Calculus-Based Modeling in Intro Physics: From Kinematics to Fields and ODEs

A guided, multi-course journey that builds this skill from the ground up. Work through the chapters in order, or jump to any lesson.

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Motion as Slopes and Areas on Graphs Velocity as the Time-Derivative of Position in 1D Acceleration as the Time-Derivative of Velocity Displacement as the Integral of Velocity With Initial Conditions Velocity Change as the Integral of Acceleration Constant-Acceleration Motion Derived From Integrals When High-School Kinematics Shortcuts Break

Displacement, Velocity, and Acceleration as Vectors Components, Unit Vectors, and Coordinate Choices Differentiating and Integrating Vector-Valued Functions Projectile Motion From a Constant Acceleration Vector Time of Flight and Range as Consequences of r(t)When a 1D Shortcut Is Allowed and When It Fails Dot Products, Projections, and Directionality Tools

Forces as Causes of Acceleration: The Modeling Question Newton’s Second Law in Vector and Component Form Constraints and Contact Forces: Gravity and the Normal Force Kinetic Friction as a Directional Model, Not a Minus Sign Initial Conditions and Integration Constants in Constant-Force Motion Position-Dependent Force: The Spring as a Non-Constant-Acceleration Model Diagnosing Non-Constant Acceleration From Force Dependence Units, Setup-Only Modeling, and the Impulse Preview

Work as Force Accumulated Along a Displacement The Constant-Force Work Shortcut and Its Validity Conditions The Work–Energy Theorem From Newton’s Second Law in 1D Spring Work and the Meaning of the Sign Solving Variable-Force Motion With the Work–Energy Method Power as the Time Rate of Doing Work Conservative vs Non-Conservative Forces as Path (In)dependence Choosing Between Newton’s Laws and Energy Methods

Rate Proportional to Amount as a Physical Model Separation of Variables as an Operator Workflow Time Constants and Initial Conditions in Exponential Decay Differential vs Integral Statements of Change Deriving the RC Differential Equation From Circuit Relations Solving RC Charging and Discharging With Separation of Variables Unit and Limit Checks for Exponential Models Setup-First ODE Modeling for RC Scenarios

A Spring–Mass System as a Physical Model Deriving the SHM Differential Equation From Newton’s Second Law Solving m x¨ + kx = 0 and Defining ω Applying Initial Conditions to Fix Amplitude and Phase Period and Frequency From ω Energy Exchange as a Consistency Check for SHM Diagnosing SHM Mistakes and Previewing Damped Motion

Electric Field and Electric Potential Difference as Distinct Quantities Potential Difference as the Accumulation of Field in One Dimension Field From Potential in One Dimension: The Meaning of a Spatial Derivative The Line-Integral Idea: Dot Product Along a Path Conservative Fields and the Meaning of Path Independence Partial Derivatives and the Gradient as Local Spatial Slopes Reading Field Direction From Contours of Constant Potential Sign Conventions, Scope Boundaries, and Gradient-Strength Synthesis

From Weighted Sums to Center of Mass Integrals Building dm From Linear Density for a Non-Uniform Rod Symmetry and Limiting Cases as Setup Checks Moment of Inertia as a Rotational Weighted Average Moment of Inertia of a Rod About Different Axes Moment of Inertia of a Disk Using Area Density Common Setup Errors for Center of Mass and Moment of Inertia Capstone Synthesis and Exit Checklist