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Vincent Selhorst-Jones

Vincent Selhorst-Jones

Collisions & Linear Momentum

Slide Duration:

Table of Contents

I. 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
What's Different About Multiple Dimensions?
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
What About Acceleration?
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
II. 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
Newton's 2nd Law: Advanced Examples

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
A Quadruple Pulley Lifting System
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
Friction is Not About Weight
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
III. 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
IV. 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
V. 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
VI. 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
Long Radio Waves & Radio Waves
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
VII. 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
Wait! What About Pressure?
22:30
Wait! What About Pressure?
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
Radiation
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
The Mole and Avogadro's Number
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
VIII. 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
Faraday Cage
8:47
Introduction to Faraday Cage
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
IX. 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
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Lecture Comments (7)

1 answer

Last reply by: Professor Selhorst-Jones
Sun Nov 8, 2015 4:33 PM

Post by Gautham Padmakumar on October 30, 2015

Why are we assuming velocity of pendulum and system are equal ?

1 answer

Last reply by: Professor Selhorst-Jones
Sun Mar 10, 2013 7:20 PM

Post by Norman Cervantes on March 9, 2013

29:05 where does -40v1 come from?

2 answers

Last reply by: Tanveer Sehgal
Sat Nov 24, 2012 6:13 AM

Post by Tanveer Sehgal on November 22, 2012

Hey,

Which calculator would you recommend I purchase? I am planning to study physics and math at a higher level.

Collisions & Linear Momentum

  • If there is no net external force that acts on a system, momentum is conserved:

    p
     

    sys,  start 
    =

    p
     

    sys,  end 
    .
    This is called the conservation of momentum.
  • We can still apply the idea of conservation of momentum in collisions where there are external forces. Generally, collisions happen so quickly and external forces tend to be very small (compared to the forces in the collision). This means that the change in momentum caused by external forces is usually negligible if we look at the instant before and the instant after the collision.
  • Conservation of energy and conservation of momentum are very different. Just because one is conserved does not necessarily mean the other is as well. We categorize collisions based on this:
    • Elastic: Energy and momentum are conserved:
      ∆Esys = 0     &     ∆

      p
       

      sys 
      = 0.
    • Inelastic: Energy is not conserved, momentum is conserved:
      ∆Esys ≠ 0     &     ∆

      p
       

      sys 
      = 0.
    • Completely inelastic is just like inelastic above, except the objects stick together after the collision.

Collisions & Linear Momentum

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
  • Investigating Collisions 0:45
    • Momentum
    • Center of Mass
  • Derivation 1:56
    • Extending Idea of Momentum to a System
    • Impulse
  • Conservation of Linear Momentum 6:14
    • Conservation of Linear Momentum
  • Conservation and External Forces 7:56
    • Conservation and External Forces
  • Momentum Vs. Energy 9:52
    • Momentum Vs. Energy
  • Types of Collisions 12:33
    • Elastic
    • Inelastic
    • Completely Inelastic
    • Everyday Collisions and Atomic Collisions
  • 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

Transcription: Collisions & Linear Momentum

Hi, welcome back to educator.com, today we are going to be talking about collisions and linear momentum.0000

One really great importance of Physics is, being able to understand the interactions between the objects, the way things work between one another in the world.0006

One of the most basic forms of interaction is collision.0015

When two things hit together, it is useful to be able to know how things are going to play out at that moment.0018

If you can model it, we have done all sorts of useful things.0024

Think of all the things that we can do with it.0026

We can investigate a car crash, you could dock a shuttle with a space station, you could look at the collisions between atomic particles.0027

There is all sorts of stuff, stuff bounces off one another, where stuff hit one another and sticks together, you will be able to understand it a lot better if you have momentum and how it is conserved, how it works for momentum.0034

Investigating collisions!0046

If you wanted to talk more about collisions, what ideas would be useful?0048

We can all figure out at this point , momentum is going to be what we are going to be talking about.0052

We just introduced the idea of momentum, and the reason why we did that was to set ourselves up to be talking about conservation of momentum, and how it works with collisions.0058

If we are going to investigate collisions, if we are going to investigate how motion changes, and what is happening as things go on, we want to be able to talk about how motion is working.0066

One of the best things we have for talking about motion is momentum, p = mv.0076

If we got that under our belts, that is going to really help us.0080

But that is not quite enough, we need one more thing.0084

The other thing we need, is the centre of mass.0087

Talking about the centre of mass allows us to explore the motion of a system, how things work together, and that is what we are going to need to be able to talk about collision.0090

Two objects working together, hitting each other, that is the system, you look at those two objects and the way they interact together, they are going to have to interact as a system because we have decided to make them a system.0097

If you look at them as a system, we will be able to know things, the way their momenta are working, and the way their centre of mass is doing things.0106

We are going to use to use centre of mass to learn indirectly more about momentum.0113

We are going to derive this.0117

We can extend the idea of momentum to a system, by looking at the total of the momenta in the system, makes up the momentum of the system.0118

This symbol right here mean, ADD things together. (Might be a little confusing if you have not seen it before.)0126

That says, add up all the momenta in the system.0135

Say we are looking at a pretty basic system that has some p1, and p2, a two object system.0139

Then the momentum of that system is adding them together, and that makes sense, the final momentum of the system is just all of them added together.0146

Just like we said that the energy of the system is all of the energies involved, you look at the energy for, potential gravity, kinetic energy, spring energy, you added them all to find out what the energy of the system was.0153

Exact same thing here: To find out the momentum of a system, you just add all the momenta together.0167

How do we know for sure though, that the mass of the system, (all masses summed up) times the velocity of the centre of mass, is equal to the momentum of the system (all added together)?0173

We can actually prove that.0185

MsysvCM = (m1+m2)(m1v1+m2v2)/(m1+m2) (in 2 particle system, what happens when two objects interact with one another, this works with an arbitrary number of things in our system, but we are going to be looking at the collision between just two things, to understand how this came to be, also note that velocity of centre of mass is the weighted average.)0186

We have got something that cancels off pretty easily, and from there, (m1v1+m2v2) = p1+p2 = Σp.0255

So, all the momenta in the system added up together, is equal to the momentum of the system, which turns out to be, = MsysvCM.0290

We look at the thing as a whole, it is the mass of the system times the movement of the system of a whole, or we look at the pieces that make it up, it is all those pieces added together, great intuition, works out perfectly.0299

Similarly, we are not going to show this mathematically, but we can do this with impulse.0312

The change in momentum of the system, Δpsys = jsys = Fextt.0316

That makes sense, if you are going to have a change in the system, you are going to put an impulse on the system, just like we did when we were working with single object at a time, but it is also going to be the case that the force on it is external, because remember, form our work with centre of mass, internal forces, because of newton's third law, every action has an equal and opposite reactionary force,0322

So, if it is an internal force in the system, sure, stuff is going to happen inside of the system, but the ultimate change to the centre of mass, is going to get a force going one way, but it is going to get the exact same force going the other way, they will cancel out.0343

Centre of mass is going to experience nothing.0355

So, form our work in centre of mass, we know that internal forces have no effect on the system's centre of mass, so no effect on the system's momentum.0356

So, Δpsys = jsys = Fextt.0362

Now we are ready to make something really useful.0372

Conservation of linear momentum: If there are no external forces acting on the system, there is no net external force acting on the system, and there should not be external forces acting on the system, but as long as they are going to cancel out, at static equilibrium, we are going to have the change in momentum of the system = zero.0375

That means, system's momentum at the beginning = system's momentum at the end.0394

If we know that there is no net external force, we know that we are going to have a constant momentum, a conserved linear momentum.0401

That is really handy, because that is going to give us a whole bunch of power, when we are looking at new kinds of problems.0409

Once again, notice that there can be internal forces, it does not matter if there are internal forces, they will have no effect on the system's momentum as a whole.0416

For example, if we had a bomb, or some explosive things, and we blew it up into a bunch of pieces.0422

Each one of those pieces would fly away.0436

But, since it is all internal forces in that system, the system's momentum is going to wind up being that same steady state when it was at rest, it is going to be zero.0439

Of course, there is also gas in there, so it is pretty complicated thing to model, you might have to deal with the gas moving in every which way as well, but the total system's momentum, when we take into account the pieces flying every way, the gas in the centre expanding out rapidly, the end of it is going to wind up also being zero momentum.0450

All sorts of things are going in opposite directions, but when we add them all up together, they ultimately cancel each other, and we get back to rest.0470

Conservation and external force:0477

This means, if we have an external force acting on our system, we require to have no external force to allow the conservation of momentum, so fi there are external forces, we fling up our hands, and go, 'Oh, we cannot do this problem'?0479

No, we do not have to do that, because we can still examine a collision, even when there is an external force. Why?0491

External forces are normally pretty small compared to collision forces.0496

Think about, if a car was crashing into another car, there is going to be little bit of friction on the ground, but for the amount of time that the crash occurs, friction on the road, like small amount of rolling resistance, small amount of internal friction inside the engine, it is going to be very little compared to the massive change in forces involved in two cars crashing together.0501

Those massive forces are going to be so large compared to all the other forces in effect, that we can basically forget about those other forces, we can forget about those external forces, because we are going to be looking at a brief period of time.0527

We can take a snap shot just before the incident of impact, a snap shot just after the incident of impact, because the impact is going to be more than an instant, but it will be something very brief, order of (1/10) of a second or less.0542

We can take a look just before the impact happens, and just after the impact is complete, the external forces are likely to be pretty small compared to the forces involved in the collision, so we can still effectively use conservation of momentum.0555

It is not perfectly correct, but it is so close as to be plenty good.0570

We are going to be totally be able to use it, and it will still make a lot of sense.0576

As long as we consider the instant just before and just after the collision, we can treat it as the momentum is conserved during the collision.0578

It will not be perfectly conserved, but it will be such a small change, that it is as if it was conserved, so it is good enough for us.0584

Finally, momentum versus energy:0593

It is really important to notice that momentum and energy are totally different.0596

mv is very different from (1/2)mv2, conservation of momentum does not imply conservation of the other.0599

In fact, in almost all every day collisions, some energy is lost to heat via friction.0607

When, for example, you drop a ball, the ball hits the ground and rebounds.0611

If it had the same exact amount of energy that it started with, the amount of energy in that ball would have to be the same as what it was when it was dropped in the beginning.0617

That means, that it would be able to bounce and come up to the same height, again and again, but it does not happen.0625

For anybody who has dropped a ball, if you drop a ball, it does not keep bouncing forever, it eventually stops moving.0632

If it stops moving, the energy is being sapped.0642

Momentum is conserved, when you consider the earth and the ball interacting with one another, but the energy is not conserved in the ball-earth system.0644

The ball-earth system winds up dissipating energy.0654

That energy does get dissipated, conservation of energy does still work, but it winds up being dissipated into things that we cannot really watch, like heat energy, sound energy, so we do not really have a good way of doing it.0658

So from our point of view, mechanical energy is not conserved, the way that we know how to deal with energy, just is not conserved.0669

So we cannot use it, we can only keep momentum in that case.0675

The system's momentum will remain constant through a collision, but the energy of the system has no such restrictions.0678

When we are dealing with collisions, we know for sure, that momentum is going to stay the same.0685

But, energy might change.0689

Similarly, when we throw an object, we know that the energy in the object is going to remain the same through its motion.0693

However, since there are external forces, like gravity acting on the object, we know that the momentum of the object is not going to be the same.0699

Sometimes energy will be conserved, with momentum not being conserved, sometimes momentum will be conserved with energy not being conserved.0706

It depends on what you are looking at, you want to think about it as how short is the time frame, what external forces are acting on it, think in terms of collisions, means momentum you can trust, and no external forces means that we can trust in energy.0712

And external forces and energy, remember, gravity is not an external force for energy, because we have taken care of it within our energy formula.0730

But for momentum, there is no such formulae for these other things, like gravity or friction.0737

Still, finally, there are some collisions where energy is not lost.0742

So, we are going to have to categorize our collisions into different types.0747

When momentum is kept, when energy is kept; when momentum is kept, when energy is not kept.0749

Fist one, elastic: Collision is called elastic when energy is conserved in addition to momentum.0754

The change in the energy of the system = zero, and the change in the momentum of the system = zero.0760

These are really uncommon in everyday activities.0764

Stuff that we are used to in reality, in real life, it pretty much never happens, if not completely never.0767

Inelastic: Collisions that we do see in everyday life.0774

A collision is called elastic when energy is not conserved during the collisions.0777

So, we cannot trust the energy of the system to be equal to the energy of the system at the end.0780

Beginning system energy, not equal to ending system energy.0784

But, the change in the momentum of the system, is trustable, we can keep that.0790

The beginning of the collision, the end of the collision, they are going to wind up having the same momentum on both sides.0794

Those are called inelastic collisions.0799

When energy is not conserved, inelastic.0802

Finally, in addition, when object involved sticks together after the collision, when they hit each other and they just keep moving as just one thing, it is called completely inelastic.0805

They just do not hit each other, and lose some energy, they hit each other, and get turned into one object, or something that can be treated as one single object.0814

Finally, almost all everyday collisions are inelastic, sometimes it is useful to approximate some collisions as elastic.0823

However, in the atomic world, particles often have elastic collisions.0830

In the atomic world, elastic collisions are actually really common and standard.0833

But, form the sort of thing that we see in everyday life, we are almost never going to see an elastic collision.0836

However there are certain things that we can approximate as elastic collisions, and learn some more stuff about.0841

On to the examples.0847

Two cars are being driven directly towards one another.0849

They are remote controlled, so nobody is hurt in the process of this example!0851

First car has mass, m1 = 1000 kg, and velocity, v1 = 30 m/s.0855

Second car has mass, m1 = 2000 kg, and traveling in the opposite direction, so, -20 m/s.0861

On impact, they stick together, so we are having a completely inelastic collision.0866

What is the velocity of the twisted hulk of metal they now make just after they collide?0870

We know that, psys(beginning) = psys(end).0875

p1(initial) + p2(initial) = p1(final)+p2(final).0886

What about the things on the right, we no more have two separate objects, we have one, stuck together.0900

In reality, this is not going to be that, it is going to the momentum of the twisted hulk of metal, momentum of the wreck, momentum of the whole.0911

So, m1v1 (initial) + m2v2(initial) = (m1+m2)vhulk.0930

So, 1000×30 + 2000×(-20) = 3000vhulk, so, -10000 kg m/s / 3000 = vhulk = -3.33 m/s, approximately.0957

That is what is left, that is what the speed that is left, actually velocity, since we know that it is going to move in the direction of the heavier car, heavier car was going to slower, it had more momentum, so it winds out in the end.1000

We are going at a -3.33 m/s velocity.1013

Example 2: Two billiard balls of mass m1 and m2, same mass of 0.16 kg, are on a pool table.1020

One of them is moving horizontally with speed of v1 = 5 m/s towards the other.1029

After the collision, the second ball is moving with a speed of v2 = 3 m/s, at 30 degrees below the horizontal.1033

What is the first's velocity?1040

In this case, we need to figure out, how much is going to the right, and how much is going up.1042

Now, we can figure out what the momentum is for the first ball, the ball that is moving.1052

We can figure out, the second one, we know what the momentum in the second one is.1059

We can figure out how much it is moving this way, and how much it is moving this way, and once we know that, we can go back and we can figure out, in this one, these two added up, are going to have to wind up equaling this one's vertical motion, which turns out to be nothing.1062

And we know that these two added up, are going to have to wind up equaling the horizontal motion.1092

We are now going to start working on it.1104

Velocity for the ball that is first moving before impact is 5 m/s to the right, o m/s up and down.1111

v2 = 3 m/s, 30 degrees below the horizontal.1119

Let us figure out what is v2(final) in terms of component vector.1124

v2(final) = cos(30)×3 m/s (since this side is adjacent), and the other side will be, sin(30)×3 m/s.1132

So, (2.598, -1.5).1157

That is an important thing that we have to catch specifically.1166

It is not just sin(30), it is actually, sin(-30), because it is below the horizontal, and we probably want to make going up positive, and going to the right positive, because that is what we are used to as standard.1168

So, it is not just sin(30), it is sin(-30) because it is below the horizontal, it is up to us to pay attention to the signs, we have to be understanding what we are saying with this.1181

So, (2.598,-1.5) m/s.1194

Now we know what the velocity of the second thing is.1198

From there, we can figure out what its momentum is.1200

First off, we know that the momentum at the beginning of the system is going to be equal to the momentum at the end of the system.1203

So, m1v1 = m1v1(after impact) + m2v2(after impact).1249

Remember, m1 and m2 are the same thing, so divide everything by that m1, m2 combination, since m1 = m2 = 0.16 kg.1275

So, velocity = v1(final) + v2(final), (5,0) = (v1x,v1y) + (2.598,-1.5).1292

At this point, we can add things together, so, remember, we are dealing with vectors, each one of the pieces of vector, x component is totally separate from the y component, so now we can break this into two separate equations.1335

We got, 5 = v1x + 2.598, and, 0 = v1y + (-1.5).1346

So, 1.5 = v1y, and , 5-2.598 = 2.402 = v1x.1362

Putting these together, v1 = (2.402,1.5) m/s, and that makes good intuitive sense.1380

If you hit the ball, the other ball winds up moving away as well, you wind up getting something that is moving slower now.1399

If it is going down, if it was going horizontally first, and the other thing bounces down, it is going to have to get rebounded upwards.1405

So, it makes good intuitive sense, passes the sanity check, and works out.1413

We are able to start these things moving, because of conservation of momentum, and we can make it a little bit easier by canceling out the masses, because we know that they all have the same mass, so it is a little bit easier for us, and from there we break one of them into its component pieces, and from there we are able to work in factors, because vectors, we can just deal with the component pieces.1415

Example 3: Two rubber balls are on a flat surface.1434

The first one has a mass, m1 = 2 kg, and the second has a mass, m2 = 1 kg.1437

The first one is moving at a velocity 5 m/s.1442

The first ball collides with the second ball, and we are going to be able to treat this collision as elastic.1445

I know, normal everyday collisions do not wind up being elastic, they are almost all inelastic, but it is good for us to get some practice with an elastic one, and rubber balls do manage to keep a lot of their energy.1449

What is the velocity of each ball after the collision?1459

Notice, this ball is going to be moving, and this ball is going to be moving after the collision.1462

One thing to notice is, which ball has to be moving faster.1467

The ball on the right is moving slower than the ball on the left, that is not going to make sense, because that means that somehow collision happened and then this ball managed to move through the other ball.1471

That does not make sense, we know that these are solid objects that are moving in one direction, it cannot move through it, so that is going to guarantee the fact that the velocity of the second ball is going to have to be greater than the velocity of the first ball.1483

In this problem, we are also going to wind up ditching the vector notation on the top, because we are working only in one dimension.1496

It is little bit easier than normal , because we are only working in one dimension, and we got a whole lot of writing coming up to do in an elastic problem, so we are going to make it a little easier by ditching the arrows, but it is important to remember that we are working on the basis of the idea of vectors, we are still going to care about direction and all those things.1503

For the beginning, we know that v2 > v1.1525

Now, let us look at, what is momentum going to tell us about this!1530

We know that, psys(beginning) = p1(final) + p2(final), (one dimension, so ditching arrows on top, but all the ideas of vectors are still there, we really care about direction of momentum.)1532

m1v1 (initial) = m1v1(final) + m2v2(final).1584

So, 2 kg×5 m/s = 2×v1 + 1×v2.1617

Sp, 10 = 2v1 + v2.1630

Now, that i snot enough to solve, we got one equation and two unknowns, so we are not able to do it.1637

But luckily, this is an elastic collision, so now we can bring energy.1642

We know, Energy(initial) = Energy(final), and the final energy of the system, since one thing is moving at the beginning, it is just that one thing's motion.1645

The final energy of the system is going to be both the things moving together, so, E1 + E2.1655

What is the energy at the beginning? (1/2)×2×(vi)2 = (1/2)m1v12+(1/2)m2v22.1663

We cancel (1/2)'s all the way across, substitute things in, 2×52 = 2v11+v22.1680

So, 50 = 2v12+v22.1699

At this point, we have got two unknowns over in this one.1708

How do we do this? Go back, and substitute one equation into the other one.1711

We get, 10 - 2v1 = v2.1717

Move that over here, so, 50 = 2v12+(10-2v1)2.1721

Simplify that out, 50 = 2v12+100-40v1+4v12.1736

Rearranging, 6v12-40v1+50 = 0. (quadratic).1750

If you are really good at factoring, you might be able to factorize this, but, most of us are not that great at factoring, so what do we use now?1770

We got two options: (1) calculator, graph it and see the roots, because we will be able to treat this as a single function and when that function equals zero, you would have found the answer to this equation.1776

Alternately, we could just throw down with the quadratic formula, and we will be able to figure out what it is.1789

What is the quadratic formula? (if you want to do advanced mathematics, or even physics, you could buy a graphing calculator, but it is not necessary.)1794

[-b +/- sqrt(b2 - 4ac)]/2a.1821

Plug things in, [40 +/- sqrt(402 - 4×6×50)]/(2×6) = (40 +/- 20)/12, gives us two possible answers.1829

The two answers are, 60/12 = 5 m/s, and 20/12 = 1.67 m/s, approximately.1861

We have got two answers, so it is up to us to figure out which one makes sense, which one is going to allow v2to be greater than v1.1881

If it is 5 m/s for v1, was it not 5 m/s beforehand?1887

That means that it hits the ball and then it keeps going at the same speed, that does not make any sense.1889

It manages to just go through the ball, remember, math is your friend, but it is not something that just spits out your answer.1905

It is up to you to keep it tamed, you have to understand what you are doing with it.1911

We brought this quadratic equation to bare, but it is up to us to understand how to interpret the results that it gives us.1915

It gives us two answers that work mathematically, the energy will be conserved, and the momentum will be conserved if that other ball was never there, it just manages to pass through it.1921

But, we know that it cannot be the truth, it did not just pass through it, it has to contact that ball.1929

So, the answer that has got to be, is this one tight here.1934

So we have got, v1 = 1.67 m/s.1938

If that is the case, we can then plug this back in, and we get, 10 - 2×(1.67) = v2, so, 6.66 m/s = v2, approximately.1943

There we go, we are able to figure out what the other velocity has got to be, because we are able to bring all these equations bare.1964

We bring what we know about energy formula, what we know about momentum, and we are able to do both of them, because we got two unknowns and two equations, we are able to solve it.1971

And there are our answers.1979

We know that v1 is 1.67 m/s, still in the positive direction, and v2 is positive at 6.66 m/s, and it turns out it is going to be faster than the initial object actually was.1980

That is the stuff that happens in elastic collisions, things do not behave necessarily quite like you expected, because energy is conserved, so we got that v2 thing going on, so some weird stuff can happen sometimes.1997

Well, not weird, because it is physics, it is reality, so it cannot be weird in that way, but it is certainly not what we necessarily expect.2008

Example 4: A ballistic pendulum is one way to measure the velocity of a bullet fired from a gun.2028

Ballistic pendulum is a big heavy thing sitting on a rod, and there is this thing here, and a bullet will come in here, and it is going to impart its momentum, so the thing will swing up.2032

It winds swinging up, just like we got here, bullet goes in here, launches itself in, and put some momentum into it, and the ballistic pendulum will continue to swing up.2050

It is one way to measure the velocity of a bullet fired form a gun.2061

A bullet is fired into a large mass at the end of a pendulum where it launched itself.2062

By measuring how high the pendulum rises, we will be able to figure out the velocity of the bullet before impact. How do we do that?2068

Denote the bullet's mass as mb, the pendulum's mass as mp, assume the pendulum's rod's mass is negligible, so all we have to worry about is the mass of the block.2074

The height that the block achieves will be 'h'.2086

Using all these things, (in real life, we could build something like this, and using a pencil we could mark how high it went up, and we will be able to weigh the mass of the bullet and the pendulum, we will be able to find a fairly light weight strong rod that was able to hold it in place without being much compared to the mass of the pendulum, and we will actually be able to do this, this is what people did in the 1700's).2091

What is the trick that we will be able to use to solve for this?2123

Momentum is not going to be conserved throughout the entire thing.2126

The momentum of the bullet, as it goes up, we have got gravity that is dealing against it.2130

So, gravity is an external force from the point of view of the momentum, if it is going up, it is going to have to have some force moving it up, so we have got external forces on the bullet.2135

But at the same time, energy is not going to be conserved.2147

We got this impact, we never said it was elastic, this is an inelastic collision, this is definitely an inelastic collision.2149

We cannot conserve energy throughout, we cannot conserve momentum throughout, how do we do this?2160

We can conserve them one piece at a time.2165

Notice, the instant, if we break this into three things, the bullet goes in, and at this moment, when it is just after the collision occurs, before it has the chance to swing up, momentum is conserved.2168

So, pi = pf, then from here, once it is swinging up, there is no more collisions, it is just a basic pendulum rod going up, so we can use energy.2187

So, for here, Ei = Ef.2199

We are able to break this into two pieces: the momentum is going to be conserved for just the collision, and after the collision, energy will be conserved.2203

We break it into two pieces: beforehand and afterwards.2212

If pi = pf, what is in the initial momentum?2215

mbvb, there is nothing else moving, we got the pendulum at rest.2222

Once again, we are going to nick those vectors, since we are dealing with linear quantities.2227

Wait a second! I hear you ask, is it not going to move up and to the right when it swings up!2237

Yes, that is true, but we will be dealing with that as energy, so when we get to that point, we only have to worry about speed.2242

For the collision, as far as we are concerned, it hits and goes in, and it is still basically moving in just one dimension, because it launches itself very quickly, and the collision happens very quickly compared to the movement of the swinging.2247

So, mbvb = (mb+mv)vsys, for ease, we will denote, vsys after collision as, vp.2258

That makes it a little bit easier to write, because we are going to have to be talking about two very different world, the world of the impact, and the world after impact where it is swinging.2297

This will make it a little bit easier, vp, it is important to pay attention to what wee are doing here.2303

We can figure out what the velocity of the bullet is, in terms of, (mb+mp)/mb, but are not able to solve for this, we still need to figure out what the velocity of the pendulum is, we do not know what the velocity of the pendulum is.2309

Remember, we only have the height.2328

Now, start looking in terms of energy.2330

Ei = Ef.2332

The energy of the system at the beginning of the swing, does it have any height?2336

No, we can consider that as its base line height. Is it moving?2340

Yes, it is definitely moving, there are no spring involved, and the only thing that we got there is the kinetic mechanical energy, kinetic energy.2342

(1/2)(mb+mp)vp2 = Mgh (at the top of its arc, it has got to just finish moving, then the only thing it has is stored gravitational potential energy.)2349

'M' in this case, is not just 'M', it is the mass of the whole thing, mb+mp, so, mb+mpgh.2356

Cancel out mb+mp on both sides, so, vp2 = 2gh, vp = sqrt(2gh).2387

Substitute that back in, vb = (mb+mp)/(mb×sqrt(2gh)), there we are.2402

By just measuring the height that the thing goes up to, by already knowing the constant of gravity, by weighing the mass of the bullet, weighing the block, we are able to figure out the velocity of the bullet, which is pretty great.2427

These are all the things that we were able to do in the 1700's.2437

Even without fancy technologies like LASER, and all these other cool stuff, we can actually come up with a really solid way to measure the velocity of a bullet by what we know about momentum and what we know about energy.2440

Hope you enjoyed this.2452

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