AP Physics C Mechanics Formula Sheet — Complete Equation Reference (2026)
AP Physics C: Mechanics is calculus-based — and the formula sheet reflects that. Here's every equation you'll need, organized by topic, with notes on when to use each one.
The Official Formula Sheet
College Board provides a formula sheet for AP Physics C: Mechanics covering the major topics. Unlike AP Physics 1, Physics C uses calculus notation throughout. The sheet is available for both multiple choice and free response.
Important: The sheet gives you formulas. It does not tell you which one to apply to a given situation. That judgment is entirely yours.
Kinematics
For AP Physics C, kinematics problems often require integration or differentiation instead of the plug-and-chug equations from Physics 1.
| Equation | Notes |
|---|---|
| v = dx/dt | Velocity is derivative of position |
| a = dv/dt = d²x/dt² | Acceleration is derivative of velocity |
| x = x₀ + v₀t + ½at² | Only valid for constant acceleration |
| v² = v₀² + 2a(x - x₀) | Also constant acceleration only |
| v = v₀ + at | Constant acceleration |
When acceleration is not constant, you must integrate:
- v(t) = ∫a(t)dt
- x(t) = ∫v(t)dt
Newton's Laws and Forces
| Equation | Variables | When to use |
|---|---|---|
| ΣF = ma = dp/dt | F = net force, p = momentum | Newton's second law |
| F_friction = μN | μ = friction coefficient, N = normal force | Kinetic or static friction |
| F_spring = -kx | k = spring constant, x = displacement | Hooke's law |
| F_gravity = Gm₁m₂/r² | G = 6.67×10⁻¹¹ N·m²/kg², r = distance between centers | Newton's law of gravitation |
| g = GM/r² | M = mass of planet | Gravitational field |
Work, Energy, Power
| Equation | Variables | Notes |
|---|---|---|
| W = ∫F·dx | W = work | General work formula (use when F varies) |
| W = Fd cosθ | Constant force case | |
| K = ½mv² | K = kinetic energy | |
| U_grav = mgh | Near Earth's surface | |
| U_grav = -Gm₁m₂/r | Gravitational PE | Negative! Zero at infinity |
| U_spring = ½kx² | Spring potential energy | |
| W_net = ΔK | Work-energy theorem | |
| P = dW/dt = F·v | P = power |
Key concept: Potential energy is defined as U = -∫F·dx. The negative sign means force points in the direction of decreasing PE.
Momentum and Impulse
| Equation | Variables | Notes |
|---|---|---|
| p = mv | p = momentum | |
| J = Δp = ∫F dt | J = impulse | |
| p_total = constant (if ΣF_ext = 0) | Conservation of momentum | |
| e = (v₂f - v₁f)/(v₁i - v₂i) | e = coefficient of restitution | e=1 elastic, e=0 perfectly inelastic |
Rotation
This is where AP Physics C diverges most from Physics 1.
Rotational Kinematics
| Linear | Rotational Equivalent |
|---|---|
| x | θ (angle) |
| v = dx/dt | ω = dθ/dt (angular velocity) |
| a = dv/dt | α = dω/dt (angular acceleration) |
| v = v₀ + at | ω = ω₀ + αt |
| x = x₀ + v₀t + ½at² | θ = θ₀ + ω₀t + ½αt² |
Rotational Dynamics
| Equation | Variables | Notes |
|---|---|---|
| τ = r × F = rF sinθ | τ = torque | |
| Στ = Iα | I = moment of inertia, α = angular acceleration | Rotational Newton's 2nd law |
| L = Iω = r × p | L = angular momentum | |
| dL/dt = Στ | Angular impulse-momentum | |
| K_rot = ½Iω² | Rotational kinetic energy | |
| W_rot = ∫τ dθ | Rotational work |
Moments of Inertia (on the formula sheet)
| Object | Axis | I |
|---|---|---|
| Point mass | Distance r from axis | mr² |
| Solid sphere | Through center | (2/5)mr² |
| Hollow sphere | Through center | (2/3)mr² |
| Solid cylinder/disk | Through center | (1/2)mr² |
| Thin rod | Through center | (1/12)mL² |
| Thin rod | Through end | (1/3)mL² |
Parallel axis theorem: I = I_cm + md²
This is provided and essential. Use it whenever the rotation axis doesn't pass through the center of mass.
Oscillations (Simple Harmonic Motion)
| Equation | Variables | Notes |
|---|---|---|
| a = -ω²x | ω = angular frequency | Defining equation of SHM |
| x(t) = A cos(ωt + φ) | A = amplitude, φ = phase | General solution |
| T = 2π/ω | T = period | |
| T_spring = 2π√(m/k) | Period of spring-mass system | |
| T_pendulum = 2π√(L/g) | L = length | Small angle approximation |
| E = ½kA² = ½mv²_max | Total energy in SHM |
Gravitation
| Equation | Notes |
|---|---|
| F = Gm₁m₂/r² | Newton's law of gravitation |
| U = -Gm₁m₂/r | Gravitational PE (note: negative) |
| v_orbit = √(GM/r) | Orbital speed for circular orbit |
| T² = (4π²/GM)r³ | Kepler's third law |
| v_escape = √(2GM/r) | Escape velocity |
What's NOT on the Formula Sheet
These are frequently needed but not provided:
- Condition for rolling without slipping: v = Rω (memorize this)
- Center of mass: x_cm = Σ(mᵢxᵢ)/Σmᵢ — not always on the sheet
- Kepler's first and second laws — conceptual, not equations
- Direction rules for cross products — must know right-hand rule
- Gravitational field inside a uniform sphere — g decreases linearly with r
AP Physics C Mechanics vs AP Physics 1 Formula Sheets
| AP Physics 1 | AP Physics C: Mechanics | |
|---|---|---|
| Math level | Algebra | Calculus (derivatives, integrals) |
| Rotation | Basic | Full rotational dynamics |
| SHM | Conceptual | Full equations |
| Gravitation | Basic | Full Newtonian gravitation |
| Provided equations | More | Similar, but calculus form |
AP Physics C: Mechanics Score Calculator
Use our AP Physics C: Mechanics Score Calculator to estimate your AP score.
Score cutoffs (2026):
| AP Score | Composite Range |
|---|---|
| 5 | 55–90 |
| 4 | 44–54 |
| 3 | 33–43 |
| 2 | 22–32 |
| 1 | 0–21 |
Note: AP Physics C: Mechanics has a 35% five-rate — the highest of any AP exam — because the student pool is highly self-selected (calculus students who chose an advanced physics course).