  Dan Fullerton

Describing Motion I

Slide Duration:

Section 1: Introduction
What is Physics?

7m 12s

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

1h 51s

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

23m 47s

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

36m 47s

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

30m 34s

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

30m 24s

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

23m 57s

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

23m 57s

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

20m 41s

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

32m 10s

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

20m 31s

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

24m 58s

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

37m 34s

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

28m 4s

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

54m 56s

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

16m 44s

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

13m 9s

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

46m 30s

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

28m 26s

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

21m 36s

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

32m 52s

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

24m

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

26m 9s

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

56m 58s

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

33m 2s

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

1h 1m 12s

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

34m 59s

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

28m 11s

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

28m 11s

Intro
0:00
Question 1
0:15
Part A: I
0:16
Part A: II
0:46
Part A: III
1:13
Part B
1:40
Part C
2:49
Part D: I
4:46
Part D: II
5:15
Question 2
5:46
Part A: I
6:13
Part A: II
7:05
Part B: I
7:48
Part B: II
8:42
Part B: III
9:03
Part B: IV
9:26
Part B: V
11:32
Question 3
13:30
Part A: I
13:50
Part A: II
14:16
Part A: III
14:38
Part A: IV
14:56
Part A: V
15:36
Part B
16:11
Part C
17:00
Part D: I
19:56
Part D: II
21:08
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• ## Related Books 1 answer Last reply by: Professor Dan FullertonThu Oct 10, 2019 10:55 AMPost by Cyrus Seyrafi on October 10, 2019Does that mean that average speed is not equal to the magnitude of average velocity? 1 answer Last reply by: Professor Dan FullertonFri Jun 9, 2017 6:25 PMPost by K Lee on June 9, 2017For 3.5Q on the AP Physics C Companion: Mechanics book, I think you meant t=1s and t=6s, not t=1s and t=5s, unless I am misreading. I didn't know where else to post this. Thanks! 1 answer Last reply by: Professor Dan FullertonTue Dec 1, 2015 8:25 AMPost by Jim Tang on November 30, 2015In Example 7, Q2, how come you took "distance traveled" to mean displacement, even though it's understood somewhat. I thought we were supposed to distinguish between them, as you pointed out earlier in the lecture. 1 answerLast reply by: Jesse LeflerWed Sep 9, 2015 6:11 AMPost by David Schaller on September 8, 2015At 19:24 why is it 2tcubed over three?  I don't quite follow. 1 answer Last reply by: Professor Dan FullertonMon Sep 7, 2015 5:09 PMPost by Shehryar Khursheed on September 7, 2015Where are the practice problems at the end of the lesson? That is, problems that I can do individually rather than you going over it in the lectures. 1 answer Last reply by: Professor Dan FullertonThu May 7, 2015 5:47 AMPost by Joshua Bowen on May 6, 2015Hey i can not see the lecture it says error 2302 1 answerLast reply by: SH LSun Mar 15, 2015 11:40 AMPost by Lily Lau on March 15, 2015At 12:19 for instantaneous velocity. Should the graph have the axis with respect to V instead of X ? 1 answer Last reply by: Professor Dan FullertonSat Mar 14, 2015 11:43 AMPost by Hlulani Rikhotso on March 14, 2015where can i get ap physics c past question papers 1 answer Last reply by: Professor Dan FullertonThu Jan 8, 2015 12:46 PMPost by Isaac Martinez on January 8, 2015Download lectures? 1 answer Last reply by: Professor Dan FullertonSat Dec 27, 2014 12:39 PMPost by Jaime De Vizcarra on December 27, 2014In the example VII: Tortoise and Hare, shouldn't the acceleration of the hare at 40s be a negative value since it is decelerating? {(2.5(m/s)-8(m/s)}/(40s)= -0.1375 m/s^2 1 answer Last reply by: Professor Dan FullertonSun Dec 7, 2014 3:12 PMPost by Dawud Muhammad on December 4, 2014hey professor,other than these vids, what else can i do to sharpen my skills..?? 1 answer Last reply by: Professor Dan FullertonSat Sep 20, 2014 9:08 PMPost by hasan lopez on September 20, 2014Can you post some practice problems so we can apply what we learned

### Describing Motion I

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

### Describing Motion I

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

### Transcription: Describing Motion I

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

I am Dan Fullerton and in this lesson we are going to start our study of kinematics, looking at the descriptions of motion.0003

Our objectives include understanding the general relationships among position, velocity, and acceleration for the motion of a particle.0011

Using kinematic equations to solve problems of motion that constant acceleration and that will be carried over into the second half of our lesson.0020

Writing an appropriate differential equations and solving it for velocity in cases when acceleration is a specified function of velocity and time.0027

Position vs. Displacement.0040

An objects position is its location at some given point in time.0042

The vector from the origin of the coordinate system to the objects position is known as the position vector which is r.0046

Sometimes it is written as rs we are going to use interchangeably in here.0053

If an object moves, its position changes.0057

This change in position is called displacement δ r or δ s.0060

Position and displacement are both vectors.0066

They have a direction as well as a magnitude.0069

In one dimension position is given by the x coordinate and the displacement oftentimes written as δ x.0071

A couple examples about the differences between these.0080

A deer walks 1300m E to a creek for a drink.0083

The deer then walks 500m W of the berry patch for dinner.0087

Before running 300m W it is startled by allowed angry fierce nasty evil raccoon.0091

What distance did the deer travel?0097

The distance the deer traveled 1300 + 500 + 300 = 2100m but what is the deer's displacement?0101

That is where how far it is from that starting point.0112

It went 1300m E and 500m W now it is only 800m E and then it went 300m W.0115

It is 500m E from where it started.0121

Δ x is its displacement is 500m E and because it is a displacement, it is a vector it needs a direction as well.0124

Average speed is the distance traveled divided by the time it took to travel that distance.0136

¯V oftentimes depicts average speed which is distance travel ÷ time.0141

Average speed is a scalar and is measured in m/s.0148

Remember that speed S is a scalar S.0153

If we look at velocity on the other hand, the velocity is the rate at which position changes0159

and position as a vector so the rate at which it changes is also a vector velocity.0163

Now average velocity is the displacement during a time interval divided by the time interval, not the distance.0169

The displacement is average velocity.0174

Average velocity is a vector, it has a direction but it also has units of m/s.0177

They are awfully easily to confuse taking the same symbol V.0183

Velocity V is a vector, speed S is a scalar S.0187

Let us take a look.0195

The deer walks 1300m E to the creek, 500m W to the berry patch before running 300m W when it is startled by that loud evil angry nasty raccoon.0197

The entire trip took 600s or 10 min.0207

What is the deer’s average speed?0211

V average is distance over time it traveled 2100m was its distance / 600s.0215

The average speed was 3.5 m/s.0224

The deer’s average velocity however average velocity is δx /t which was 500m E.0229

Its displacement divided by the time 600s or 0.83 m/s E.0239

Notice how subtle these are in their differences.0251

Average speed you are worried about distance.0254

Average velocity you need displacement.0256

Let us do an example with our dear friend Chuck the hungry squirrel.0263

Pork chop travels 4m E and then 3m N in search of an acorn.0267

The entire trip takes him 20s.0272

Find how far Chuck travelled.0274

That is easy.0277

The distance travelled is 4m + 3m which is 7m.0278

Chuck's displacement is however is a little trickier.0284

Chuck traveled 4m E and 3m N.0288

Displacement for a straight line distance from where you start to where you finish so that is a 345 triangle that must be 5m.0295

Chuck’s displacement is 5m NE and if we wanted to we can find the angle to be even more specific0305

that would be the inverse tan for calling that angle θ of our opposites over the adjacent 3m /4 m which is about 36.9°.0312

Chuck's average speed or average speed is going to be distance ÷ time or 7m /20 s which is 0.35 m/s.0326

Chuck’s average velocity is a little bit different δx/t how far his displacement divided by time?0340

It was 5m NE/ 20s which can be 0.25 m/s and it is a vector.0347

It needs a direction NE.0357

Subtle difference is you really have to know what you are talking about and read carefully0361

when you get in the distance displacements being velocity considerations.0364

Let us take a look at acceleration that is the rate at which velocity changes.0370

Acceleration is change in velocity overtime.0375

It is a vector and it has a direction and the units of acceleration are m/s or m/s².0378

That can be confusing to a lot of folks the first time you see it.0386

If velocity changes you may go from a velocity of 10 m/s to 20 m/s.0390

If that time it takes you to change your velocity by 10 m/s is 1s you went from 20 m/s to 10 m/s to 20 m/s in 1s.0395

Your acceleration was 10 m/s every second or 10 m/s².0406

As we are talking about the relationship to velocity and acceleration is the derivative of velocity with respect to time or would be written as V prime.0413

Acceleration problem.0427

Monty the monkey accelerates from rest to velocity of 9 m/s and a time span of 3s.0428

Find Monty’s acceleration.0434

Acceleration is change in velocity ÷ time = final - initial velocity over time which is 9 m/s - 0 m/s all in 3s or 3 m/s².0438

Let us take a look at the position vector.0460

If we have a half in space and we have a particle that is moving along that path we could define its position of various points in time.0462

Its first position at some time t1 we can define its position by a vector from the origin to that point.0473

Let us call that the position vector at time t1.0482

A little while later, it is over here at time t2 so we can define the position vector at t2 as such.0486

There is its position vector at time t2.0495

What happened between t1 and t2?0500

That was when we had a change in position δr which would be that position at t2 - the position at t1.0503

If we could define S going from that point to that point there is δr.0514

While we are doing this we can look at the position function.0524

The function of time and realize that our x value changes as a function of time so we have x(t)0527

and the I ̂ direction and the unit vector direction along the x axis + y value is a function of time in the y direction, the unit vector in a y direction J ̂ .0537

If you prefer you could write this as x is a function of time for the x coordinate, y is a function of time for the y coordinate.0550

Of course you can extend at the three dimensions as well.0559

Then our average velocity vector is just going to be change in r over some time interval.0563

If we take that and go further in the average velocity, we have a particle traveling along the path find the average velocity the between 1 and 6 seconds?0574

The average velocity of the change in position divided by time before looking in the x that is going to be x - x is 0/t or we are just traveling in one dimension.0584

It is a time vs. distance traveled graph which is going to be our final value to about 5m - 2.5m/ 6s it is about 5 or 6s – 1s or 2.5/5 which is going to be about 0.5 m/s.0596

Interestingly though we could also look at the slope here.0628

If we go when we try and take the slope for those two points, we will pick a couple points on our graph.0632

It looks like an easy one to pick will be 0, 2 and we will also go over here to 6s and say that we are at 5.0644

Our slope is rise over run is going to be 5m -2m/6s -0 s or 3m/6s is 0.5 m/s.0650

The slope of the position time graph gave us the same thing as velocity it does give you the velocity.0670

Looking at instantaneous velocity.0679

Average velocity observed over an infinitely small time interval.0682

As you make that time interval smaller and smaller until it becomes infinitesimally small you get the instantaneous velocity at that exact point in time.0686

Instantaneous velocity is the derivative of position with respect to time.0696

Velocity as a function of time is the derivative of position with respect to time or you could write that S prime.0702

We wanted to know what the absolute instantaneous velocity was here 3s,0712

we would find tangent to the curve the slope right at that point in time.0717

That is 2 m/s is our slope that is the instantaneous velocity at that point in time.0722

Now we also have the xt graph, the area under velocity time graph is the displacement during that time interval.0729

If we make this instead we go to a velocity time graph.0739

This shows our velocity is a function of time.0743

The area under it, if you integrate that integral of velocity or just find the area that your change in position for that time interval.0746

Acceleration is the rate which the velocity changes.0758

Acceleration is the limit is Δ t goes to 0 and Δ v/Δ t.0762

Usually make that time interval shorter and shorter or the derivative of velocity with respect to time.0766

Since velocity is a derivative of position that is the 2nd derivative of the x respect the time.0773

And note that we write that as d² x/dt² really squaring anything this is talking about the 2nd derivative.0780

The derivative and then to take the derivative of the first derivative.0787

The average acceleration is Δ v/t so the velocity time graph we could take the slope here and I get something that looks kind like this I think take the slope at that point.0792

If we take the slope of that line, the slope of the velocity time graph at a specific point that would be about -.18m/s².0810

Which means that the acceleration at time t= 4s right where that point lines up would be -.18 m /s².0822

The slope gives you the acceleration at that point in time.0833

We can also look at some graph transformations.0838

If you have a position time graph and you take the slope you can get a velocity time graph.0841

Take the slope of the velocity time graph you can get an acceleration time graph.0847

Or going the other way start with acceleration time graph, if you take the area under the graph you get the change in velocity.0852

If you have a velocity time graph and you take the area of the integral you get the change in position.0860

You can go from one graph to another based on what you are trying to find using slopes and areas, derivatives, and integrals.0866

Alright the velocity, acceleration in two dimensions.0876

Our velocity vector is the limit as Δ t approaches 0 with a time interval gets infinitesimally small.0880

Δ r/Δ t or we wrote that as the dr dt.0890

The derivative of r with respect to time and if r is in multiple dimensions and that would be the x component derivative of the x component in the x direction0897

+ derivative of the y component in the y dimension for however many the dimensions you might have.0909

Or in bracket notation the dx dt for the x, dy dt for the y.0916

Acceleration then is the derivative of velocity with respect to time which would be the 2nd derivative of x with respect to t in the x direction0927

+ the 2nd derivative of y with respect to t in the y direction or in bracket notation again t ⁺2x/dt², 2nd derivative of y with respect to t.0941

You can keep expanding upon that for however many dimensions you need.0961

Typically we are going to be working with 2 and 3 dimensions in this course.0965

Let us talk a little bit more about derivatives.0970

If we have some function x as a constant × t to some exponent N, the derivative of x with respect to is that power N × our constant × t to the N – 1.0974

The basic polynomial derivative formula.0987

We are going to be using that a bit so let us practice for second.0990

The position of the particles as a function of time is 2 -40 + 2 t² -3 t³.0994

Find a velocity and acceleration of the particles as a function of time.1002

Velocity is a function of time is the derivative of x with respect to the time or you could write it as x prime.1007

You might even see that written as x with dot over it.1015

That is going to be, I am going to start at this side just because I like the bigger exponent first that will be -9t²+ 40 -4.1019

To find our acceleration as a function of time that is the derivative of velocity with respect time.1035

Or the 2nd derivative of x with respect to time or we could write this as V prime or V with the dot,1043

or x double prime or x with two dots they all mean the same thing.1054

Eventually we take the derivative of our velocity and I would come up with -18t + 4.1061

Alright more examples, an object moving in a straight line has a velocity V in m/s that varies with time t.1072

According to this function 3 +2t², find the acceleration of the object in 1s.1080

Acceleration is just the derivative of the velocity with respect to time which is going to be derivative to the velocity is going to be 4t.1087

Since we know in this problem that T = 1s acceleration is just going to be equal to 4 × 1 or 4m/s².1098

For part B, determine the displacement of the object between t = 0 and t = 5s.1114

We want to know the displacement Δ x that is going to be the integral from T = 0 to 5s.1120

We can start using definite integrals here because we are given some limits on the time.1127

Our velocity with respect to time which will be the integral from T = 0 to 5s of 3 +2t² Tt which will be 2t³ /3 + 3t all evaluated from 0 to 5s.1131

Remember what this means, that means we are going to plug the 5 in for the t first so we would get 2 × 5³ /3³ /3 + 3 × 5 – 0³ + 3 × 0 = 0.1162

What I come up with here was Δ x is going to be 5 × 5 = 25 × 5 = 125 × 2= 250 ÷ 3 + 15= 98m.1183

There we go the displacement of the object between 0 and 5s.1209

Taking a look at another example.1215

The velocity of time curve for the tortoise and hare traveling the straight line is shown below.1216

I will color the tortoise here in orange and our hare in blue.1222

What happens at time T= 30s?1227

Interpreting these can get a little tricky so let us take our time and go right through it.1231

At T=30s it looks like the tortoise and the hare have the exact same value for speed.1235

A tortoise and hare have the same speed.1244

How do you know the two have travelled the same distance at time T = 60s?1256

Let us see.1262

At T=60s if we look here we are given the velocity time graph.1263

If we want to know distance travelled that is the area under the graph.1269

For the tortoise, that would be the area of this rectangle which is 60 × 4 =240m.1272

For the hare, it is the area of this triangle which you can probably see visually is the same or use ½ base × height you can find that the area for the hare is 240m.1283

At T =60s, the area under each curve is the same so Δ x must be the same.1295

What is the acceleration of the hare at T= 40s?1307

What I would do is I would go over here to T=40s and take a look and say the slope of the line there should give you the acceleration.1322

Acceleration is slope or change in velocity over change in time that looks like we are going from the slope of that line is -8 m/s / 60s is -4/30 – 2/15m/s².1334

That would be the hare’s acceleration.1355

One last one, which of the following pairs of graphs best shows the distance travelled vs. time in speed1360

and speed vs. time for our car accelerating down the hill from rest?1367

Let us take a look at the first one.1373

If distance and time, it looks like it is moving the same amount every time and the speed is constant.1374

If it is accelerating that does not make any sense, it cannot be A.1382

It looks like the further it goes, its distance travelled for each of the time is getting bigger and bigger.1387

The slope of this with the different points gives you your speed graph getting bigger at a constant rate.1392

B looks like it is accelerating your speed is constantly increasing at a constant rate.1397

This does not make sense because the slope of this does not match your speed curve.1403

Same here the distance is increasing linearly while speed is going up quadratically.1408

Hopefully that gets you a start on describing motion.1416

We will go a little bit further with this in our next lesson describing motion part 2.1420

Thank you so much for watching www.educator.com and make it a great day.1424

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