Vincent Selhorst-Jones

Multi-Dimensional Kinematics

Slide Duration:

Section 1: Motion
Math Review

16m 49s

Intro
0:00
The Metric System
0:26
Distance, Mass, Volume, and Time
0:27
Scientific Notation
1:40
Examples: 47,000,000,000 and 0.00000002
1:41
Significant Figures
3:18
Significant Figures Overview
3:19
Properties of Significant Figures
4:04
How Significant Figures Interact
7:00
Trigonometry Review
8:57
Pythagorean Theorem, sine, cosine, and tangent
8:58
Inverse Trigonometric Functions
9:48
Inverse Trigonometric Functions
9:49
Vectors
10:44
Vectors
10:45
Scalars
12:10
Scalars
12:11
Breaking a Vector into Components
13:17
Breaking a Vector into Components
13:18
Length of a Vector
13:58
Length of a Vector
13:59
Relationship Between Length, Angle, and Coordinates
14:45
One Dimensional Kinematics

26m 2s

Intro
0:00
Position
0:06
Definition and Example of Position
0:07
Distance
1:11
Definition and Example of Distance
1:12
Displacement
1:34
Definition and Example of Displacement
1:35
Comparison
2:45
Distance vs. Displacement
2:46
Notation
2:54
Notation for Location, Distance, and Displacement
2:55
Speed
3:32
Definition and Formula for Speed
3:33
Example: Speed
3:51
Velocity
4:23
Definition and Formula for Velocity
4:24
∆ - Greek: 'Delta'
5:01
∆ or 'Change In'
5:02
Acceleration
6:02
Definition and Formula for Acceleration
6:03
Example: Acceleration
6:38
Gravity
7:31
Gravity
7:32
Formulas
8:44
Kinematics Formula 1
8:45
Kinematics Formula 2
9:32
Definitional Formulas
14:00
Example 1: Speed of a Rock Being Thrown
14:12
Example 2: How Long Does It Take for the Rock to Hit the Ground?
15:37
Example 3: Acceleration of a Biker
21:09
Example 4: Velocity and Displacement of a UFO
22:43
Multi-Dimensional Kinematics

29m 59s

Intro
0:00
0:07
Scalars and Vectors
0:08
A Note on Vectors
2:12
Indicating Vectors
2:13
Position
3:03
Position
3:04
Distance and Displacement
3:35
Distance and Displacement: Definitions
3:36
Distance and Displacement: Example
4:39
Speed and Velocity
8:57
Speed and Velocity: Definition & Formulas
8:58
Speed and Velocity: Example
10:06
Speed from Velocity
12:01
Speed from Velocity
12:02
Acceleration
14:09
Acceleration
14:10
Gravity
14:26
Gravity
14:27
Formulas
15:11
Formulas with Vectors
15:12
Example 1: Average Acceleration
16:57
Example 2A: Initial Velocity
19:14
Example 2B: How Long Does It Take for the Ball to Hit the Ground?
21:35
Example 2C: Displacement
26:46
Frames of Reference

18m 36s

Intro
0:00
Fundamental Example
0:25
Fundamental Example Part 1
0:26
Fundamental Example Part 2
1:20
General Case
2:36
Particle P and Two Observers A and B
2:37
Speed of P from A's Frame of Reference
3:05
3:22
Acceleration Shows the Change in Velocity
3:23
Acceleration when Velocity is Constant
3:48
Multi-Dimensional Case
4:35
Multi-Dimensional Case
4:36
Some Notes
5:04
Choosing the Frame of Reference
5:05
Example 1: What Velocity does the Ball have from the Frame of Reference of a Stationary Observer?
7:27
Example 2: Velocity, Speed, and Displacement
9:26
Example 3: Speed and Acceleration in the Reference Frame
12:44
Uniform Circular Motion

16m 34s

Intro
0:00
Centripetal Acceleration
1:21
Centripetal Acceleration of a Rock Being Twirled Around on a String
1:22
Looking Closer: Instantaneous Velocity and Tangential Velocity
2:35
Magnitude of Acceleration
3:55
Centripetal Acceleration Formula
5:14
You Say You Want a Revolution
6:11
What is a Revolution?
6:12
How Long Does it Take to Complete One Revolution Around the Circle?
6:51
Example 1: Centripetal Acceleration of a Rock
7:40
Example 2: Magnitude of a Car's Acceleration While Turning
9:20
Example 3: Speed of a Point on the Edge of a US Quarter
13:10
Section 2: Force
Newton's 1st Law

12m 37s

Intro
0:00
Newton's First Law/ Law of Inertia
2:45
A Body's Velocity Remains Constant Unless Acted Upon by a Force
2:46
Mass & Inertia
4:07
Mass & Inertia
4:08
Mass & Volume
5:49
Mass & Volume
5:50
Mass & Weight
7:08
Mass & Weight
7:09
Example 1: The Speed of a Rocket
8:47
Example 2: Which of the Following Has More Inertia?
10:06
Example 3: Change in Inertia
11:51
Newton's 2nd Law: Introduction

27m 5s

Intro
0:00
Net Force
1:42
Consider a Block That is Pushed On Equally From Both Sides
1:43
What if One of the Forces was Greater Than the Other?
2:29
The Net Force is All the Forces Put Together
2:43
Newton's Second Law
3:14
Net Force = (Mass) x (Acceleration)
3:15
Units
3:48
The Units of Newton's Second Law
3:49
Free-Body Diagram
5:34
Free-Body Diagram
5:35
Special Forces: Gravity (Weight)
8:05
Force of Gravity
8:06
Special Forces: Normal Force
9:22
Normal Force
9:23
Special Forces: Tension
10:34
Tension
10:35
Example 1: Force and Acceleration
12:19
Example 2: A 5kg Block is Pushed by Five Forces
13:24
Example 3: A 10kg Block Resting On a Table is Tethered Over a Pulley to a Free-Hanging 2kg Block
16:30
Newton's 2nd Law: Multiple Dimensions

27m 47s

Intro
0:00
Newton's 2nd Law in Multiple Dimensions
0:12
Newton's 2nd Law in Multiple Dimensions
0:13
Components
0:52
Components
0:53
Example: Force in Component Form
1:02
Special Forces
2:39
Review of Special Forces: Gravity, Normal Force, and Tension
2:40
Normal Forces
3:35
Why Do We Call It the Normal Forces?
3:36
Normal Forces on a Flat Horizontal and Vertical Surface
5:00
Normal Forces on an Incline
6:05
Example 1: A 5kg Block is Pushed By a Force of 3N to the North and a Force of 4N to the East
10:22
Example 2: A 20kg Block is On an Incline of 50° With a Rope Holding It In Place
16:08
Example 3: A 10kg Block is On an Incline of 20° Attached By Rope to a Free-hanging Block of 5kg
20:50

42m 5s

Intro
0:00
Block and Tackle Pulley System
0:30
A Single Pulley Lifting System
0:31
A Double Pulley Lifting System
1:32
2:59
Example 1: A Free-hanging, Massless String is Holding Up Three Objects of Unknown Mass
4:40
Example 2: An Object is Acted Upon by Three Forces
10:23
Example 3: A Chandelier is Suspended by a Cable From the Roof of an Elevator
17:13
Example 4: A 20kg Baboon Climbs a Massless Rope That is Attached to a 22kg Crate
23:46
Example 5: Two Blocks are Roped Together on Inclines of Different Angles
33:17
Newton's Third Law

16m 47s

Intro
0:00
Newton's Third Law
0:50
Newton's Third Law
0:51
Everyday Examples
1:24
Hammer Hitting a Nail
1:25
Swimming
2:08
Car Driving
2:35
Walking
3:15
Note
3:57
Newton's Third Law Sometimes Doesn't Come Into Play When Solving Problems: Reason 1
3:58
Newton's Third Law Sometimes Doesn't Come Into Play When Solving Problems: Reason 2
5:36
Example 1: What Force Does the Moon Pull on Earth?
7:04
Example 2: An Astronaut in Deep Space Throwing a Wrench
8:38
Example 3: A Woman Sitting in a Bosun's Chair that is Hanging from a Rope that Runs Over a Frictionless Pulley
12:51
Friction

50m 11s

Intro
0:00
Introduction
0:04
Our Intuition - Materials
0:30
Our Intuition - Weight
2:48
Our Intuition - Normal Force
3:45
The Normal Force and Friction
4:11
Two Scenarios: Same Object, Same Surface, Different Orientations
4:12
6:36
Friction as an Equation
7:23
Summing Up Friction
7:24
Friction as an Equation
7:36
The Direction of Friction
10:33
The Direction of Friction
10:34
A Quick Example
11:16
Which Block Will Accelerate Faster?
11:17
Static vs. Kinetic
14:52
Static vs. Kinetic
14:53
Static and Kinetic Coefficient of Friction
16:31
How to Use Static Friction
17:40
How to Use Static Friction
17:41
Some Examples of μs and μk
19:51
Some Examples of μs and μk
19:52
A Remark on Wheels
22:19
A Remark on Wheels
22:20
Example 1: Calculating μs and μk
28:02
Example 2: At What Angle Does the Block Begin to Slide?
31:35
Example 3: A Block is Against a Wall, Sliding Down
36:30
Example 4: Two Blocks Sitting Atop Each Other
40:16
Force & Uniform Circular Motion

26m 45s

Intro
0:00
Centripetal Force
0:46
Equations for Centripetal Force
0:47
Centripetal Force in Action
1:26
Where Does Centripetal Force Come From?
2:39
Where Does Centripetal Force Come From?
2:40
Centrifugal Force
4:05
Centrifugal Force Part 1
4:06
Centrifugal Force Part 2
6:16
Example 1: Part A - Centripetal Force On the Car
8:12
Example 1: Part B - Maximum Speed the Car Can Take the Turn At Without Slipping
8:56
Example 2: A Bucket Full of Water is Spun Around in a Vertical Circle
15:13
Example 3: A Rock is Spun Around in a Vertical Circle
21:36
Section 3: Energy
Work

28m 34s

Intro
0:00
Equivocation
0:05
Equivocation
0:06
Introduction to Work
0:32
Scenarios: 10kg Block on a Frictionless Table
0:33
Scenario: 2 Block of Different Masses
2:52
Work
4:12
Work and Force
4:13
Paralleled vs. Perpendicular
4:46
Work: A Formal Definition
7:33
An Alternate Formula
9:00
An Alternate Formula
9:01
Units
10:40
Unit for Work: Joule (J)
10:41
Example 1: Calculating Work of Force
11:32
Example 2: Work and the Force of Gravity
12:48
Example 3: A Moving Box & Force Pushing in the Opposite Direction
15:11
Example 4: Work and Forces with Directions
18:06
Example 5: Work and the Force of Gravity
23:16
Energy: Kinetic

39m 7s

Intro
0:00
Types of Energy
0:04
Types of Energy
0:05
Conservation of Energy
1:12
Conservation of Energy
1:13
What is Energy?
4:23
Energy
4:24
What is Work?
5:01
Work
5:02
Circular Definition, Much?
5:46
Circular Definition, Much?
5:47
Derivation of Kinetic Energy (Simplified)
7:44
Simplified Picture of Work
7:45
Consider the Following Three Formulas
8:42
Kinetic Energy Formula
11:01
Kinetic Energy Formula
11:02
Units
11:54
Units for Kinetic Energy
11:55
Conservation of Energy
13:24
Energy Cannot be Made or Destroyed, Only Transferred
13:25
Friction
15:02
How Does Friction Work?
15:03
Example 1: Velocity of a Block
15:59
Example 2: Energy Released During a Collision
18:28
Example 3: Speed of a Block
22:22
Example 4: Speed and Position of a Block
26:22
Energy: Gravitational Potential

28m 10s

Intro
0:00
Why Is It Called Potential Energy?
0:21
Why Is It Called Potential Energy?
0:22
Introduction to Gravitational Potential Energy
1:20
Consider an Object Dropped from Ever-Increasing heights
1:21
Gravitational Potential Energy
2:02
Gravitational Potential Energy: Derivation
2:03
Gravitational Potential Energy: Formulas
2:52
Gravitational Potential Energy: Notes
3:48
Conservation of Energy
5:50
Conservation of Energy and Formula
5:51
Example 1: Speed of a Falling Rock
6:31
Example 2: Energy Lost to Air Drag
10:58
Example 3: Distance of a Sliding Block
15:51
Example 4: Swinging Acrobat
21:32
Energy: Elastic Potential

44m 16s

Intro
0:00
Introduction to Elastic Potential
0:12
Elastic Object
0:13
Spring Example
1:11
Hooke's Law
3:27
Hooke's Law
3:28
Example of Hooke's Law
5:14
Elastic Potential Energy Formula
8:27
Elastic Potential Energy Formula
8:28
Conservation of Energy
10:17
Conservation of Energy
10:18
You Ain't Seen Nothin' Yet
12:12
You Ain't Seen Nothin' Yet
12:13
Example 1: Spring-Launcher
13:10
Example 2: Compressed Spring
18:34
Example 3: A Block Dangling From a Massless Spring
24:33
Example 4: Finding the Spring Constant
36:13
Power & Simple Machines

28m 54s

Intro
0:00
Introduction to Power & Simple Machines
0:06
What's the Difference Between a Go-Kart, a Family Van, and a Racecar?
0:07
Consider the Idea of Climbing a Flight of Stairs
1:13
Power
2:35
P= W / t
2:36
Alternate Formulas
2:59
Alternate Formulas
3:00
Units
4:24
Units for Power: Watt, Horsepower, and Kilowatt-hour
4:25
Block and Tackle, Redux
5:29
Block and Tackle Systems
5:30
Machines in General
9:44
Levers
9:45
Ramps
10:51
Example 1: Power of Force
12:22
Example 2: Power &Lifting a Watermelon
14:21
Example 3: Work and Instantaneous Power
16:05
Example 4: Power and Acceleration of a Race car
25:56
Section 4: Momentum
Center of Mass

36m 55s

Intro
0:00
Introduction to Center of Mass
0:04
Consider a Ball Tossed in the Air
0:05
Center of Mass
1:27
Definition of Center of Mass
1:28
Example of center of Mass
2:13
Center of Mass: Derivation
4:21
Center of Mass: Formula
6:44
Center of Mass: Formula, Multiple Dimensions
8:15
Center of Mass: Symmetry
9:07
Center of Mass: Non-Homogeneous
11:00
Center of Gravity
12:09
Center of Mass vs. Center of Gravity
12:10
Newton's Second Law and the Center of Mass
14:35
Newton's Second Law and the Center of Mass
14:36
Example 1: Finding The Center of Mass
16:29
Example 2: Finding The Center of Mass
18:55
Example 3: Finding The Center of Mass
21:46
Example 4: A Boy and His Mail
28:31
Linear Momentum

22m 50s

Intro
0:00
Introduction to Linear Momentum
0:04
Linear Momentum Overview
0:05
Consider the Scenarios
0:45
Linear Momentum
1:45
Definition of Linear Momentum
1:46
Impulse
3:10
Impulse
3:11
Relationship Between Impulse & Momentum
4:27
Relationship Between Impulse & Momentum
4:28
Why is It Linear Momentum?
6:55
Why is It Linear Momentum?
6:56
Example 1: Momentum of a Skateboard
8:25
Example 2: Impulse and Final Velocity
8:57
Example 3: Change in Linear Momentum and magnitude of the Impulse
13:53
Example 4: A Ball of Putty
17:07
Collisions & Linear Momentum

40m 55s

Intro
0:00
Investigating Collisions
0:45
Momentum
0:46
Center of Mass
1:26
Derivation
1:56
Extending Idea of Momentum to a System
1:57
Impulse
5:10
Conservation of Linear Momentum
6:14
Conservation of Linear Momentum
6:15
Conservation and External Forces
7:56
Conservation and External Forces
7:57
Momentum Vs. Energy
9:52
Momentum Vs. Energy
9:53
Types of Collisions
12:33
Elastic
12:34
Inelastic
12:54
Completely Inelastic
13:24
Everyday Collisions and Atomic Collisions
13:42
Example 1: Impact of Two Cars
14:07
Example 2: Billiard Balls
16:59
Example 3: Elastic Collision
23:52
Example 4: Bullet's Velocity
33:35
Section 5: Gravity
Gravity & Orbits

34m 53s

Intro
0:00
Law of Universal Gravitation
1:39
Law of Universal Gravitation
1:40
Force of Gravity Equation
2:14
Gravitational Field
5:38
Gravitational Field Overview
5:39
Gravitational Field Equation
6:32
Orbits
9:25
Orbits
9:26
The 'Falling' Moon
12:58
The 'Falling' Moon
12:59
Example 1: Force of Gravity
17:05
Example 2: Gravitational Field on the Surface of Earth
20:35
Example 3: Orbits
23:15
Example 4: Neutron Star
28:38
Section 6: Waves
Intro to Waves

35m 35s

Intro
0:00
Pulse
1:00
Introduction to Pulse
1:01
Wave
1:59
Wave Overview
2:00
Wave Types
3:16
Mechanical Waves
3:17
Electromagnetic Waves
4:01
Matter or Quantum Mechanical Waves
4:43
Transverse Waves
5:12
Longitudinal Waves
6:24
Wave Characteristics
7:24
Amplitude and Wavelength
7:25
Wave Speed (v)
10:13
Period (T)
11:02
Frequency (f)
12:33
v = λf
14:51
Wave Equation
16:15
Wave Equation
16:16
Angular Wave Number
17:34
Angular Frequency
19:36
Example 1: CPU Frequency
24:35
Example 2: Speed of Light, Wavelength, and Frequency
26:11
Example 3: Spacing of Grooves
28:35
Example 4: Wave Diagram
31:21
Waves, Cont.

52m 57s

Intro
0:00
Superposition
0:38
Superposition
0:39
Interference
1:31
Interference
1:32
Visual Example: Two Positive Pulses
2:33
Visual Example: Wave
4:02
Phase of Cycle
6:25
Phase Shift
7:31
Phase Shift
7:32
Standing Waves
9:59
Introduction to Standing Waves
10:00
Visual Examples: Standing Waves, Node, and Antinode
11:27
Standing Waves and Wavelengths
15:37
Standing Waves and Resonant Frequency
19:18
Doppler Effect
20:36
When Emitter and Receiver are Still
20:37
When Emitter is Moving Towards You
22:31
When Emitter is Moving Away
24:12
Doppler Effect: Formula
25:58
Example 1: Superposed Waves
30:00
Example 2: Superposed and Fully Destructive Interference
35:57
Example 3: Standing Waves on a String
40:45
Example 4: Police Siren
43:26
Example Sounds: 800 Hz, 906.7 Hz, 715.8 Hz, and Slide 906.7 to 715.8 Hz
48:49
Sound

36m 24s

Intro
0:00
Speed of Sound
1:26
Speed of Sound
1:27
Pitch
2:44
High Pitch & Low Pitch
2:45
Normal Hearing
3:45
Infrasonic and Ultrasonic
4:02
Intensity
4:54
Intensity: I = P/A
4:55
Intensity of Sound as an Outwardly Radiating Sphere
6:32
Decibels
9:09
Human Threshold for Hearing
9:10
Decibel (dB)
10:28
Sound Level β
11:53
Loudness Examples
13:44
Loudness Examples
13:45
Beats
15:41
Beats & Frequency
15:42
Audio Examples of Beats
17:04
Sonic Boom
20:21
Sonic Boom
20:22
Example 1: Firework
23:14
Example 2: Intensity and Decibels
24:48
Example 3: Decibels
28:24
Example 4: Frequency of a Violin
34:48
Light

19m 38s

Intro
0:00
The Speed of Light
0:31
Speed of Light in a Vacuum
0:32
Unique Properties of Light
1:20
Lightspeed!
3:24
Lightyear
3:25
Medium
4:34
Light & Medium
4:35
Electromagnetic Spectrum
5:49
Electromagnetic Spectrum Overview
5:50
Electromagnetic Wave Classifications
7:05
7:06
Microwave
8:30
Infrared and Visible Spectrum
9:02
Ultraviolet, X-rays, and Gamma Rays
9:33
So Much Left to Explore
11:07
So Much Left to Explore
11:08
Example 1: How Much Distance is in a Light-year?
13:16
Example 2: Electromagnetic Wave
16:50
Example 3: Radio Station & Wavelength
17:55
Section 7: Thermodynamics
Fluids

42m 52s

Intro
0:00
Fluid?
0:48
What Does It Mean to be a Fluid?
0:49
Density
1:46
What is Density?
1:47
Formula for Density: ρ = m/V
2:25
Pressure
3:40
Consider Two Equal Height Cylinders of Water with Different Areas
3:41
Definition and Formula for Pressure: p = F/A
5:20
Pressure at Depth
7:02
Pressure at Depth Overview
7:03
Free Body Diagram for Pressure in a Container of Fluid
8:31
Equations for Pressure at Depth
10:29
Absolute Pressure vs. Gauge Pressure
12:31
Absolute Pressure vs. Gauge Pressure
12:32
Why Does Gauge Pressure Matter?
13:51
Depth, Not Shape or Direction
15:22
Depth, Not Shape or Direction
15:23
Depth = Height
18:27
Depth = Height
18:28
Buoyancy
19:44
Buoyancy and the Buoyant Force
19:45
Archimedes' Principle
21:09
Archimedes' Principle
21:10
22:30
22:31
Example 1: Rock & Fluid
23:47
Example 2: Pressure of Water at the Top of the Reservoir
28:01
Example 3: Wood & Fluid
31:47
Example 4: Force of Air Inside a Cylinder
36:20
Intro to Temperature & Heat

34m 6s

Intro
0:00
Absolute Zero
1:50
Absolute Zero
1:51
Kelvin
2:25
Kelvin
2:26
Heat vs. Temperature
4:21
Heat vs. Temperature
4:22
Heating Water
5:32
Heating Water
5:33
Specific Heat
7:44
Specific Heat: Q = cm(∆T)
7:45
Heat Transfer
9:20
Conduction
9:24
Convection
10:26
11:35
Example 1: Converting Temperature
13:21
Example 2: Calories
14:54
Example 3: Thermal Energy
19:00
Example 4: Temperature When Mixture Comes to Equilibrium Part 1
20:45
Example 4: Temperature When Mixture Comes to Equilibrium Part 2
24:55
Change Due to Heat

44m 3s

Intro
0:00
Linear Expansion
1:06
Linear Expansion: ∆L = Lα(∆T)
1:07
Volume Expansion
2:34
Volume Expansion: ∆V = Vβ(∆T)
2:35
Gas Expansion
3:40
Gas Expansion
3:41
The Mole
5:43
Conceptual Example
5:44
7:30
Ideal Gas Law
9:22
Ideal Gas Law: pV = nRT
9:23
p = Pressure of the Gas
10:07
V = Volume of the Gas
10:34
n = Number of Moles of Gas
10:44
R = Gas Constant
10:58
T = Temperature
11:58
A Note On Water
12:21
A Note On Water
12:22
Change of Phase
15:55
Change of Phase
15:56
Change of Phase and Pressure
17:31
Phase Diagram
18:41
Heat of Transformation
20:38
Heat of Transformation: Q = Lm
20:39
Example 1: Linear Expansion
22:38
Example 2: Explore Why β = 3α
24:40
Example 3: Ideal Gas Law
31:38
Example 4: Heat of Transformation
38:03
Thermodynamics

27m 30s

Intro
0:00
First Law of Thermodynamics
1:11
First Law of Thermodynamics
1:12
Engines
2:25
Conceptual Example: Consider a Piston
2:26
Second Law of Thermodynamics
4:17
Second Law of Thermodynamics
4:18
Entropy
6:09
Definition of Entropy
6:10
Conceptual Example of Entropy: Stick of Dynamite
7:00
Order to Disorder
8:22
Order and Disorder in a System
8:23
The Poets Got It Right
10:20
The Poets Got It Right
10:21
Engines in General
11:21
Engines in General
11:22
Efficiency
12:06
Measuring the Efficiency of a System
12:07
Carnot Engine ( A Limit to Efficiency)
13:20
Carnot Engine & Maximum Possible Efficiency
13:21
Example 1: Internal Energy
15:15
Example 2: Efficiency
16:13
Example 3: Second Law of Thermodynamics
17:05
Example 4: Maximum Efficiency
20:10
Section 8: Electricity
Electric Force & Charge

41m 35s

Intro
0:00
Charge
1:04
Overview of Charge
1:05
Positive and Negative Charges
1:19
A Simple Model of the Atom
2:47
Protons, Electrons, and Neutrons
2:48
Conservation of Charge
4:47
Conservation of Charge
4:48
Elementary Charge
5:41
Elementary Charge and the Unit Coulomb
5:42
Coulomb's Law
8:29
Coulomb's Law & the Electrostatic Force
8:30
Coulomb's Law Breakdown
9:30
Conductors and Insulators
11:11
Conductors
11:12
Insulators
12:31
Conduction
15:08
Conduction
15:09
Conceptual Examples
15:58
Induction
17:02
Induction Overview
17:01
Conceptual Examples
18:18
Example 1: Electroscope
20:08
Example 2: Positive, Negative, and Net Charge of Iron
22:15
Example 3: Charge and Mass
27:52
Example 4: Two Metal Spheres
31:58
Electric Fields & Potential

34m 44s

Intro
0:00
Electric Fields
0:53
Electric Fields Overview
0:54
Size of q2 (Second Charge)
1:34
Size of q1 (First Charge)
1:53
Electric Field Strength: Newtons Per Coulomb
2:55
Electric Field Lines
4:19
Electric Field Lines
4:20
Conceptual Example 1
5:17
Conceptual Example 2
6:20
Conceptual Example 3
6:59
Conceptual Example 4
7:28
8:47
8:48
Why Does It Work?
9:33
Electric Potential Energy
11:40
Electric Potential Energy
11:41
Electric Potential
13:44
Electric Potential
13:45
Difference Between Two States
14:29
Electric Potential is Measured in Volts
15:12
Ground Voltage
16:09
Potential Differences and Reference Voltage
16:10
Ground Voltage
17:20
Electron-volt
19:17
Electron-volt
19:18
Equipotential Surfaces
20:29
Equipotential Surfaces
20:30
Equipotential Lines
21:21
Equipotential Lines
21:22
Example 1: Electric Field
22:40
Example 2: Change in Energy
24:25
Example 3: Constant Electrical Field
27:06
Example 4: Electrical Field and Change in Voltage
29:06
Example 5: Voltage and Energy
32:14
Electric Current

29m 12s

Intro
0:00
Electric Current
0:31
Electric Current
0:32
Amperes
1:27
Moving Charge
1:52
Conceptual Example: Electric Field and a Conductor
1:53
Voltage
3:26
Resistance
5:05
Given Some Voltage, How Much Current Will Flow?
5:06
Resistance: Definition and Formula
5:40
Resistivity
7:31
Resistivity
7:32
Resistance for a Uniform Object
9:31
Energy and Power
9:55
How Much Energy Does It take to Move These Charges Around?
9:56
What Do We Call Energy Per Unit Time?
11:08
Formulas to Express Electrical Power
11:53
Voltage Source
13:38
Introduction to Voltage Source
13:39
Obtaining a Voltage Source: Generator
15:15
Obtaining a Voltage Source: Battery
16:19
Speed of Electricity
17:17
Speed of Electricity
17:18
Example 1: Electric Current & Moving Charge
19:40
Example 2: Electric Current & Resistance
20:31
Example 3: Resistivity & Resistance
21:56
Example 4: Light Bulb
25:16
Electric Circuits

52m 2s

Intro
0:00
Electric Circuits
0:51
Current, Voltage, and Circuit
0:52
Resistor
5:05
Definition of Resistor
5:06
Conceptual Example: Lamps
6:18
Other Fundamental Components
7:04
Circuit Diagrams
7:23
Introduction to Circuit Diagrams
7:24
Wire
7:42
Resistor
8:20
Battery
8:45
Power Supply
9:41
Switch
10:02
Wires: Bypass and Connect
10:53
A Special Not in General
12:04
Example: Simple vs. Complex Circuit Diagram
12:45
Kirchoff's Circuit Laws
15:32
Kirchoff's Circuit Law 1: Current Law
15:33
Kirchoff's Circuit Law 1: Visual Example
16:57
Kirchoff's Circuit Law 2: Voltage Law
17:16
Kirchoff's Circuit Law 2: Visual Example
19:23
Resistors in Series
21:48
Resistors in Series
21:49
Resistors in Parallel
23:33
Resistors in Parallel
23:34
Voltmeter and Ammeter
28:35
Voltmeter
28:36
Ammeter
30:05
Direct Current vs. Alternating Current
31:24
Direct Current vs. Alternating Current
31:25
Visual Example: Voltage Graphs
33:29
Example 1: What Voltage is Read by the Voltmeter in This Diagram?
33:57
Example 2: What Current Flows Through the Ammeter When the Switch is Open?
37:42
Example 3: How Much Power is Dissipated by the Highlighted Resistor When the Switch is Open? When Closed?
41:22
Example 4: Design a Hallway Light Switch
45:14
Section 9: Magnetism
Magnetism

25m 47s

Intro
0:00
Magnet
1:27
Magnet Has Two Poles
1:28
Magnetic Field
1:47
Always a Dipole, Never a Monopole
2:22
Always a Dipole, Never a Monopole
2:23
Magnetic Fields and Moving Charge
4:01
Magnetic Fields and Moving Charge
4:02
Magnets on an Atomic Level
4:45
Magnets on an Atomic Level
4:46
Evenly Distributed Motions
5:45
Unevenly Distributed Motions
6:22
Current and Magnetic Fields
9:42
Current Flow and Magnetic Field
9:43
Electromagnet
11:35
Electric Motor
13:11
Electric Motor
13:12
Generator
15:38
A Changing Magnetic Field Induces a Current
15:39
Example 1: What Kind of Magnetic Pole must the Earth's Geographic North Pole Be?
19:34
Example 2: Magnetic Field and Generator/Electric Motor
20:56
Example 3: Destroying the Magnetic Properties of a Permanent Magnet
23:08

• ## Related Books & Services

### Multi-Dimensional Kinematics

• In multiple dimensions, we realize some of the concepts we talked about before are actually vectors: displacement, velocity, and acceleration.
• Position is still location, we just now need to expand our coordinate system to have the right number of dimensions for whatever we are working on.
• Distance is still just a measure of length, but displacement is a vector describing the difference between two locations on our coordinate grid.
• Speed is still a question of "how fast", but velocity has to show direction as well as speed, so it must be a vector.
• Acceleration describes the change in velocity, so it must also be a vector.
• Gravity is an acceleration, so we now express it as a vector: g=(0,  −9.8) [(m/s)/s]. Notice how there is no lateral gravity!
• Our formulas from before now must be modified to take vectors into account:
 →d (t) = 1 2 →a t2 + →vi t + →di .
• In one dimension we had vf 2 = vi 2 + 2a(∆d). Since it's meaningless to square a vector (remember, we can't multiply vectors together), we have to do this component-wise in multiple dimensions:
 vfx   2 = vix   2 + 2ax(∆dx)     &     vfy   2 = viy   2 + 2ay(∆dy)

### Multi-Dimensional Kinematics

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
• What's Different About Multiple Dimensions? 0:07
• Scalars and Vectors
• A Note on Vectors 2:12
• Indicating Vectors
• Position 3:03
• Position
• Distance and Displacement 3:35
• Distance and Displacement: Definitions
• Distance and Displacement: Example
• Speed and Velocity 8:57
• Speed and Velocity: Definition & Formulas
• Speed and Velocity: Example
• Speed from Velocity 12:01
• Speed from Velocity
• Acceleration 14:09
• Acceleration
• Gravity 14:26
• Gravity
• Formulas 15:11
• Formulas with Vectors
• Example 1: Average Acceleration 16:57
• Example 2A: Initial Velocity 19:14
• Example 2B: How Long Does It Take for the Ball to Hit the Ground? 21:35
• Example 2C: Displacement 26:46

### Transcription: Multi-Dimensional Kinematics

Hi, welcome back to educator.com. Today we are going to be talking about multi-dimensional kinematics.0000

What happens when we are moving in more than just one dimension!0005

So, previously we dealt with everything as if we were only one-dimensional, always just one dimension.0009

Otherwise, everything is a scalar, a single number on its own.0011

But, now we are going to see some of the concepts we were dealing with before, actually are vectors in hiding.0017

All the vectors, all the ones we were talking about, are getting from one place to another place, because a place can be many dimensions.0022

We are used to living in a three-dimensional world.0030

You do not just go to the store on a direct path, not everything is always going to be a straight line, if you look at two paths or more paths.0031

So, we are going to have to talk about things using multiple dimensions, either in x-y axes or in x-y-z coordinate system, some kind of coordinate system that has more than one dimension, if we going to be talking about the real world.0039

All of our vectors are going to be the ideas that we have a change, displacement, when we are moving from one location to another location.0045

Velocity, moving at a certain speed.0057

But, more than just speed, it is going to talk about the way we are moving.0060

Acceleration, when we are changing the velocity.0064

Scalars, on the other hand were the things that were just raw length, if we changed it, it is really one-dimensional.0066

The distance between two points is, what it would be if you take a tape measure between those two points.0072

It does not care what the angle is, it just cares where, how far it is from 'a' to 'b'.0076

The displacement on the other hand, would care how did you get there, it is just not the length, there is an entire circle you could go.0082

If you were talking about some length, that length could be pointing in any direction on a circle.0090

You need more than that if you are going to really talk about displacement. Speed is similar to distance.0095

It is just distance/time, if you are looking for the average speed. That is how fast you are traveling, but once again, it is not going to say anything about where you are actually heading.0100

Finally, time.0109

We are going to treat time as a scalar, however I do want to say that if you are to get in to more heavier Physics, you would wind up seeing that time can actually be treated as another part of the space vector, where we are located.0110

That is going to have to do with the idea of space-time, that is beyond what we are talking about right now.0123

So, we are going to be able to treat time as just a single dimensional quantity all the time.0128

First, before we get started, a quick note on vectors.0134

When we are indicating a vector, I like to use a little arrow, like this guy here.0136

Other people, they prefer to use a bold font. If you are looking in a text book do not be surprised if you are seeing bold fonts everywhere, I am writing little arrows.0140

Or, if you look in one text book, and it has bold fonts, and another has little arrows, now you know why!0149

They are both ways of indicating vectors.0154

Sometimes, when we already know we are definitely talking about vectors, it is just assumed that we write it like that.0156

Finally , when we are writing it with handwriting, I tend to write it like this, because frankly, I am a little bit lazy.0161

I could write it like that, but that takes little bit extra effort, so, instead I make this little harpooned hat that lands on top of it.0166

So, little harpoon on top makes that little arrow, and that is how I write it when we are dealing with a vector, when we are writing it up.0174

But when it is written, it has that actual arrow above it. Alright! So, position.0180

Like said before, position is just the location of an object at a given moment in time.0185

But instead of having position be just along a single string, it is no longer just a single dimensional coordinate axis, we now would have to have some sort of grid, may be two dimensions, may be three dimensions.0189

We are going to have to talk about more than just one dimension though.0200

Here, we would be able to talk about the point (3,2), just the x axis distance, and the y axis distance.0203

We go over 3, and then we go up 2. Simple as that.0210

Distance and displacement. So, like before, distance is a measure of length.0216

We got two points 'a' and 'b', the distance is just that line, that straight line distance from one point to the other.0220

It is just the length that you would measure if you were using a tape measure, or walking it out with your feet.0228

The displacement on the other hand is a vector that indicates the change occurring between those two points.0232

So, 'a' to 'b' would be very different vector than 'b' to 'a', and even if we are dealing with the same length over here, but pretend it is the same length, I think it is about the same length, If we had 'c' over here, 'a' to 'c' would be a completely different vector than 'a' to 'b', even though these are the same length.0238

So, the length 'ab' = the length 'ac' = the length 'ca' = the length 'ba'.0256

But, each one of those would be a totally different vector.0262

We are talking about taking a different path, We have to go there in a different way.0265

We 6ake the same number of steps, so to speak, to get there, but we are taking a totally different path to get there.0270

Because we are going to a different final location.0276

For example, say we walk 3 km North of the house, and then 4 km East.0280

We start off at home, we go up 3 km, and then we go East another 4 km.0285

So, what would our displacement be? Our displacement will be from where we started, to where we ended.0293

So, we go like that, and that would be our displacement vector.0298

Our displacement vector would be (4,3).0301

And also notice, we are not saying this explicitly, but we almost always assume 'up' as positive, and 'right' is positive.0311

Because, that is what they are on the x-y axis on the normal Cartesian coordinate system.0321

That is what we are used to in Algebra, we tends to translate over.0325

Once in a while we might want to change which we consider to be positive, and which we consider to be negative, but for the most part we are going to treat going North as positive, going South as negative, going East, to the right as positive, going West, as negative.0329

And if we are dealing with on a flat ground, may be on a table, and had a box, if the box moved to the right, that would be positive, if it moves to the left, that would be negative.0342

If the box moved up, that would be positive, if the box moved down, that would be negative.0351

It is up to us to impose a coordinate system.0357

It is always an important thing to keep in mind.0359

It is us humans who impose a coordinate system on the world and make sense where things are, by giving them assigned values.0360

The assigned values can vary, but we have to choose how we are going to start with to make the window frame to look at the world with.0369

So, our displacement would be this vector right here, (4,3) km.0377

We traveled 4 to the East, we traveled 3 North.0382

It does not matter which direction we have done it.0385

We could have alternatively done it, it does not matter which order we do it in. We could have alternatively done like this, we would have landed at the same spot.0387

But what is the distance between where you started and where you ended? We have got a right triangle.0394

By Pythagorean theorem, we know that 32 + 42 has to be equal to, here is the symbol that we use to show distance, when we want to show how long that vector is, we use the magnitude or the absolute value as you are used to seeing in Math.0400

So, this would be the absolute value squared, so 9 + 16 = the distance squared, so we get 25, which is going to wind up becoming 5.0405

And we would technically get +/- 5, but when we take the square root, we know there is no such thing as a negative value in distance, so we know that we got to have a positive value.0429

We can just forget about the negative when we are doing this.0439

And finally, what distance did we travel? We noticed that there is a difference between the distance from beginning to end, a path a bird might take, versus the path that we actually took with our feet.0442

In this case, if we measure the feet path, we have to go 3 km North, to the first point, and then change to 4 km East.0453

So, we would be 3 + 4 =7 for the total km, for the total distance traveled.0464

Each one of these are being different things.0476

Displacement is the vector that says how do you get from where you started, to where you ended. Which path you have to take.0478

You have to, no matter how you do it, no matter what path you wind up taking, you could walk like this, then walk like this, then walk like this, then walk like this, and then walk like this, and get to the point, the same starting point.0485

That is going to wind up being up being the displacement vector of zero.0496

Because you did not ultimately displace yourself.0499

In this case, we did ultimately displace ourselves, we displaced ourselves 4 km to the East, 3 km to the North. Compare that to the distance.0502

We wound up being in a different location, we are now away from our original location by 5 km.0510

And finally, the feet steps, we had to actually walk using our feet.0517

How far we actually traveled by foot, is going to be the total distance we traveled, 3+4 = 7 km.0522

Three very different ideas, but important to remember that we can talk about each one of these things, and each one of these is going to get used, at different times.0531

Speed and Velocity.0538

Just like before, speed is how fast, it is just how fast you are going some time.0540

So, it is that length that you have traveled, divided by the time it took you to do that travel.0544

But velocity is based on displacement. It asks how you got there, not just how fast you moved to get there, but how you got there.0549

Did you go there at this angle, did you go there at this angle, did you go there at this angle.0557

Speed, since they are all the same length, if they were all the same length, that would be the exact same speed, but traveling in three very different directions.0563

We are going to have different meeting.0573

If you are traveling 60 km/h, to the North, that is very different from if you are traveling 60 km/h to the East.0575

They would be the same speed, in the pedometer in your car, but they are going to be very different velocities, because you are traveling in a different path, you are traveling in different way, different direction.0584

Velocity is based off displacement. Velocity is a vector as well.0585

Velocity is the displacement divided by the time, so v = change in displacement/time, Δ d/t.0598

Let us go back and look at the example we just had, we started in a house, we traveled 3 km to the North, and then we travel 4 km to the east.0607

We do this in 2 hours.0619

If it is 2 hours, our displacement, is equal to (4,3) km.0623

So, what is our average velocity? Average velocity is the change in our displacement.0640

Displacement is the change in the locations, we denote with d.0648

We can talk about displacement, as d, like this, but we could also talk as the change in location.0652

Once again, we have this thing where location, displacement, sometimes they get used for the same letter, so there is little bit of confusion here.0661

But we know we need to talk about how far we traveled.0667

We traveled (4,3) km as a vector.0670

If we want to find out what the velocity is, velocity = (4,3) km / 2 hours, so velocity is going to be equal to (2,3/2) km.0673

Now, if we want to know what our average speed was, we need to see what distance did we travel.0690

Distance = 5 km .0695

So the average speed is going to be, we are going to look at the magnitude, the size of that speed vector and it is going to be 5/2 = 2.5 km/h .0701

Here is an important note. Speed from velocity.0722

As we just saw, on the previous example, if you know something's velocity, we can easily figure out its speed.0724

You could figure out how far it traveled total and divide it by the amount of time to get there.0730

But you can even do better than that. Speed is the length of the velocity vector.0735

We are going to figure it out by looking at how long is the velocity vector.0741

If we want the speed, we are going to look at the distance that we traveled and divide it by the time, or we could look at the velocity vector, which is displacement/time.0746

So, if we just look at the length of the velocity, we already wound up dividing by the time, and it is going to work out.0756

Last time, we got that the average speed for that trip was, 2.5 km/h.0761

But velocity vector, v = (2,3/2) km/h .0770

So, if we want to make it a little bit faster, we just need to see what the magnitude of this is, and it is going to wind up being the exact same as this right here.0779

Let us double check that.0785

If we do sqrt(22 + (3/2)2), we get sqrt(4 + (9/4)), i.e. sqrt(25/4), which is equal to 5/2, exact same thing.0789

So, it is just a question of do we wind up figuring out the distance, and then divide it by the time, or do we wind up figuring out the velocity, which already involves dividing by the time.0818

And then just figure out how long that vector is on its own. So, two ways to do it.0826

Normally it is going to be easier if we want to find out the speed of something, we know its velocity vector, we just toss it in to this.0830

(vx)2 + (vy)2, we can easily figure out what our speed is from that.0836

Acceleration just comes from velocity. Since velocity is vector, acceleration must be a vector.0850

Acceleration = change in velocity / time .0856

So, if we have two different velocities in two different times, we find out what the difference between them is, and then we divide it by the time.0860

Gravity is going to be just as the same as before, but we remember it is only going to be effective only in one axis.0868

There is no lateral gravity on the Earth.0872

If we jump in to the air, you do not get shifted to the right or to the left by gravity.0874

So, we are not going to have any x-axis shifting and since when we normally talk about coordinate systems, this tends to be down, this tends to be up, and these are right and left.0878

Occasionally we will also look from the top-down when we talk about North and South and East and West.0890

But if we are talking about something that is lateral and moving vertically, then we are going to normally do it as up and down being the y-axis.0895

If up and down is the y-axis, then we need -9.8 m/s/s, because we are going down at 9.8 m/s/s .0901

There is not going to any real changes from the formulae before.0914

This is the exact same as it was before, except now are winding up looking at it with vectors.0919

So, everything is now working in terms of the vectors here.0924

Displacement, the location at time t = 1/2 × a t2 + vit + the initial location.0927

This one is a little bit special, in that we are going to wind up having to break it down into its components.0942

Remember, before we had, vf2 = vi2 + 2a × displacement.0948

That was what happened when we were looking in one dimension.0956

But if we are now looking in two dimensions, this equation right here applies in each dimension, so we wind up applying it over the x's and then over the y's.0959

So we just need to separate it and break it into each one.0972

We cannot talk about a vector squared, that does not mean anything.0974

We cannot just scale it, because who is going to multiply who?0977

Instead we have to break it into its components, then we easily square the components of that vector.0984

So, we just keep the x axis, the x components separate from the y components, the y axis.0991

They each do their own thing, as long as the acceleration and the distance, they are not changing at all.0996

As long as you got a steady acceleration, then we are safe, and we can talk about it this way.1001

Acceleration = change in velocity, so now we are just working in terms of vectors, and same as velocity = change in location, or the displacement.1006

Lets us look at some examples:1017

If we have a car driving North at 30 m/s, and then it turns right, so this is vi, turns right, and the final, goes East at 30 m/s.1019

If the car takes 5 s to complete the turn, what is the average acceleration that the car has to have on it for the duration of the turn?1037

The first thing to note here, is we have to be talking in multiple dimensions.1044

We are talking in multiple dimensions, because the car is moving North and then East, so we doing in two totally different directions.1047

We cannot just do this in one coordinate axis.1054

So, what we really want to do is, we want to convert these directions and lengths into actual vectors.1056

First we are going 30 m/s to the North, so that is going to be positive, so we get, (0,30) m/s is vi.1064

vf = 30 m/s to the East.1078

If we want to find out what the acceleration is, acceleration = (change in velocity vectors)/(time involved).1086

That is, (final - initial)/time = ((30,0) - (0,30)) / 5 = (30,-30) / 5 = (6,-6) m/s/s , because it is changing, for every second it goes, it winds up changing either 6 m/s faster to the East, or -6 m/s 'faster' to the North.1094

What that really means is, 6 slower.1138

-6, we mean as we are now being accelerated to the South, which means since it is already moving to the North, it is going to lose some of its speed to that southern acceleration.1143

Second example is going to be a long one, it is going to create a bunch of different ideas that we are all going to hook together to help us understand how the stuff works.1155

We have a ball being thrown out of a window from a height of 10 m, with an initial speed of 20 m/s angled 30 degrees above the horizontal, we can ask some questions.1162

What is the initial velocity vector v for the ball?1170

Then, ignoring air resistance, how long does it take the ball to hit the ground?1173

Finally, ignoring air resistance once again, what is the ball's displacement from its starting point, and what is its distance?1177

We have got some building, and a window, somebody throws a ball out of that window.1185

We know, if we were to set a horizontal straight line, this is going to be an angle of 30 degrees, and this ball comes out at 20 m/s.1194

Down here is the ground, and as time goes on, the ball is going to fly forward, and then gravity is going to take more and more of a share of its velocity, and it is going to eventually land, hit the ground somewhere.1204

We want to figure out how long is it going to be flying through the air, how long is it going to take before its y location hits zero.1218

Once we know that, we can figure out, how far is it going to make it to the right.1228

First, we are going to have to know, what is its initial velocity vector.1232

If we have got, something that is 20 long on this side, and 30 degrees here, we can figure out what the other sides have to be, just use trig.1234

So, this side, since it is the side opposite, is going to be sin(30) × 20.1247

This one, would be cos(30) × 20.1254

So, cos(30) × 20 = sqrt(3)/2 × 10 = 17.3 m/s, approximately.1259

And sine is going to be 10 m/s.1272

That is going to give us initial velocity vector of (17.3,10) m/s. (right, up).1276

Now we are ready to move on, because now we have got a vector, and we can make use of this vector, and break each component off, and work with each component separately.1289

If we want to figure out how long does it take the ball to hit the ground, we do not care what its x component is, it is the same thing if the ball hits the ground here, or here, or here.1296

All that matters is, when does it have that zero.1306

When does the ball make contact at the ground, when is the y axis location zero.1310

If we have got, d(t) = (1/2)at2 + vit + initial location, we can then just break this into looking at the y's throughout.1316

We just look at the y components throughout, and we can figure out when it is going to hit the ground.1333

What is its acceleration?1338

The acceleration is, -9.8, so (1/2) × (-9.8) × t2 + 10t + 10 .1343

We want to solve this for when does its location becomes zero.1363

We want to make that left side set to zero, and we want to see what 't' will allow that zero to appear on the left side.1366

So, -4.9t2 +10t +10.1375

How do we solve something like this, we got a couple of choices.1379

Right here, we got a polynomial.1383

If we have got a polynomial, one of the first things we can do is factor it.1386

To me, that does not look really easy to factor.1390

The easiest thing after that, is to chuck it into the quadratic formula, make that machine go through it.1392

(-b +/- sqrt(b2 - 4ac)) / 2a , and that is going to be the solutions when time is going to make it equal to zero.1402

That gives us one way to do it, one other way, you could just plug that into a computer or a powerful calculator.1414

But let us go through the quadratic formula, because we could all just work through it that way.1423

The answers, t = (-10 +/- sqrt(102 - 4 × (-4.9) × 10)) / (2× (-4.9))1427

Keep working through that, we get, (-10 +/- sqrt(100 + 196)) / (-9.8) = (-10 +/- sqrt(296))/(-9.8) = (-10 +/- 17.2) / (-9.8)1455

Now, we are going to have to ask ourselves, are we going to go with the plus, or are we going to go with the minus.1490

Because mathematically, both of those are correct answers, both of them are times when this equation is going to be fulfilled, it will be fulfilled, that equation will be fulfilled both at the plus and the minus.1493

But, we can see logically that the ball only hits the ground one side, it does not have both a forward time and a negative time.1511

So, what we want to do, is we know our answer can only be a positive time, because this equation is not true at t < 0.1520

Before this moment, we do not know where the ball was, the ball was sitting in some apartment before somebody picked it up, and decided to through it out the window.1528

At that moment, that equation right here, was not true before time = 0.1534

Once time = 0, that is the moment the ball is actually thrown, we know we can start using this equation.1540

So, the only solution that will work, is the one where time > 0 .1546

We look at this, and we are going to have to find a way for us to have a negative number on top, because (-10 + 17.2) will give us a positive on top, and then we divide by a negative, we wind up getting a negative thing.1549

We are going to actually have to go with the negative answer right here, not the negative answer, but the negative of the plus-minus, because it is the only one which would work.1562

If we were to solve it out and get both answers, we would be able to get 2 of them, one positive answer and a negative answer.1571

But then the negative answer we know, cannot work, so we have to chose which sign is going to give us a time that is positive.1579

We chose the negative one, so knock out the positive one, so we are going to get, 2.78 s.1587

So that ball has 2.78 s flight time, before it hits the ground.1597

With that, we can now figure out what its x location is, when it hits the ground.1602

If it has got 2.78 s flight time, we have see how far it has managed to travel in the x, before it hits the ground.1606

Its location in the x is just going to be its 17.3 × 2.78 .(velocity × time)1614

But, do we have to worry about acceleration?1639

The formula that we normally work with, (1/2)at2, remember, there is no lateral gravity, we do not have anything accelerating the ball once it is thrown, so all we have to worry about is the initial velocity.1642

We multiply those out, we get, 48.1 m is the x location, so if we want to combine that, we have got initial location vector is at where, where does it start?1653

Does it start at (0,0)?1673

No!, remember, the house starts up here, 10 above the ground.1674

So we will make the x axis have it be zero at the point of the window it comes out of.1678

So zero for the initial location, but it starts at +10.1683

Final location, it hits the ground, and it is at 48.1, and then it has got 0 here.1688

So the change, the displacement that it experiences between those two locations, is going to be (48.1,0) - (0,10).1697

So we are going to get (48.1,-10), because it traveled down 10 m before it hits the ground, is the displacement it experiences.1711

If we want to know what the distance it experiences is, remember, we throw it out of the window, and it lands over here, if we want to know what its distance is, we just have to measure, there is the displacement, so we just need to know what is this length.1722

So that length is going to be, the size of the displacement vector.1737

Size of the displacement vector, sqrt((48.1)2 + (-10)2).1744

Punch that out, we get, 49.1 m1754

So the ball has a displacement vector of 48.1 to the right, and down 10, so -10 on the y axis.1761

But the distance between where it lands, and where it started moving is 49.1 m.1769

If we want to know how far its travel was, we do not have the mathematical technology yet, we will be able to figure that out.1773

We need to go learn some more calculus before we will be able to figure out what its travel path was, how long it had to move through to get where it landed.1781

But we can figure what is the distance from where it landed to where it started.1790

Hope that all made sense, see you later.1795

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