AP Physics C: Mechanics — Formula Sheet

Kinematics

1D Motion (constant acceleration)

Velocityv = v₀ + at

Positionx = x₀ + v₀t + ½at²

Velocity-positionv² = v₀² + 2aΔx

Average velocityv̄ = Δx / Δt

Calculus definitions

Velocityv = dx/dt

Accelerationa = dv/dt = d²x/dt²

DisplacementΔx = ∫v dt

Velocity changeΔv = ∫a dt

Projectile Motion (a_x = 0, a_y = −g)

Horizontal positionx = v₀cosθ · tconstant velocity

Vertical positiony = v₀sinθ · t − ½gt²

Vertical velocityv_y = v₀sinθ − gt

Range (flat ground)R = v₀²sin(2θ) / gmax at θ = 45°

Max heightH = v₀²sin²θ / 2g

Circular Motion

Arc lengths = rθθ in rad

Tangential speedv = rω

Centripetal accel.a_c = v²/r = ω²rinward

PeriodT = 2πr/v = 2π/ω

Relative Motion

Relative velocityv_A/B = v_A − v_BA rel. to B

Relative positionr_A/B = r_A − r_B

Galilean transformv_A/C = v_A/B + v_B/C

Newton's Laws & Dynamics

    1st Law: An object remains at rest or in uniform motion unless acted on by a net force.  
    3rd Law: For every action there is an equal and opposite reaction (F_AB = −F_BA).
  

Newton's 2nd Law

Net forceΣF = ma

Net force (general)ΣF = dp/dt

WeightW = mgg = 9.8 m/s²

Normal (on incline)N = mg cosθ

Friction

Static (max)f_s ≤ μ_s N

Kineticf_k = μ_k N

Relationshipμ_s > μ_kalways

Circular Dynamics

Centripetal net forceΣF_c = mv²/r = mω²r

Frequencyf = 1/T

Angular velocityω = 2πf

Variable Forces & Drag

Spring (Hooke's Law)F = −kxrestoring force

Linear dragF_d = −bvb = drag coeff.

Terminal velocityv_t = mg/bwhen ΣF = 0

Variable force workW = ∫F(x) dx

Work & Energy

Work

Constant forceW = Fd cosθ = F⃗ · d⃗

Variable forceW = ∫F(x) dx

Work–energy theoremW_net = ΔKE

PowerP = dW/dt = F · v

Average powerP_avg = W / Δt

Kinetic Energy

TranslationalKE = ½mv²

RotationalKE_rot = ½Iω²

Rolling (total)KE = ½mv² + ½Iω²

Potential Energy

GravitationalU_g = mgh

Elastic (spring)U_s = ½kx²

Conservative forceF = −dU/dx

Conservation of Energy

Isolated systemKE_i + PE_i = KE_f + PE_fno non-cons. forces

With frictionΔE_mech = −W_frictionenergy lost to heat

General formΔKE + ΔPE = W_ncW_nc = non-cons. work

Momentum & Collisions

Linear Momentum

Momentump = mv

ImpulseJ = Δp = FΔt

Impulse (integral)J = ∫F dt

ConservationΣp_i = Σp_f (if ΣF=0)

Center of Mass

Discrete massesx_cm = Σmᵢxᵢ / M

Continuous bodyx_cm = ∫x dm / M

Velocity of cmv_cm = Σmᵢvᵢ / M

Net forceΣF_ext = M a_cm

Collisions

Elasticp conserved AND KE conservede = 1

Inelasticp conserved, KE loste < 1

Perfectly inelasticm₁v₁ + m₂v₂ = (m₁+m₂)v_f

Elastic v₁' (1D)v₁' = (m₁−m₂)v₁ / (m₁+m₂)m₂ initially at rest

Elastic v₂' (1D)v₂' = 2m₁v₁ / (m₁+m₂)m₂ initially at rest

Rotation & Angular Momentum

Rotational Kinematics

Angular velocityω = dθ/dt

Angular accel.α = dω/dt

ω (const. α)ω = ω₀ + αt

θ (const. α)θ = ω₀t + ½αt²

ω² relationω² = ω₀² + 2αΔθ

Rotational Dynamics

Torqueτ = r × F = rF sinθ

Newton's 2nd (rot.)Στ = Iα = dL/dt

Moment of inertiaI = Σmᵢrᵢ² = ∫r² dm

Parallel axis thm.I = I_cm + Md²

Common Moments of Inertia

Solid sphereI = 2/5 MR²

Hollow sphere (thin)I = 2/3 MR²

Solid disk / cylinderI = 1/2 MR²

Hollow cylinder (thin)I = MR²

Rod (about center)I = 1/12 ML²

Rod (about end)I = 1/3 ML²

Angular Momentum

Angular momentumL = Iω

Point massL = r × p = mvr sinθ

Newton's 2ndΣτ_net = dL/dt

ConservationL_i = L_fwhen Στ = 0

Rotational KEKE_rot = ½Iω² = L²/2I

Universal Gravitation

Gravitational Force

ForceF = GMm/r²

G6.674 × 10⁻¹¹ N·m²/kg²

Surface gg = GM/R²

Grav. fieldg⃗ = −GM/r² r̂

Potential Energy & Speed

Grav. PEU = −GMm/rU→0 as r→∞

Orbital speedv_orb = √(GM/r)

Escape speedv_esc = √(2GM/R)

Total orbital EE = −GMm/2r

Kepler's Laws

1st LawOrbits are ellipses; the sun is at one focus.

2nd LawEqual areas swept in equal times (angular momentum is conserved).

3rd LawT² = (4π²/GM) · r³circular orbit

3rd Law (general)T² ∝ a³a = semi-major axis

Simple Harmonic Motion & Oscillations

SHM Equations

Defining equationd²x/dt² = −ω²x

Positionx(t) = A cos(ωt + φ)

Velocityv(t) = −Aω sin(ωt + φ)

Accelerationa(t) = −Aω² cos(ωt + φ)

Max speedv_max = Aωat x = 0

Max accel.a_max = Aω²at x = ±A

Speed at position xv = ω√(A² − x²)

Spring–Mass System

Angular frequencyω = √(k/m)

PeriodT = 2π√(m/k)

Total energyE = ½kA²

Energy split½mv² + ½kx² = ½kA²

Pendulums

Simple pendulum ωω = √(g/L)

Simple pendulum TT = 2π√(L/g)

Physical pendulum TT = 2π√(I/mgd)

Small angle approx.sinθ ≈ θθ < ~15°

    Phase φ: Determined by initial conditions. If x(0) = A → φ = 0. If x(0) = 0, v > 0 → φ = −π/2.  
    Frequency: f = ω/2π,   T = 1/f = 2π/ω.
  

Variable & Constant Reference

Translational mechanics

x, y position (m)

v, v₀ velocity (m/s)

a acceleration (m/s²)

t time (s)

m, M mass (kg)

F force (N)

N normal force (N)

f friction force (N)

μ friction coefficient

k spring constant (N/m)

p momentum (kg·m/s)

J impulse (N·s)

W work (J)

KE kinetic energy (J)

U, PE potential energy (J)

P power (W)

r radius (m)

θ angle (rad or °)

Rotational & oscillatory

ω angular velocity (rad/s)

α angular accel. (rad/s²)

τ torque (N·m)

I moment of inertia (kg·m²)

L angular momentum (kg·m²/s)

T period (s)

f frequency (Hz)

A amplitude (m)

φ phase constant (rad)

G grav. constant

R planet radius (m)

d pivot distance (m)

b drag coefficient

e coefficient of restitution

Key Constants

Gravitational constantG = 6.674 × 10⁻¹¹ N·m²/kg²

g (Earth surface)g = 9.8 m/s² ≈ 10 m/s² (approx.)

Mass of EarthM_E = 5.97 × 10²⁴ kg

Radius of EarthR_E = 6.37 × 10⁶ m

Mass of SunM_S = 1.99 × 10³⁰ kg