AP Physics C: Mechanics
Calculus-Based Mastery

Where Calculus Meets Classical Mechanics

The most rigorous AP science exam — and one of the most rewarding. Master every integral, every differential equation, and every FRQ type with Dr. Arjun Patel and SofAI. From kinematics to Kepler's laws, calculus is your superpower.

• › Single-select questions — calculus-based quantitative and conceptual
• › No calculator permitted — must know key integrals and derivatives
• › Tests all 6 units (Kinematics through Oscillations & Gravitation)

• › 3 substantial FRQs — each typically worth ~15 points across multiple parts
• › Calculator permitted — show all calculus steps for full credit
• › Requires clear derivations: set up integrals, evaluate, include units

Multiple Choice Target: ≥ 70% (~25 of 35 questions correct — no calculator needed)

FRQ 1 Target: Full credit — show complete integral setup and evaluation with units

FRQ 2 Target: Full credit — free body diagram + Newton's law equation + correct algebra

FRQ 3 Target: Full credit — identify all energy terms and apply correct conservation law

Master the work integral first. Every energy problem on the exam reduces to ∫F·dx. Practice evaluating this integral for linear, spring (F = -kx), and gravity (F = -GMm/r²) forces until it becomes automatic.

Know all three kinematic derivative relationships cold: v = dx/dt, a = dv/dt = d²x/dt². In the reverse direction: x = ∫v dt, v = ∫a dt. This unlocks every kinematics and Newton's Law FRQ.

Rotation is Newton's 2nd Law in disguise. Just as F = ma, we have τ_net = Iα. When you see rotation, immediately write the rotational analog of every translational equation you know.

The SHM differential equation d²x/dt² = -(ω²)x always yields x(t) = A cos(ωt + φ). Know how to match initial conditions to find A and φ, and that ω = √(k/m) for a spring and ω = √(g/L) for a pendulum.

Angular momentum conservation (τ_net = 0 → L = constant) is the rotation version of linear momentum conservation. When a problem says 'no external torque,' write L_i = L_f and substitute Iω for each phase.

The work-energy theorem W_net = ΔKE always holds. For conservative forces, also use conservation of mechanical energy: K_i + U_i = K_f + U_f. Know when each approach is faster — energy methods often beat Newton's Laws for finding speeds.

FRQ Mastery Suite

AP Physics C FRQs demand full calculus derivations — the setup, the integral or differential equation, and the step-by-step evaluation. This is where every point is earned or lost.

Phase 1: Calculus Foundation + Kinematics + Newton's Laws

• Review derivatives and integrals (chain rule, product rule, separation of variables)
• Master kinematic equations via derivatives: v = dx/dt, a = dv/dt
• Build free body diagram fluency — every problem starts with a FBD
• FRQ practice: one kinematics derivation problem per week (timed: 15 min)

Phase 2: Energy, Momentum, and Rotational Mechanics

• Deep dive: work integral ∫F·dx for spring, gravity, and variable force problems
• Master center of mass integrals and impulse-momentum theorem
• Start rotation: derive I for ring, disk, rod from I = ∫r²dm
• FRQ practice: one rotation problem and one energy problem per week

Phase 3: Oscillations, Gravitation, and FRQ Mastery

• Solve the SHM differential equation from scratch: d²x/dt² = -ω²x → x(t) = A cos(ωt + φ)
• Master orbital mechanics: derive orbital speed and period from Newton's law
• Write 2 full FRQs per week under timed conditions (15 min each)
• Complete 3 full past AP Physics C exams (45 min MC + 45 min FRQ)

Phase 4: Full Exam Simulation and Score Optimization

• One full timed practice exam per week with Dr. Arjun review via SofAI
• Review every wrong MC answer and trace the calculus error
• Final review: build a one-page 'master equation sheet' (not for exam — for internalization)
• Score 5 mindset: every FRQ answer must show integral/derivative setup before the result

Your Physics Skills Are an Academic Superpower — Use Them in AP Seminar

AP Physics C builds exactly the skills AP Seminar demands: quantitative reasoning, evidence-based argumentation, and the ability to model complex systems. See how Jiang combined STEM rigor with academic research to build an outstanding portfolio recognized at the national level.