"AP Physics C equation sheet" is searched about 5,900 times a month in the United States, by students who want the official table to practise with and, closer to May, by students who want to know what it does not contain. The College Board's 2025 Mechanics table of information runs to four pages and about 60 equations, and it is given to every candidate in both sections. This page reproduces its structure, annotates every equation with what it is for and what the symbols mean, lists what is missing and must be memorised, explains how to use it on the day, and gives a printable annotated version. (The Electricity and Magnetism exam has a separate sheet, which this page does not cover.)
Download the annotated Mechanics equation sheet (PDF) (A4, the translational and rotational equations annotated on page 1, the tables, what is not on the sheet and a practice set on page 2, free). The official sheet itself is on AP Central as the "AP Physics C: Mechanics Exam Reference Information".
What the sheet gives: constants, prefixes, units, trig values
| Section | Contents | Notes |
|---|---|---|
| Constants | G = 6.67 × 10⁻¹¹ m³/(kg·s²) = 6.67 × 10⁻¹¹ N·m²/kg²; g = 9.8 m/s² = 9.8 N/kg | only two constants: there is no mass of the Earth, no radius, no speed of light on the Mechanics sheet; a question that needs them gives them |
| Prefixes | tera 10¹², giga 10⁹, mega 10⁶, kilo 10³, centi 10⁻², milli 10⁻³, micro 10⁻⁶, nano 10⁻⁹, pico 10⁻¹² | |
| Unit symbols | hertz Hz, joule J, kilogram kg, metre m, newton N, second s, watt W | |
| Trig values | sin, cos and tan for 0°, 30°, 37°, 45°, 53°, 60°, 90° | the 37° and 53° columns are the 3-4-5 triangle: sin 37° = 3/5, cos 37° = 4/5; incline problems use them so the arithmetic stays clean |
| Conventions | the frame of reference is inertial, air resistance is negligible, springs and strings are ideal, unless stated | these save a line in every free-response answer |
The mechanics equations, annotated: translational (the left column)
| Equation | What it is | Symbols | Use it when |
|---|---|---|---|
| vₓ = vₓ₀ + aₓt | velocity after time t at constant acceleration | v velocity, a acceleration, t time, subscript 0 initial | constant-acceleration motion; the first of the three kinematics equations |
| x = x₀ + vₓ₀t + ½aₓt² | position after time t at constant acceleration | x position | you have time but not final velocity |
| vₓ² = vₓ₀² + 2aₓ(x − x₀) | velocity from displacement at constant acceleration | you have distance but not time | |
| Δx = ∫vₓ(t) dt | displacement as the integral of velocity | velocity varies with time: given v(t) or a v-t graph (area) | |
| Δvₓ = ∫aₓ(t) dt | change in velocity as the integral of acceleration | acceleration varies with time | |
| x_cm = Σmᵢxᵢ ÷ Σmᵢ | centre of mass of point masses | mᵢ masses at positions xᵢ | discrete objects; do each coordinate separately |
| r_cm = ∫r dm ÷ ∫dm | centre of mass of a continuous body | dm a mass element | rods and plates of varying density; dm = λ dℓ for a rod |
| λ = dm/dℓ | linear mass density | λ mass per length | non-uniform rods: λ(x) given, integrate for mass, centre of mass or I |
| a_sys = ΣF ÷ m_sys = F_net ÷ m_sys | Newton's second law for a system | ΣF the sum of external forces | every free-body problem; a = F_net/m, written for the whole system |
| F_g | = G m₁m₂ ÷ r² | Newton's law of gravitation | |
| F_f | ≤ μ | F_N | |
| F_s = −kΔx | Hooke's law for an ideal spring | k spring constant, Δx extension | springs; the minus sign makes it a restoring force |
| a_c = v² ÷ r = rω² | centripetal acceleration | ω angular speed | anything moving in a circle: cars on bends, satellites, conical pendulums |
| T = 1 ÷ f | period and frequency | T period, f frequency | circular motion and oscillation |
| K = ½mv² | kinetic energy | energy methods | |
| W = ∫F·dr (from a to b) | work as the integral of force along a path | the dot product: only the component along dr | variable forces (springs, gravity at large r); for a constant force it reduces to Fd cos θ |
| ΔK = ΣWᵢ = ΣF∥,ᵢdᵢ | the work-energy theorem | F∥ the parallel component | total work by all forces equals the change in kinetic energy |
| ΔU = −∫F_cf(r)·dr (from a to b) | potential energy change from a conservative force | cf conservative force | defining U from F: gravity, springs, any given F(x) |
| Fₓ = −dU/dx | force as the negative gradient of potential energy | U(x) given as a function or graph: force is minus the slope; equilibrium where the slope is zero | |
| U_s = ½k(Δx)² | elastic potential energy | springs in energy conservation | |
| U_G = −G m₁m₂ ÷ r | gravitational potential energy, zero at infinity | orbits, escape speed, energy at altitude (never mgh for large heights) | |
| ΔU_g = mgΔy | gravitational potential energy change near the surface | Δy change in height | small heights where g is constant |
| P_avg = W ÷ Δt = ΔE ÷ Δt | average power | energy delivered over an interval | |
| P_inst = dW/dt | instantaneous power | equals F·v for a force on a moving object (not printed; derive it) | |
| p = mv | momentum | collisions, impulse | |
| F_net = dp/dt | Newton's second law in momentum form | variable-mass or variable-force problems; reduces to ma for constant m | |
| J = ∫F_net(t) dt (t₁ to t₂) = Δp | impulse-momentum theorem | J impulse | force-time graphs (area), collisions |
| v_cm = Σpᵢ ÷ Σmᵢ = Σmᵢvᵢ ÷ Σmᵢ | velocity of the centre of mass | collisions and explosions: the centre of mass moves at constant velocity when the net external force is zero |
The mechanics equations, annotated: rotational (the right column)
| Equation | What it is | Translational twin | Use it when |
|---|---|---|---|
| ω = dθ/dt | angular velocity | v = dx/dt | θ(t) given |
| α = dω/dt | angular acceleration | a = dv/dt | ω(t) given |
| ω = ω₀ + αt | angular velocity at constant α | vₓ = vₓ₀ + aₓt | constant angular acceleration: a wheel spinning up or down |
| θ = θ₀ + ω₀t + ½αt² | angular position at constant α | x = x₀ + vₓ₀t + ½aₓt² | revolutions turned in a time |
| ω² = ω₀² + 2α(θ − θ₀) | angular velocity from angle turned | vₓ² = vₓ₀² + 2aₓΔx | no time given |
| v = rω | tangential speed of a point at radius r | linking the rim of a wheel to its spin; rolling without slipping (v_cm = rω) | |
| a_T = rα | tangential acceleration | a point on a spinning-up disc; the centripetal part is rω², separately | |
| τ = r × F | torque as a cross product | magnitude rF sin θ; the lever-arm form; direction by the right-hand rule | |
| I_tot = ΣIᵢ = Σmᵢrᵢ² | rotational inertia of point masses or combined bodies | m | adding the inertias of parts about the same axis |
| I = ∫r² dm | rotational inertia of a continuous body | deriving I for a rod or disc: the free-response favourite | |
| I′ = I_cm + Md² | the parallel-axis theorem | the axis is not through the centre of mass: a rod pivoted at its end (1/12 + 1/4 = 1/3 ML²) | |
| α_sys = Στ ÷ I_sys = τ_net ÷ I_sys | Newton's second law for rotation | a = F_net/m | pulleys with mass, falling rods, yo-yos |
| K_rot = ½Iω² | rotational kinetic energy | K = ½mv² | rolling bodies have both: ½mv² + ½Iω² |
| W = ∫τ·dθ | rotational work | W = ∫F·dr | a torque through an angle |
| L = r × p = Iω | angular momentum, for a particle and for a rigid body | p = mv | conservation when the net torque is zero: skaters, collapsing stars, a bullet hitting a door |
| ΔL = ∫τ dt | angular impulse | J = ∫F dt | torque over time |
| Δx_cm = rΔθ | distance rolled by a body rolling without slipping | the rolling condition in displacement form | |
| T = 2π ÷ ω = 1 ÷ f | period from angular frequency | every oscillation and rotation | |
| T_s = 2π√(m ÷ k) | period of a mass on a spring | SHM of springs; independent of amplitude | |
| T_p = 2π√(ℓ ÷ g) | period of a simple pendulum (small angles) | pendulums; independent of mass | |
| T_phys = 2π√(I ÷ (mgd)) | period of a physical pendulum | a swinging rod, plate or hoop: d is the pivot-to-centre-of-mass distance | |
| x = x_max cos(ωt + φ) | displacement in simple harmonic motion | SHM position; differentiate for v and a (not printed): v = −ωx_max sin(ωt + φ), a = −ω²x |
The two columns are deliberately parallel: every rotational equation is a translational one with m → I, F → τ, x → θ, v → ω, a → α, p → L. Students who see the pairing need to learn half as much.
The geometry, vector, calculus and identity tables
| Table | Contents | Why it is there |
|---|---|---|
| Geometry and trigonometry | areas of a rectangle (bh), triangle (½bh), circle (πr², C = 2πr, arc s = rθ); volumes of a rectangular solid (ℓwh), cylinder (πr²ℓ, surface 2πrℓ + 2πr²) and sphere (4/3 πr³, surface 4πr²); the right triangle (a² + b² = c², sin, cos, tan) | densities, centre of mass of shapes, arc length in rotation |
| Vectors | A·B = AB cos θ; | A × B |
| Calculus | the chain rule df/dx = (df/du)(du/dx); d/dx of xⁿ, eᵃˣ, ln(ax), sin(ax), cos(ax); the integrals of xⁿ (n ≠ −1), eᵃˣ, 1/(x + a), cos(ax), sin(ax) | everything with varying forces, densities or velocities; the integral of 1/(x + a) is for drag and rockets |
| Identities | log(a·bˣ) = log a + x log b; sin²θ + cos²θ = 1; sin 2θ = 2 sin θ cos θ; tan θ = sin θ ÷ cos θ | projectile range (the sin 2θ), energy in SHM (sin² + cos²) |
What is not on the sheet (and must be known)
| Missing | What to know | Where it bites |
|---|---|---|
| Newton's laws in words | inertia; F = ma as the sum of external forces; action-reaction pairs act on different bodies | every free-body diagram justification |
| Standard rotational inertias | thin rod about its centre 1/12 ML², about one end 1/3 ML²; solid disc or cylinder ½MR²; hoop or thin ring MR²; solid sphere 2/5 MR²; hollow sphere 2/3 MR² | rolling races, pendulums, pulleys; often the question says "derive" and you use I = ∫r² dm |
| Rolling without slipping | v_cm = rω, a_cm = rα; K = ½mv² + ½Iω²; friction does no work; a solid sphere beats a disc beats a hoop down a ramp (a = g sin θ ÷ (1 + I/mr²)) | ramp and rolling questions every year |
| Orbits | circular orbit speed √(GM ÷ r); orbital kinetic energy ½GMm/r; total energy −GMm/2r; escape speed √(2GM ÷ r); Kepler's third law T² = 4π²r³ ÷ GM | satellite and planet questions |
| Drag and terminal velocity | F = −bv or −cv²; terminal velocity when drag equals weight (mg/b); setting up and solving m dv/dt = mg − bv | the differential-equation free-response question |
| Static equilibrium | ΣF = 0 and Στ = 0 about any point | ladders, beams, signs on brackets |
| Collisions | momentum conserved in all; kinetic energy conserved only in elastic; in a 1-D elastic collision the relative speed reverses (v₁ − v₂ = −(v₁′ − v₂′)) | collision questions in both sections |
| SHM derivatives | v = −ωx_max sin(ωt + φ), a = −ω²x; v_max = ωx_max; energy ½kx_max² constant | SHM graph and energy questions |
| Relative velocity, projectiles | components; range R = v² sin 2θ ÷ g on level ground | projectile questions, with the sin 2θ identity from the sheet |
| The derivations | where every sheet equation comes from | "derive an expression" is the commonest free-response verb |
How to use the sheet, and the exam format
Practise with the official sheet from week one, so that by the exam you know every equation's position and what is absent. In the exam use it to check, not to find: a sign (the minus in F_s = −kΔx and in F = −dU/dx), an exponent, which period formula has ℓ and which has m. Reading the sheet to discover an approach costs minutes and usually yields the wrong equation (the constant-acceleration kinematics applied to a varying force is the classic error; the integral forms are directly above them for that reason).
The 2025-26 exam: Section I, 42 multiple-choice questions in 85 minutes, 50 percent; Section II, 4 free-response questions in 95 minutes, 50 percent (mathematical routines, translation between representations, experimental design and analysis, qualitative-quantitative translation); the exam is hybrid digital, with multiple choice in Bluebook and free responses handwritten. The College Board has said the number of multiple-choice questions and the section timings will change from the May 2027 exam; check the course page for the current figures. A calculator (scientific or graphing) is allowed throughout, and the sheet is provided in both sections.
Where to get help
A 1-on-1 online physics class in US after-school hours with a tutor who teaches AP Physics C Mechanics and Electricity and Magnetism, AP Physics 1 and 2 and A-level and IB physics every week: the sheet used from the first lesson, every derivation done by hand until "derive an expression" is the easy question, and the missing list above drilled as flashcards. The AP Physics C page explains how the classes run; the free 30-minute first class is a real lesson on a current problem set. Related pages: the AP Physics 1 score calculator, the five equations of motion and potential energy.
Questions parents ask
1Do you get an equation sheet on the AP Physics C exam?
2What is on the AP Physics C Mechanics equation sheet?
3What is not on the AP Physics C equation sheet?
4Is the AP Physics C equation sheet the same as the AP Physics 1 sheet?
5How should you use the equation sheet in the exam?
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