033

Our x component is going to be V cos 30° because we are looking for the adjacent side of that right triangle.1042

It is going to be about 8.66 m/s.1051

The vertical component Vy is going to be V sin 30°.1055

We got the opposite side there which is going to be about 5 m/s.1063

And of course if we ever want to put these back together, if we have the components in one of the whole we can just use the Pythagorean theorem.1069

Let us take a look at the second example.1076

An airplane flies with the velocity of 750 kph 30° South East.1080

What is the magnitude of the planes eastward velocity?1086

Lets draw a diagram again and we would call that north and south so this must be east and west.1090

It is traveling the velocity 750 kph 30° South of East in this basic direction.1109

Our V = 750 kph and an angle of 30° South of East.1123

The magnitude of the planes eastward velocity, it looks like we are after an x component here.1133

Let us draw that in, we are after just this piece, the x component.1139

Vx equals that is the adjacent sides and that going to be V cos θ which is going to be 750 kph.1148

The magnitude of our entire vector × the cos 30° and that can give you about 650 kph.1157

How about the problem with the magnitude of vector?1170

A dog walks a lady 8m due north and then 6m due east.1175

You probably all seen that before.1179

It is a really big dog and little even lady the dogs doing the controlling.1181

Determine the magnitude of the dog’s total displacement.1185

The way I do that is looks like we have a couple of vectors that we can add up.1190

Dogs walk lady 8m due north so I draw vector magnitude 8m due north and then 6m due east.1194

Determine the magnitude of the dog’s total displacement.1211

Displacement being the straight line distance from where you start to where you finish.1214

The way I will do that then so we are going to from the starting point of our first to the ending point of the last.1219

That red vector represents the total displacement.1228

How do I find the magnitude of that?1233

It is a right triangle I can use that Pythagorean theorem.1236

A² + B²=C².1239

Our hypotenuse is going to be =√(a² + b² ) which is the √(8² + 6²).1246

64 + 36 is going to be 100 m² which implies then that C is the square root of that which is going to be 10 m.1257

If we want to find the angle, let us define an x axis.1271

What if we wanted to find that angle θ?1275

We could if we wanted to θ equals inverse tan of the opposite side of a right triangle divided by1278

the adjacent side that could be the inverse tan of.1286

The opposite side is this piece here that is not shown but we can see that is 8m ÷ the adjacent side that is going to be the length of 6m which comes out to be about 53.1°.1290

We are not asked for that but if we were we would know how to go calculate it.1305

Let us look at the vector addition problem.1310

A frog hops 4m angle of 30° North of East.1312

He then hops 6m angle of 60° North of West.1316

What was the frog total displacement from his starting position?1320

Alright we are getting a little bit more challenging here.1324

Let us draw what we have north, east, and west.1328

Frog hops 4m angle of 30° North of East1346

He goes 4m North of East.1351

Let us called about 4m assuming that is 30° and I'm just estimating these.1359

He then hops 6m an angle of 60° North of West.1366

To figure out what that is let us draw x and y here.1372

At 60° North of West that is going to be roughly this direction and it goes 6m that way and we will say that is something like that.1375

There is our second piece of 60°.1400

What is the frog’s total displacement from his starting position?1403

Thought displacement goes from the starting point of our first to the ending point of our last.1407

We could be fairly accurate if we are doing this with the protractor we will call that C, A, and B but it is a lot easier to do analytically.1414

Let us take a look at how we add A and B up to get C in vector notation.1426

A vector has an x component that is going to be 4m cos 30° and the y component that is going to be 4 m sin 30°.1431

4m and our angle is 30° we can break that to x and y components.1448

Our B vector the 6m and angle of 60° North of West is just going to be, since we can tell it is going left to1453

begin the x piece is going to be -6 m cos 60° in the y component 6m sin 60°.1462

If we then want to find out our total C, our C vector is just going to be the sum of those.1477

That is going to be our x components 4m cos 30° + -6m cos 60° for the x and for the y we have 4m sin 30° +6m sin 60°.1487

And if we put that all together I find that our C vector is about 0.46m, 7.2m.1515

Our estimation here of ½ m to the right, 7.2 to the left, it is roughly in the ballpark for just a eyeballing that one.1524

If we want to know the magnitude of our answer the magnitude of C like that is going to be.1533

Well we have these two components x and y it is going to be the √ x component 4.6m² + 7 ⁺2m = 7.21m for the magnitude.1542

If want the angle from the origin we can go back to our trig.1562

Θ is the inverse tan of the opposite over the adjacent which is going to be our 7.2m ÷ 4.6m or about 86.3° North of East.1569

You can see how valuable, how useful these vectors can be and breaking them up into components in order to manipulate them.1587

Find the angle θ depicted by the blue vector below given the x and y components.1596

Let just hit this again because it is often times a trouble spot as we are getting started.1602

We know the opposite side and the adjacent sides so θ is going to be the inverse tangent of the opposite side over the adjacent.1607

Put that in your calculator and making sure it is a degree mode for a question like this where we want an answer in degrees and I come up with 60°.1615

Let us go hit vector notation a little bit more.1626

Unit vector notation we said can be written as A vector = x × I ̂ the unit vector in x direction + its y value × j ̂ + z value × k ̂.1629

The vector component notation we would write that S or bracket notation x, y, z.1646

You will see vectors written in many different ways in many different textbooks.1654

Some of the standard ones I used most often are the capital letter with a line over it or lower case letter with the line over it is the vector.1658

Sometimes you will see just a very old letter in something like a textbook, if it is bold that usually means it is a vector.1670

If you want the magnitude of the vector you have the vector symbol inside absolute value signs that would be a magnitude.1679

In other books you will see double lines surrounded to indicate magnitude.1687

Still in other if you see A written bold for A vector.1692

A that is not bold may indicate the magnitude of a vector if you do not add that extra symbology to explain it is a vector could be magnitude.1696

Take a look at your book that you are using and try and find out what it is using1706

and maintain some consistency with that throughout the course.1710

We will use a couple of these just you get used to all the different forms of notation.1713

We can add vectors we can subtract vectors we can also multiply vectors.1718

But there are two types of vector notation.1724

The Dot product also known as a scalar product takes two vectors you multiply them and what you get is an output as a scalar.1727

The cross products or vector product gives you a vector as the output of the multiplication.1734

We will talk about these different types starting with the Dot product and the Scalar product.1740

What it does is it tells you the component of a given a vector in the direction of the second vector really multiplied by the magnitude of that second vector.1745

You would write it as A • B and the result is the component of vector A that is in the direction of vector B multiplied by the size of vector B.1754

How can we define that a little bit better?1765

A • B = to the magnitude of A magnitude × magnitude of B × cos of the angle between those two vectors.1768

Another definition that you are really going to want to know.1777

Let us see how we can calculate this.1781

In unit vector notation let us assume we have some vector A where A = x component of A × i ̂ the unit vector in x direction + the y component of A × the unit vector1784

in the y direction j ̂ + the Z component of A × the unit vector in Z direction known as k ̂.1799

We are going to also define vector B as the x component of B I ̂ + y component of B j ̂ + the Z component of B k ̂.1810

Therefore A • B we want to do these two together that is just going to be Ax Bx multiply the two x components together + ay by.1826

Multiply the y components together + az bz.1847

And no unit vectors because it is scalar.1853

The dot product gives you a scalar output.1856

In vector component notation if we written A as Ax Ay Az and written B as Bx By Bz then A • B would still be AxBx + AyBy + AzBz.1860

Depending on the type of notation you prefer still get the dot product of the exact same formula.1896

Let us do it.1904

Find the dot product of the following vectors A and B where A is 123 in the x, y, and z directions B is 321.1906

Couple ways we could do this.1915

We will start off doing it the easy way.1918

A • B is going to be the x components multiplied 3 + the y components multiplied 4 + z components multiplied 3 or 10.1925

We also mentioned that you can find that by AB cos θ.1940

Let us do that while you are here but first we need to find the magnitude of A.1945

The magnitude of that vector A we can use our Pythagorean theorem that is going to be the square root of the component squared.1949

1² + 2² + 3² which would be √14.1956

The magnitude of B is going to be √(3² + 2² + 1²) also √14.1965

We could also look at A • B as AB cos θ their magnitude × cos of the angle between them1978

which implies that we know A • B is 10 must be equals √14 × √14 is this going to be 14 cos θ.1993

Or 10 = 14 cos θ therefore θ = the inverse cos of 10/14 which is 44.4° and we just found the angle between A and B.2006

Couple different ways you can do this.2025

Let us take a look at a couple of special dot products.2030

The dot product of perpendicular vectors is always 0 because there is no component of 1 that lie on the other.2033

If their dot product is 0 they are perpendicular.2040

The dot product of parallel vectors is just the product of their magnitudes.2043

One way we could look at this is if we are talking about A • B is AB cos θ.2049

If the angle between θ is 0 cos of 0 is 1 you just get the product of their magnitudes.2061

If however they are perpendicular write that specifically if θ = 0 they are parallel.2069

AB cos θ however if they are perpendicular and θ = 90° cos 90 is going to be 0 therefore you would get 0 for your dot product.2076

A couple of dot product properties.2091

First off the commutative property A • B = B • A that works it is commutative.2094

You have A + B vectors .C = A • C + B • C associative property.2108

If you are taking a derivative, the derivative of A • B = to the derivative of A • B + the derivative of B • A.2125

Let us do a couple more examples here.2146

If A -2, 3 and B is 4, by find a value of By such A and B are perpendicular vectors.2150

The way to start this is recognizing that if they are perpendicular then their dot product A • B must be = 0.2158

A • B =0 let us take a look what happens when we do our dot product.2170

The x components multiplied we get -8 + 3By = 0 or 3By = 8.2177

By = 8/3 if they are going to be perpendicular.2191

That is the dot product.2201

The cross product of two vectors gives you a vector perpendicular to both because magnitude is equal2204

to the area of the parallelogram defined by the two initial vectors.2209

Sounds complicated but let us say that we are talking about a couple of vectors where A cross x symbol with B gives you some vector C.2213

The area of a parallelogram defined by those vectors let us see if we can scope that out a little bit.2230

I will draw something kind of like that and the area of that parallelogram defined by AB is the magnitude of your vector C.2236

And C going to be perpendicular to both A and B where its direction is given by the right hand rule.2252

We have to do as is if you have A cross B take the fingers of your right hand, point them in the direction of vector A2258

bend them in the direction of vector B and your thumb points in the direction that is positive for C.2266

A cross with B gives you C a right-hand rule for cross products.2274

And that is going to come up in this course multiple times as well as in the ENM course.2279

Now interesting to note the cross product of parallel vectors is 0 which it has to be because they cannot define parallelogram.2284

Defining the cross product.2295

A × B the magnitude of A cross B is AB sin θ.2297

The direction given by right-hand rule where is before A • B is value was AB cos θ.2302

The cross product only the magnitude of the vector is AB sin θ still have to worry about direction2309

because the cross product it outputs a vector not a scalar.2315

If we look at the cross product with unit vector notation A cross B = Ay Bz –Az By on the I ̂ direction the x component.2321

The y component Az Bx – Ax Bz in the y direction the j ̂ component.2335

The z component is Ax By - Ay Bx so you could memorize that formula that is one way to do it because calculating cross product is considerably more complex than dot products.2344

Or the way I tend to do it is with some linear algebra looking at the determinant.2357

What we are going to do is we are going to take the determinant of these vectors such that if we have A cross B where is x is Ax Ay Az the components of B are Bx By Bz.2363

I would start by drawing A 3 × 3 matrix I ̂ at the top, j, ̂ and k ̂ is my first row.2375

My second row Ax Ay Az and my third row Bx By Bz.2387

And that is we are going to take the determinant of.2405

When you do that, the way I do this to help me understand and to help do this a little bit more easily is2409

I also repeat these over to the right and left.2414

Ax Ay Az what comes next in the pattern would be Ax Ay and down here we have Bx By.2417

I also need that over here to the left so over here I will have Az before we get Ax.2427

We will have Ay there, we have a By, and the Bz here.2433

To take this determinant when I do it is I startup here and for the I ̂ direction I am going to go down into the right and those are going to be positives.2442

I am going to start with Ay Bz × I ̂ - Az By and all of those are multiplied by I ̂ + for the j component I started to j I go down to the right.2452

I have Az Bx – Ax Bz × j ̂ + Ax By – Ay Bx J ̂.2481

That is another way you can come up with the formula.2519

And typically this is a lot easier for me to remember how to do than memorizing that entire previous formula.2521

An example, find a cross product of the following vectors if we are given A and B we want to find C which is A cross B.2530

A couple ways we can do this.2539

First let us start with looking at the magnitude of A and pretty easy to see2541

that the magnitude of A is just going to be 2 or you could go on the Pythagorean theorem √0² + 2² + 0² still give you 2.2546

B is written in a slightly different notation but that is just the equivalent to 2, 0, 0 and the magnitude of B must be 2.2555

If we want to know the magnitude of C, magnitude of C is going to be magnitude of A.2571

Magnitude of B × the sin of the angle between them is going to be 2 × 2 × the sin of the angle between them.2579

If this is in the y direction this is in the x direction they are perpendicular that is 90° sin 90 is 1 so that is just going to be 4.2589

We know that we are going to have a magnitude of 4 on our answer.2600

Using the right-hand rule we got to be perpendicular to both i and j, to both x and y that means it is in the z direction.2605

If I we are to graph this out quickly.2613

Let us put our x here y, z, if we are doing this we start off with the y.2618

We are bending our fingers in the direction of x that tells me that down is going to be the direction of my positive for my z by right-hand rule.2619

I could just by thinking through this one state that z going to be 0, 0, and that is going to be -4 because of our right-hand rule.2637

A little bit shaky on doing that so instead let us do the determinant.2648

Let us find out analytically how we can do that.2652

We start off with i ̂, j ̂, k ̂.2656

Our first vector is going to be 0, 2, 0 and then for our B vector we have 2, 0, 0.2662

We are going to take our determinant and repeat our pattern.2673

0, 2, 2, 0 we would have a 0, 2 over here and the 0, 0.2675

Our C vector is going to start at i ̂ that is going to be 0-0 that is easy.2684

J ̂ 0 -0 that is easy.2695

K ̂ 0-4 k ̂ C is just -4 k ̂ or 0, 0, -4.2699

Couple of ways we can go about solving them.2712

Let us take a look at a couple of cross product properties.2715

A cross with B = -B × A.2720

A × B + C= A × B + A × C.2732

If we have a constant some C × A cross B = C × A cross B or = A cross C × B.2749

If we take the derivative of a cross product the derivative of A cross B is equal to the derivative of A cross B + A cross B.2772

I think that is good on vector math for the time being.2796

Let us talk for a few minutes about units.2799

The fundamental units in physics there are 7 of them.2802

Our length which is measured in meters, mass in kilograms, time is in seconds, temperature is in kelvins.2806

The amount of the substance is measured in moles, you might remember that from chemistry.2815

Electrical current is in the amperes and luminosity is in candela.2819

And here we are talking about mechanics we are mostly going to be dealing with meters, kilograms, and seconds.2826

All over other units are derived units they are combinations of these fundamental units.2834

Given units we can oftentimes use not to help us check our answers you will see if we have done things right.2842

If displacement is in meters, time is in seconds, velocity would be derived unit meters per second2848

Acceleration the meter per second, per second or meter per second squared, force is measured in Newton’s which is really a kg m/s².2854

The gravitational constant capital G is in N m/s² / kg².2863

You can go and see if all the units match up, if they do not you have probably made a mistake.2868

For example this first one does this dimension correct or are there errors?2873

A meter per second is equal to a meter per second times a second + m/s².2877

No, it does not work.2882

If you came up with that formula you probably messed up somewhere.2884

Distance displacement in meters is equal to a meter per second of velocity times a second squared.2888

No, that is not going to work out because that will cancel.2896

Meter equals meter per second is not going to work out so that can not be right.2900

This one over here though the force which is a kg m/s² is n equivalent to gravitational constant2906

which is a N m²/ kg² × mass which is a kilogram × mass which is a kilogram divided by a distance squared.2914

Let us see kg², kg², m², m².2925

Newton which is a kg m/s²= N.2930

Yes, this one works so that formula would be valid.2934

A great way to check your answers as you are going through and doing these problems called dimensional analysis.2937

The part you all have been waiting for calculus.2944

AP physics C is not really about calculus.2948

We will use calculus as a tool throughout the course.2952

We are going to cover just a few basic calculus applications here and2956

you might have seen some of them before you might have not.2959

You can find a much more thorough and detailed accounting of calculus2963

on the www.educator.com courses AP calculus AB and BC right here.2966

Believe it or not you have probably done a lot of calculus already.2973

Ever taken tangent line to find the slope, that is differential calculus or ever look at the area under a graph that is integral calculus.2976

You might not have known it you have probably done some calculus in your life already.2984

Let us take a look at differential calculus first.2990

Differentiation is finding the slope of a line tangent to a curve.2994

The derivative is the slope of the line tangent to a function at any given point, the result of differentiation.2998

If we have a curve here we can really do is take a point of the curve and we are going to try our best to find the slope of that line.3005

And given that function what we are doing when we take a derivative is finding the slope at a given point3015

or finding the function that tells you the slope of the original function that is differentiation.3021

If we say that we have some function a value y which is a function of x which is A some constant A × X ⁺n3028

then the first derivative of y is equal to the derivative of y with respect to x as for notation.3037

Or the first derivative of x, that function x all mean the same thing is equal to n Ax ⁺n -1.3046

This basic formula for a polynomial differentiation that you will become very familiar with throughout the course.3054

The derivative with respect to E ⁺x is just E ⁺x.3061

The derivative with respect to x, the natural log of x is 1/x.3066

The derivative of the sin is the cos, the derivative of the cos is the opposite of the sin.3071

All the things that you will become familiar with throughout the course.3077

Let us do a couple of derivatives and if this is troubling to you, you probably been learning it as you go along in the course.3080

If we wanted to know the first derivative of y or y would respect x we can write as y prime or dy dx.3091

The derivative with respect to x of our y function which is 4 x³ are all just different forms of notation for the same thing.3100

It is going to be 12 x² using that formula for polynomial from the previous slide.3107

Here we got the same basic idea y prime is going to be equal to -2.4 × 0.75 = -1.8x we will subtract 1 from that - 3.4.3116

The derivative of e ⁺2x, y prime would be 2e ⁺2x.3131

Derivative of the sin of 7x², a y prime is going to be equal to 14x through the sin is the cos of 7x².3139

and cos to 2x⁶, y prime is going to be equal to -12x sin 2x⁶.3152

If these are given you some trouble you are probably going to check out some of the calculus lessons before we get too deep into the math here.3169

We will start with very little calculus but it is going to grow as we go to the course you are going to need it as tool before we are done.3176

Integral calculus, integration is finding the area under the curve or adding up lots of little things to get a whole.3185

The integral is the area under a curve at any given point, the result of an integration.3192

Integration and derivation are inverse functions.3197

The integral is the anti derivative or the derivative is the anti integral.3201

If we have a curve like this and we want an integral between a couple points we are actually doing3206

is finding the area between those points going the opposite direction.3212

Suppose the derivative of a function is given can you determine the original function?3218

That is what integration is.3223

If you know the derivative of something is 2x you really need to think what is the original function if derivative was 2x?3225

And that would be y = x² the derivative x² is 2x.3233

You could write this in integral form y =∫ of 2x with respect to x.3238

Which when we do that is going to be x² + this constant of integration where that constant come from.3247

If we had a constant over here if this was x² + 3 the derivative of 3 is 0 so we still have a derivative 2x.3254

We do not know if there was a 3 or not there.3262

This constant says you know there could be some constant there we do not know what it is yet.3264

It could be -5 it could be 0 it could be 37,000.3269

Just being there as a piece we do not know about and we have some different tricks to make that go away3273

in order to find out what those constants are later on.3278

Let us look at a couple of common integrations.3284

The integral of x ⁺ndx follows this formula 1/n + 1 x ⁺n + 1 + some constant and n cannot be -1 or else you would have an undefined function.3287

The integral of the E ⁺x dx would just be e ⁺x.3301

The integral of the dx/x is the natural log of x.3305

The integral of the cos of x is the sin of x and the integral of the sin of x is the opposite of the cos of x.3308

Let us do a couple quick integrations for practice.3316

The integral of 3x² dx is just going to be x³ as to think about the derivative of x³ is going to be 3x².3320

We have to remember our constant of integration + C.3328

The integral of 2 cos 6x dx now what I would do there is recognize that3332

when I do this we will have to do integral of 2 cos 6x dx.3338

We are going to have a 6 in here in order to integrate I need to put 16 over there so they are maintained my same value so that is going to be = sin 6x/3 + C.3347

They are more involved integration.3363

The integral e ⁺4x dx is going to be integral of e ⁺4x dx now what this in the form e ⁺xdx.3366

We are going to need a 4 here which means I am going to have to put of 1⁄4 out there to maintain my original value.3376

That is going to be e ⁺4x / 4 + constant of integration.3383

And finally integral here with some limits.3390

When we have these limits that is really telling us what values we are integrate and allows us to get rid of that constant of integration.3393

I would write that as integral of 2xdx well that is x² evaluated from x = 0 to x = 4.3400

What that means is what you are going to do is you are going to take your x²3409

and you are going to plug your top value that 4 in the first 1 - your same thing but with this value plug in for x.3413

That is going to be equal to 4² – 0² which is 16.3427

If these are troubling the www.educator.com lessons on calculus are outstanding.3435

Let us look at this one more way.3442

If we said this was the area under the curve how could that look?3444

Let us draw a quick graph and assume that we have some y function where y = 2x.3448

There is y there is x.3461

We are going to look from the limits where x = 0 to x = 4 that means that over here at 4 the y value is going to be 8.3464

We said that integration give you the area under the curve.3477

Let us figure out what the area is under this curve.3480

That is a triangle so the area of that triangle formula that for the area triangle is ½ base × height is going to be ½ × our base 4 × height 8 which is 16.3485

Analytical version and graphical version gives you the same value.3500

What you are really doing is finding the area under that graph.3506

Let us take a look at our last example for this lesson.3511

The velocity of a particle is a function of time is given by the equation v of t = 3t².3517

The particle starts at position 0 at time 0.3523

Find the slope of the velocity time graph as a function of time?3526

That will give you the particles acceleration function.3531

Acceleration is the slope of that vt graph, Velocity vs. time which we could write is V prime of t.3535

The first derivative of V which is going to be the derivative with respect to T and derivative with respect to time of whatever that function is.3544

Vt² which happens to be 60 so the acceleration of the particle is equal to 6 times whatever the time happens to be.3553

When you know the velocity function you can find the acceleration, calculus.3562

Find the area under the velocity time graph as a function of time to give your particles position function.3568

Position we will call that r is going to be the integral of our velocity function with respect to time will be the integral of 3t² dt which should be t³ + our consonant of integration.3574

But here is a little trick to making a consonant of integration go way.3592

We know by one of our boundary conditions that the position of our particle at time 0 equals 0 because that is given up here in the problem.3596

If r=0 at time 0 that means if we plug 0 in here for time 0 r must equal z.3607

But r is 0 at that time so z must equal 0.3616

Therefore our total function is r = t³ z is 0 in this problem.3620

We are able to come up with the particles position function based on velocity.3627

We took the derivative in one direction to find the acceleration.3633

We took the integral going the other way in order to find it is change in position.3636

Alright that helpfully gets you a feel for the type of math we will be using in this course.3642

Thank you for watching www.educator.com.3646

Come back real soon and make it a great day everyone.3648

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