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Learn Calculus-Based Physics by talking it through
Calculus-Based Physics is a university-level first course that uses derivatives and integrals to describe motion, forces, energy, momentum, rotation, and oscillations. The sequence starts with vectors, units, and the calculus needed for physics, then moves through one- and two-dimensional kinematics, Newton's laws, work and energy, momentum and collisions, rotation, and gravitation. Later topics cover simple harmonic motion, damped and driven oscillators, fluids, mechanical and sound waves, and the first and second laws of thermodynamics. Each topic builds on the ones before it, so the tutor checks what you already have before adding the next idea.
In a Socratic session, the tutor asks rather than tells. For a topic such as projectile motion, you might be asked what happens to horizontal velocity when no horizontal force acts, and then how that answer changes the vertical equation. The tutor tracks whether you can explain an idea in words and whether you can carry out the procedure, such as setting up a free-body diagram or integrating a force to find work. Ideas you have met before come back in later sessions to check that they have stuck.
Who it is for
- Students taking a calculus-based first course in mechanics, such as a Physics 241 style sequence.
- Students who already know basic calculus and want to apply derivatives and integrals to physical motion.
- Parents or teachers checking how the course is organized and what it covers.
- Students reviewing for an exam who want to explain reasoning, not only memorize formulas.
What you will be able to do
- Describe motion in one and two dimensions with position, velocity, and acceleration as derivatives and integrals.
- Draw free-body diagrams and set up Newton's second law for systems with tension, friction, and inclines.
- Apply work-energy and momentum conservation to solve problems, including elastic and inelastic collisions.
- Compute moments of inertia by integration and use torque and angular momentum for rotating bodies.
- Derive and solve the equation for simple harmonic motion, and describe damping and resonance.
- Check equations with dimensional analysis and explain results in words.
A taste of a lesson on Work and the Work-Energy Theorem
The tutor asks, you answer, and it works from what you actually said. No videos to sit through.
The 27 topics in Calculus-Based Physics
Each topic builds on the ones before it. The tutor checks what you already know and starts there.
Vectors and Vector Algebra
Vectors as quantities with magnitude and direction; unit vectors, components in 2D/3D, addition/subtraction graphically and by components, scalar multiplication, dot product and cross product with geometric meaning (projection, area, torque direction). Understanding requires both interpreting vector diagrams and computing with components.
Units, Dimensional Analysis, and Estimation
SI units, unit conversion, dimensional analysis to check equations and derive relationships, and order-of-magnitude estimation. Students learn to verify that physics equations are dimensionally consistent and to sanity-check numerical answers.
Derivatives for Physics
Review of derivative concept as instantaneous rate of change and slope of tangent line; power rule, product rule, chain rule, and derivatives of trig/exponential functions applied to position, velocity, and acceleration. Emphasis on interpreting derivatives physically as well as computing them.
Integrals for Physics
Definite and indefinite integrals as accumulation and area under a curve; antiderivative techniques (power rule, substitution) and the Fundamental Theorem of Calculus, applied to recovering velocity from acceleration and position from velocity, and computing work, impulse, and center of mass.
Builds on Derivatives for Physics
One-Dimensional Kinematics
Position, displacement, velocity, and acceleration in one dimension using calculus: v = dx/dt, a = dv/dt, and their integral inverses; constant-acceleration (kinematic) equations derived from calculus and applied to free fall. Understanding includes reading and constructing position-time and velocity-time graphs.
Builds on Units, Dimensional Analysis, and Estimation, Derivatives for Physics, Integrals for Physics
Motion in Two and Three Dimensions
Position, velocity, and acceleration as vectors that change with time; projectile motion as superposition of independent horizontal and vertical motions; relative velocity between reference frames. Requires combining vector algebra with calculus of vector-valued functions.
Builds on Vectors and Vector Algebra, One-Dimensional Kinematics
Newton's Laws of Motion
Newton's first law (inertia), second law (F_net = ma, including as a differential equation), and third law (action-reaction pairs); free-body diagrams; concept of mass vs weight and inertial reference frames. Students must both explain why the laws hold and apply F=ma to solve for motion.
Builds on Motion in Two and Three Dimensions
Applications of Newton's Laws
Solving multi-body problems with tension, normal force, friction (static and kinetic), and inclined planes using free-body diagrams and systems of equations from Newton's second law. Emphasis on setting up and solving coupled equations correctly.
Builds on Newton's Laws of Motion
Uniform and Nonuniform Circular Motion
Centripetal acceleration derived from calculus of rotating position vectors, tangential and radial acceleration components, and applying Newton's second law to circular paths (banked curves, vertical circles). Understanding includes why centripetal force is not a separate force but the net inward force.
Builds on Applications of Newton's Laws
Work and the Work-Energy Theorem
Work as the integral of force over displacement (including variable forces), kinetic energy, and the work-energy theorem derived from Newton's second law using calculus. Understanding includes computing work for constant and variable forces via integration.
Builds on Integrals for Physics, Applications of Newton's Laws
Potential Energy and Conservation of Energy
Conservative vs nonconservative forces, potential energy functions derived from force via integration (and force from potential energy via differentiation), gravitational and spring potential energy, and conservation of mechanical energy with nonconservative work (friction) accounted for.
Builds on Work and the Work-Energy Theorem
Momentum and Impulse
Linear momentum, impulse as the time-integral of force, and the impulse-momentum theorem; conservation of momentum for isolated systems derived from Newton's third law. Applications to variable-mass and impulsive-force problems.
Builds on Integrals for Physics, Applications of Newton's Laws
Collisions
Elastic and inelastic collisions in one and two dimensions; using conservation of momentum (and energy for elastic collisions) to solve for post-collision velocities; center-of-mass frame concept for simplifying collision problems.
Builds on Potential Energy and Conservation of Energy, Momentum and Impulse
Center of Mass and Systems of Particles
Definition of center of mass for discrete and continuous mass distributions using integration, motion of the center of mass under external forces, and its relation to total momentum of a system.
Builds on Integrals for Physics, Momentum and Impulse
Rotational Kinematics
Angular position, velocity, and acceleration as calculus analogs of linear kinematics; relationships between linear and angular quantities (v = ωr, a_t = αr) for rigid body rotation about a fixed axis.
Builds on One-Dimensional Kinematics, Uniform and Nonuniform Circular Motion
Rotational Dynamics and Torque
Torque as the rotational analog of force (including as a cross product), moment of inertia computed by integration for continuous bodies, Newton's second law for rotation (τ_net = Iα), and the parallel axis theorem.
Builds on Vectors and Vector Algebra, Integrals for Physics, Rotational Kinematics
Rotational Energy and Angular Momentum
Rotational kinetic energy, work-energy theorem for rotation, angular momentum for particles and rigid bodies, torque as the rate of change of angular momentum, and conservation of angular momentum with applications (e.g., spinning skaters, gyroscopes).
Builds on Potential Energy and Conservation of Energy, Center of Mass and Systems of Particles, Rotational Dynamics and Torque
Statics and Rotational Equilibrium
Conditions for static equilibrium (net force and net torque zero), solving for unknown forces/torques in rigid bodies such as beams and ladders, and the concept of stability related to center of mass.
Builds on Applications of Newton's Laws, Rotational Dynamics and Torque
Newton's Law of Universal Gravitation
Newton's law of gravitation, gravitational field and potential energy derived by integration, orbital motion (circular and elliptical) via Newton's laws, Kepler's laws explained from conservation principles, and escape velocity.
Builds on Uniform and Nonuniform Circular Motion, Potential Energy and Conservation of Energy, Rotational Energy and Angular Momentum
Simple Harmonic Motion
Oscillatory motion described by a differential equation (F = -kx leading to x(t) = A cos(ωt+φ)), energy in SHM, simple and physical pendulums, and the relationship between SHM and uniform circular motion.
Builds on Potential Energy and Conservation of Energy, Rotational Dynamics and Torque
Damped and Driven Oscillations, Resonance
Effects of damping forces on oscillators (underdamped, critically damped, overdamped), driven oscillators, and resonance phenomena; qualitative and semi-quantitative treatment of the governing differential equations.
Builds on Simple Harmonic Motion
Fluid Statics and Dynamics
Pressure, Pascal's principle, Archimedes' principle and buoyancy, hydrostatic pressure variation with depth via integration, and fluid dynamics including the continuity equation and Bernoulli's equation with their derivations from work-energy principles.
Builds on Integrals for Physics, Potential Energy and Conservation of Energy
Mechanical Waves
Wave equation for transverse and longitudinal waves, wave speed on a string, superposition and interference, standing waves and resonance in strings and pipes, and the relationship between SHM of individual particles and wave propagation.
Builds on Simple Harmonic Motion, Damped and Driven Oscillations, Resonance
Sound Waves and the Doppler Effect
Sound as a longitudinal pressure wave, speed of sound in media, intensity and the decibel scale, and the Doppler effect for moving sources and observers derived from wave timing arguments.
Builds on Mechanical Waves
Temperature, Heat, and the First Law of Thermodynamics
Temperature scales, thermal equilibrium, heat as energy transfer, specific heat and latent heat, and the first law of thermodynamics (ΔU = Q - W) relating heat, work, and internal energy for simple systems.
Builds on Work and the Work-Energy Theorem
Kinetic Theory and Ideal Gases
Ideal gas law derived from microscopic kinetic theory assumptions, relationship between temperature and average molecular kinetic energy, and the Maxwell-Boltzmann distribution described qualitatively.
Builds on Temperature, Heat, and the First Law of Thermodynamics
Entropy and the Second Law of Thermodynamics
Concept of entropy as a measure of disorder/microstates, the second law of thermodynamics, heat engines and refrigerators, and the Carnot cycle as the theoretical efficiency limit.
Builds on Kinetic Theory and Ideal Gases
Practice questions in Calculus-Based Physics
A few of the kinds of questions you will work through, with answers.
Vector A = (3, 4) and B = (1, -2). What is the dot product A · B?
See the answer
-5
A · B = (3)(1) + (4)(-2) = 3 - 8 = -5. The dot product multiplies matching components and adds them.
A car's position is x(t) = 2t² meters. What is its velocity at t = 3 s?
See the answer
12 m/s
Velocity is v = dx/dt = 4t, so at t = 3 s, v = 12 m/s.
A net force F(x) = 6x N acts on a particle as it moves from x = 0 to x = 2 m. How much work is done?
See the answer
12 J
W = ∫ from 0 to 2 of 6x dx = [3x²] from 0 to 2 = 12 J.
A 2 kg cart moving at 3 m/s hits a stationary 1 kg cart and they stick together. What is their common speed after the collision?
See the answer
2 m/s
Momentum is conserved: 2(3) = (2 + 1)v, so v = 2 m/s.
A uniform rod of mass M and length L rotates about one end. What is its moment of inertia, in terms of M and L?
See the answer
I = ML²/3
Integrating r² dm from 0 to L with dm = (M/L) dr gives ML²/3.
A mass-spring system has k = 200 N/m and m = 2 kg. What is its angular frequency ω, in rad/s?
See the answer
10 rad/s
ω = √(k/m) = √(200/2) = √100 = 10 rad/s.
Questions about this subject
Do I need calculus before starting this course?
The course assumes you can differentiate and integrate basic functions. The topics Derivatives for Physics and Integrals for Physics review the calculus you need, using physical examples.
Will the tutor just give me the answer to my homework?
The tutor asks questions that lead you toward the answer, so you do the reasoning. It is built to check your understanding, not to hand over solutions.
What comes after this course?
The sequence ends with thermodynamics, including the first law, kinetic theory, and the second law with entropy and the Carnot limit. Topics like fluids, waves, and sound are also covered in the sequence.
How does the tutor know whether I understand a topic?
It tracks whether you can explain an idea in your own words, and whether you can carry out the procedure, such as solving a problem. It returns to earlier topics later to see if the ideas still hold.
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