Applications of Calculus to Mechanics

Explore a comprehensive series of lectures on the applications of calculus to mechanics. Delve into topics such as functions of displacement, gravity and escape velocity, simple harmonic motion, projectile motion, and resisted motion. Learn to interpret displacement-time graphs, understand physical models and their differential equations, and apply Newton's laws of motion. Examine various types of motion, including vertical and horizontal resisted motion, and investigate concepts like terminal velocity and quadratic drag. Work through numerous example questions and exam problems to reinforce your understanding of calculus in mechanical systems. Gain practical skills in solving complex problems related to motion, forces, and equilibrium, preparing you for advanced studies in physics and engineering.

Functions of Displacement: Example Question.
Functions of Displacement: Harder Example (1 of 3 - Finding v²).
Functions of Displacement: Prologue.
Functions of Displacement: Proof 1.
Functions of Displacement: Proof 2.
Functions of Displacement: Harder Example (2 of 3 - Integrating for x).
Functions of Displacement: Harder Example (3 of 3 - Establishing Domain for v).
Gravity & Escape Velocity (1 of 3).
Gravity & Escape Velocity (2 of 3).
Gravity & Escape Velocity (3 of 3).
Introduction to Simple Harmonic Motion.
Simple Harmonic Motion: Basic Equations.
Characteristics of Simple Harmonic Motion (1 of 2).
Characteristics of Simple Harmonic Motion (2 of 2).
Simple Harmonic Motion: Amplitude Example.
Simple Harmonic Motion: Shifted Centre Example (1 of 2).
Simple Harmonic Motion: Shifting the Centre.
Simple Harmonic Motion: Shifted Centre Example (2 of 2).
Projectile Motion: North Korean Tank (1 of 4).
Projectile Motion: North Korean Tank (2 of 4).
Projectile Motion: North Korean Tank (3 of 4).
Projectile Motion: Summary of Equations.
Projectile Motion: Glenn Mcgrath (1 of 2).
Projectile Motion: Glenn Mcgrath (2 of 2).
Projectile Motion: North Korean Tank (4 of 4).
Projectile Motion: Colliding Particles (1 of 3).
Projectile Motion: Colliding Particles (2 of 3).
Projectile Motion: Colliding Particles (3 of 3).
Tricky Projectile Question: Different Angles of Projection.
Tricky Projectile Question: Equation of Path.
Tricky Projectile Question: Restriction on Angles.
Interpreting Displacement-Time Graphs.
Integrating Motion Equations: The Tennis Ball (1 of 3).
Integrating Motion Equations: The Tennis Ball (2 of 3).
Integrating Motion Equations: The Tennis Ball (3 of 3).
Introduction to Simple Harmonic Motion: Time Equations.
Key Illustration for Understanding Simple Harmonic Motion.
Physical Models and their Differential Equations.
Simple Harmonic Motion Example Question: The Spring.
Understanding SHM by examining its graphs.
Introductory Guide to Describing Motion.
Velocity as a Function of Displacement.
Acceleration in terms of Velocity (1 of 2: Review).
Acceleration in terms of Velocity (2 of 2: Derivation & Example).
The Ball & Stone (1 of 2: Finding Time of Collision).
The Ball & Stone (2 of 2: Determining Restriction on V).
Simple Harmonic Not-Motion: Fluctuating Temperature.
Projectile Motion: Simple Worked Example (1 of 4: Resolving Initial Velocity).
Projectile Motion: Simple Worked Example (2 of 4: Developing 4 Time Equations).
Projectile Motion: Simple Worked Example (3 of 4: Understanding the Point of Impact).
Projectile Motion: Simple Worked Example (4 of 4: Equation of Path).
Projectile Motion: Aiming for a Target (1 of 2: Generating Time Equations).
Projectile Motion: Aiming for a Target (2 of 2: Determining Firing Angle).
Relationship Between High & Low Firing Angles.
Equation of Path Example Question (1 of 2): Identifying Important Features.
Equation of Path Example Question (2 of 2): Adding in a Slanted Road.
Equation of Path: Understanding Time as the Parameter.
Mid-Air Target Question (1 of 4): Time of Equal Horizontal/Vertical Displacement.
Mid-Air Target Question (2 of 4): Time of Impact.
Mid-Air Target Question (3 of 4): Implied Restriction on Firing Angle.
Mid-Air Target Question (4 of 4): Two Final Results.
Exam Problem: Simple Harmonic Motion with Auxiliary Angle.
Defining Momentum & Force.
Introduction to Mechanics.
Newton's First Law: Inertia.
Mechanics Example 1: Using F = ma to find v(t).
Newton's Second Law: Inertial Mass.
Newton's Third Law: Reactions.
Mechanics Example 2: Using F = ma to find v(x).
Mechanics Example 3: Starting from Displacement Function.
Mechanics Example 4: Calculating Total Distance of a Multi-Step Journey.
Weight.
Introduction to Resisted Motion (1 of 2: What is Resistance?).
Introduction to Resisted Motion (2 of 2: Example question).
Resistance - should it be kv or mkv? (1 of 3: Introductory thoughts).
Resistance - should it be kv or mkv? (2 of 3: Inferring from details in the question).
Resistance - should it be kv or mkv? (3 of 3: What to do when it's ambiguous).
Vertical Resistance & Gravity example question (1 of 2: Finding x(v)).
Vertical Resistance & Gravity example question (2 of 2: Proving Final Result).
Vertical Resistance & Gravity: Framing the Question.
Mechanics Example 5: Momentum, Terminal Velocity & Total Distance.
Physics vs. "Motion" in Mathematics (1 of 2: What's included?).
Physics vs. "Motion" in Mathematics (2 of 2: What's different?).
Types of Motion in HSC Mathematics (2U, Ext 1 & Ext 2).
Introduction to Simple Harmonic Motion (1 of 2: Key Features).
Introduction to Simple Harmonic Motion (2 of 2: Time Equations).
Simple Harmonic Motion Question (1 of 3: Basic Features).
Simple Harmonic Motion Question (2 of 3: Extreme Values).
Harder SHM Question (1 of 5: Interpreting the question).
Harder SHM Question (2 of 5: Setting up the equations).
Harder SHM Question (3 of 5: Determining specific times).
Simple Harmonic Motion Question (3 of 3: Other Characteristics).
Harder SHM Question (4 of 5: Examining the geometry of movement).
Harder SHM Question (5 of 5: Determining specific speed).
Alternate Forms for Simple Harmonic Motion (Example 1 of 2).
Alternate Forms for Simple Harmonic Motion (Example 2 of 2).
Motion as Functions of Displacement (1 of 2: Why it matters).
Motion as Functions of Displacement (2 of 2: Example question).
Full Derivation of Acceleration = d(½v²)/dx.
Using d(½v²)/dx without SHM (1 of 2: Understanding Velocity).
Using d(½v²)/dx without SHM (2 of 2: Reintroducing Time).
Using the d(½v²)/dx result (1 of 2: The vertical spring).
Using the d(½v²)/dx result (2 of 2: Rest Position & Max Speed).
Differential Equations for SHM (1 of 2: A curious pattern).
Differential Equations for SHM (2 of 2: Deriving v²=n²[a²-x²]).
Projectile Motion (1 of 5: Defining Conditions for Projectile Motion and how time is built into it).
Projectile Motion (2 of 5: Outlining relationship between Initial v, x and y and projection angle).
Applications of Projectile Motion (1 of 4: Proving that two separate particles collide).
Projectile Motion (3 of 5: Defining the Acceleration, Velocity and Displacement Functions for x & y).
Projectile Motion (4 of 5: Finding max height using vertical velocity and displacement equations).
Projectile Motion (5 of 5: Finding Flight Time, Horizontal Range and Impact speed and angle).
Application of Projectile Motion (4 of 4: Proving that Path taken by projectile is a parabola).
Applications of Projectile Motion (2 of 4: Calculating V2, Collision Time & Place).
Applications of Projectile Motion (3 of 4: Finding the velocity the collision occurs at).
Harder Motion (2 of 2: Finding an expression for particle B to substitute time into to find x).
Harder Projectile Motion (1 of 5: Manipulating Trig Identities and finding time as a function of x).
Harder Projectile Motion (2 of 5: Substituting time expression to find velocity in terms of d).
Harder Projectile Motion (3 of 5: Finding what happens when theta approaches alpha and π/2).
Harder Projectile Motion (4 of 5: Finding a relationship between α & θ using Stationary Point).
Harder Projectile Motion (5 of 5: Using the second derivative to find the minimum value of θ).
Mechanics (1 of 7: Introduction to Forces and Newton's First Law and its relation to Mechanics).
Mechanics (2 of 7: Introduction to Newton's Second Law and Third Law and its relation to Mechanics).
Mechanics (3 of 7: Representing Physical Motion in mathematical terms).
Mechanics (4 of 7: Finding the angle subtended by the other line with the horizontal wall).
Mechanics (5 of 7: Resolving Forces to find the Horizontal forces acting on both strings).
Mechanics (6 of 7: Finding the Forces acting on the particle vertically).
Mechanics (7 of 7: Introductory Example to Mathematical Representation of Physical Motion).
Resisted Motion - Basic Example (1 of 2: Time as a function of Velocity).
Resisted Motion: Introductory Concepts.
Resisted Motion - Basic Example (2 of 2: Further Manipulation & Conclusions).
Resisted Motion - Harder Example (1 of 2: Maximum Height).
Resisted Motion - Harder Example (2 of 2: End of the journey).
Intro to Straight Line Motion (1 of 3: Overview of language).
Intro to Straight Line Motion (2 of 3: Unpacking a basic question).
Intro to Straight Line Motion (3 of 3: Interpreting the equations).
Simple Harmonic Motion Example Question (1 of 3: Determining period of motion).
Simple Harmonic Motion Example Question (2 of 3: Solving for time).
Simple Harmonic Motion Example Question (3 of 3: Using graph symmetry).
SHM - Other Centres of Motion (1 of 2: Rearranging with trigonometric identities).
SHM - Other Centres of Motion (2 of 2: Determining attributes from the equation).
Functions of Displacement (1 of 3: Basic Simple Harmonic Motion).
Functions of Displacement (2 of 3: SHM with different centre).
Functions of Displacement (3 of 3: Straight line motion example).
Equation of Path (1 of 4: Establishing the horizontal equations).
Equation of Path (2 of 4: Deriving the Cartesian equation).
Equation of Path (3 of 4: Finding horizontal range).
Equation of Path (4 of 4: Example question).
Simple Harmonic Motion v² Equation (1 of 2: Deriving the result).
Simple Harmonic Motion v² Equation (2 of 2: Example question).
HSC Tide Question (1 of 3: Proving the time equation).
HSC Tide Question (2 of 3: Solving for time).
HSC Tide Question (3 of 3: Leaving the harbour safely).
Intro to Mechanics (1 of 4: Mathematics & physics).
Intro to Mechanics (2 of 4: Equations & kinematics).
Intro to Mechanics (3 of 4: Simple harmonic motion - foundations).
Intro to Mechanics (4 of 4: Basic SHM example).
Simple Harmonic Motion example (1 of 3: Interpreting given data).
Simple Harmonic Motion example (2 of 3: Forming an equation).
Simple Harmonic Motion example (3 of 3: Identifying the time of a given displacement).
Acceleration in terms of displacement (1 of 2: Explanation).
Acceleration in terms of displacement (2 of 2: Worked example).
Simple Harmonic Velocity via Displacement (1 of 2: Equating forms of acceleration).
Simple Harmonic Velocity via Displacement (2 of 2: Worked example).
Objects in Equilibrium (1 of 4: Comparing forces with displacement).
Objects in Equilibrium (3 of 4: Balancing vertical forces).
Objects in Equilibrium (2 of 4: Using trigonometric relationships).
Objects in Equilibrium (4 of 4: Worked exam question).
Concurrent Forces - Non-Equilibrium (2 of 3: Worked example).
Concurrent Forces - Non-Equilibrium (1 of 3: Introduction).
Concurrent Forces - Non-Equilibrium (3 of 3: Finding magnitude & direction).
Horizontal Resisted Motion (1 of 3: Introduction).
Horizontal Resisted Motion (2 of 3: Velocity in terms of displacement).
Horizontal Resisted Motion (3 of 3: Locating eventual resting place).
Plane Braking Model (3 of 3: Resetting the time variable).
Plane Braking Model (2 of 3: Considering reverse thrust).
Plane Braking Model (1 of 3: Constant frictional force).
Vertical Resisted Motion (5 of 5: How long till it returns to the ground?).
Vertical Resisted Motion (4 of 5: Finding v = f(t) by integration).
Vertical Resisted Motion (3 of 5: Determining the maximum height).
Vertical Resisted Motion (2 of 5: Finding y = f(v) by integration).
Vertical Resisted Motion (1 of 5: Introduction).
Terminal Velocity (1 of 2: Balancing forces).
Terminal Velocity (2 of 2: Determining drop height).
Resisted Projectile Motion (1 of 4: Understanding horizontal motion).
Resisted Projectile Motion (2 of 4: Understanding vertical motion).
Resisted Projectile Motion (3 of 4: Finding cartesian equation).
Resisted Projectile Motion (4 of 4: Determining equations from first principles).
Quadratic Drag (1 of 3: Evaluating the drag coefficient).
Quadratic Drag (2 of 3: Investigating horizontal motion).
Quadratic Drag (3 of 3: Determining the angle of projection).
Proving simple harmonic motion (Exam Question 1 of 10).
Finding maximum speed of SHM (Exam Question 2 of 10).
Forces on a hanging object (Exam Question 5 of 10).
Mechanics of a falling object (Exam Question 10 of 10).