Calculus-Based Physics: Kinematics, 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 GraphsVelocity as the Time-Derivative of Position in 1DAcceleration as the Time-Derivative of VelocityDisplacement as the Integral of Velocity With Initial ConditionsVelocity Change as the Integral of AccelerationConstant-Acceleration Motion Derived From IntegralsWhen High-School Kinematics Shortcuts Break
Displacement, Velocity, and Acceleration as VectorsComponents, Unit Vectors, and Coordinate ChoicesDifferentiating and Integrating Vector-Valued FunctionsProjectile Motion From a Constant Acceleration VectorTime of Flight and Range as Consequences of r(t)When a 1D Shortcut Is Allowed and When It FailsDot Products, Projections, and Directionality Tools
Forces as Causes of Acceleration: The Modeling QuestionNewton’s Second Law in Vector and Component FormConstraints and Contact Forces: Gravity and the Normal ForceKinetic Friction as a Directional Model, Not a Minus SignInitial Conditions and Integration Constants in Constant-Force MotionPosition-Dependent Force: The Spring as a Non-Constant-Acceleration ModelDiagnosing Non-Constant Acceleration From Force DependenceUnits, Setup-Only Modeling, and the Impulse Preview
Work as Force Accumulated Along a DisplacementThe Constant-Force Work Shortcut and Its Validity ConditionsThe Work–Energy Theorem From Newton’s Second Law in 1DSpring Work and the Meaning of the SignSolving Variable-Force Motion With the Work–Energy MethodPower as the Time Rate of Doing WorkConservative vs Non-Conservative Forces as Path (In)dependenceChoosing Between Newton’s Laws and Energy Methods
Rate Proportional to Amount as a Physical ModelSeparation of Variables as an Operator WorkflowTime Constants and Initial Conditions in Exponential DecayDifferential vs Integral Statements of ChangeDeriving the RC Differential Equation From Circuit RelationsSolving RC Charging and Discharging With Separation of VariablesUnit and Limit Checks for Exponential ModelsSetup-First ODE Modeling for RC Scenarios
A Spring–Mass System as a Physical ModelDeriving the SHM Differential Equation From Newton’s Second LawSolving m x¨ + kx = 0 and Defining ωApplying Initial Conditions to Fix Amplitude and PhasePeriod and Frequency From ωEnergy Exchange as a Consistency Check for SHMDiagnosing SHM Mistakes and Previewing Damped Motion
Electric Field and Electric Potential Difference as Distinct QuantitiesPotential Difference as the Accumulation of Field in One DimensionField From Potential in One Dimension: The Meaning of a Spatial DerivativeThe Line-Integral Idea: Dot Product Along a PathConservative Fields and the Meaning of Path IndependencePartial Derivatives and the Gradient as Local Spatial SlopesReading Field Direction From Contours of Constant PotentialSign Conventions, Scope Boundaries, and Gradient-Strength Synthesis
From Weighted Sums to Center of Mass IntegralsBuilding dm From Linear Density for a Non-Uniform RodSymmetry and Limiting Cases as Setup ChecksMoment of Inertia as a Rotational Weighted AverageMoment of Inertia of a Rod About Different AxesMoment of Inertia of a Disk Using Area DensityCommon Setup Errors for Center of Mass and Moment of InertiaCapstone Synthesis and Exit Checklist