Calculus-based mechanics — kinematics, Newton's laws, energy, momentum, and rotation.
Study guide last reviewed September 25, 2026 · see what changed
Format reflects the current published structure — always confirm against AP Central before test day, since formats occasionally change.
| Section | Details | Time | Weight |
|---|---|---|---|
| Section I – Multiple Choice | 35 questions | 45 min | 50% |
| Section II – Free Response | 3 questions | 45 min | 50% |
Straight answers about the AP Physics C: Mechanics exam — format, credit, dates, and this site.
Section I – Multiple Choice: 35 questions (45 min, 50% of the total score); Section II – Free Response: 3 questions (45 min, 50% of the total score). See the Exam Format tab above for the full breakdown, and always double-check the current year's format on AP Central before test day.
It's multiple-choice questions and written free-response or essay questions.
It depends entirely on the college — most award credit for a 3 or higher, but many selective schools require a 4 or a 5, and some don't grant credit at all for certain scores. Always check your target college's own AP credit policy page rather than assuming a 3 will count everywhere.
AP exams are administered during the College Board's annual testing window in the first two full weeks of May, with the exact date and time for each subject set by the College Board and announced the preceding fall. Check AP Central or your school's AP coordinator for this year's exact date.
Yes. The unit-by-unit study guide, worked examples, practice quiz, and full-length timed practice exam on this page are all free, with no login, signup, or paywall.
Unit-by-unit breakdown of what the course actually covers, in teaching order.
Position, velocity, and acceleration formally related through derivatives and integrals (v = dx/dt, a = dv/dt), extending algebra-based kinematics to handle non-constant acceleration by integrating a(t) to find v(t) and x(t), including problems where acceleration is itself a function of position or velocity.
F = ma applied with calculus for variable forces, including velocity-dependent forces like air resistance/drag that require solving a differential equation rather than using constant-acceleration kinematics, plus multi-body systems (Atwood machines, connected blocks) analyzed with Newton's laws and constraint equations.
Work as the line integral of force over a path (W = ∫F·dx), essential for variable forces like springs (F = -kx, giving W = ½kx²) where constant-force work formulas fail, and the work-energy theorem, power (P = dW/dt = F·v), and conservation of energy applied rigorously with potential energy defined as U = -∫F·dx.
Center of mass calculated via integration for continuous mass distributions (not just point masses), momentum conservation in collisions and explosions, and impulse as the integral of force over time (J = ∫F dt) for time-varying forces, extending the algebra-based treatment to non-constant forces.
Moment of inertia calculated via integration (I = ∫r² dm) for continuous rigid bodies, yielding the standard formulas for rods, disks, spheres, and hoops from first principles, plus the parallel axis theorem for shifting the axis of rotation, torque as a vector cross product (τ = r × F), and rotational kinematics/dynamics paralleling linear motion with calculus.
Simple harmonic motion derived from Newton's second law as a differential equation (d²x/dt² = -(k/m)x), yielding the sinusoidal solution and connecting angular frequency ω to period, plus damped and driven oscillations as extensions beyond the algebra-based course's scope.
Newton's law of universal gravitation (F = Gm1m2/r²) applied to orbital mechanics, deriving circular orbital velocity and period from setting gravitational force equal to centripetal force, Kepler's three laws (elliptical orbits, equal areas in equal times, T² ∝ r³) derived from Newtonian mechanics, and gravitational potential energy (U = -Gm1m2/r) for escape velocity problems.
Concrete, fully worked examples — this is what separates a page reviewers see as genuinely useful from one that just lists topics.
Variable acceleration: a particle has a(t) = 6t m/s², starting at rest at the origin. Find v(t) and x(t).
v(t) = ∫6t dt = 3t² + C; since v(0)=0, C=0, so v(t) = 3t². x(t) = ∫3t² dt = t³ + C; since x(0)=0, x(t) = t³. Always apply the initial condition to solve for the constant of integration — a bare antiderivative without it isn't a complete answer.
Velocity-dependent drag: a falling object experiences F = mg − bv. Set up (don't solve) the differential equation for velocity, and identify terminal velocity.
m(dv/dt) = mg − bv. Terminal velocity occurs when dv/dt = 0: mg = bvterm → vterm = mg/b. You can find terminal velocity without solving the full differential equation — recognizing dv/dt = 0 as the condition is the key move.
Variable force: a spring exerts F = −kx. Find the work done compressing it from x=0 to x=d.
W = ∫₀ᴸ F dx = ∫₀ᴸ (−kx)(−dx)... more directly, the work done by you compressing it equals the energy stored: W = ∫₀ᴸ kx dx = ½kd². A constant-force formula (W=Fd) would be wrong here since F changes continuously with x — that's exactly why calculus-based mechanics needs the integral.
Center of mass: find the center of mass of a thin rod of length L with linear density λ(x) = λ₀(x/L) (density increasing linearly from one end).
xcm = [∫₀ᴸ xλ(x) dx] / [∫₀ᴸ λ(x) dx] = [∫₀ᴸ λ₀x²/L dx] / [∫₀ᴸ λ₀x/L dx] = [λ₀L²/3] / [λ₀L/2] = 2L/3. The center of mass sits closer to the denser end, as expected physically — a useful sanity check on any center-of-mass integral.
Moment of inertia by integration: derive I for a uniform rod of mass M, length L, rotated about its center.
I = ∫r² dm, with dm = (M/L)dr, integrated from −L/2 to L/2: I = (M/L)∫−L/2L/2 r² dr = (M/L)[r³/3] = ML²/12 — the standard formula, derived from first principles rather than just recalled.
Orbital mechanics: derive the orbital speed of a satellite in a circular orbit of radius r around a planet of mass M.
Set gravitational force equal to centripetal force: GMm/r² = mv²/r → v = √(GM/r). Notice the satellite's own mass cancels — every satellite at a given radius moves at the same orbital speed regardless of its mass, a frequently tested conceptual point.
The errors that cost the most points across the whole exam, not just one unit.
Unofficial practice questions written for this study guide — not official AP content and not from College Board. Use them to self-check, then use the real past free-response questions below for authentic exam practice.
Unofficial practice, timed to feel like the real thing. Section I pulls from this page's full multiple-choice bank; Section II gives you free-response prompts to write out and self-score against a rubric. Bank size is still growing toward full exam length — timing is paced per-question, not pulled from this subject's official duration.