Free Physics Formulas Cheat Sheet: Quick Reference for Exams

Free Physics Formulas and Worked Examples for StudentsPhysics can feel like a language of symbols and equations. This guide collects the most important formulas across core physics topics, explains when to use them, and gives worked examples so you can see the steps and reasoning. Use it as a study reference, exam cheat sheet, or practice companion.

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Mechanics

Key concepts: motion, forces, energy, momentum.

Kinematics (one dimension)

• Displacement: x = x0 + v0 t + ⁄₂ a t^2
• Velocity (constant acceleration): v = v0 + a t
• Velocity squared: v^2 = v0^2 + 2 a (x − x0)

Worked example: A car starts from rest and accelerates at 2.5 m/s^2 for 8 s. Find its final speed and distance traveled.

• v = v0 + a t = 0 + 2.5·8 = 20 m/s
• x = 0 + 0 + 1/2·2.5·8^2 = 0.5·2.5·64 = 80 m

Newton’s Laws & Forces

• Newton’s second law: F_net = m a
• Friction (kinetic): f_k = μ_k N

Worked example: A 10 kg block on a horizontal surface with μ_k = 0.2 is pulled with 50 N. Find acceleration.

• N = m g = 10·9.81 = 98.1 N
• f_k = μ_k N = 0.2·98.1 = 19.62 N
• F_net = 50 − 19.62 = 30.38 N
• a = F_net / m = 30.38 / 10 = 3.038 m/s^2

Work and Energy

• Work: W = F · d · cosθ
• Kinetic energy: KE = ⁄₂ m v^2
• Potential energy (gravity): U = m g h
• Work–energy theorem: W_net = ΔKE
• Conservation of energy: E_total = KE + U + … (constant when non-conservative work = 0)

Worked example: A 2 kg mass is dropped from 5 m. Find speed just before hitting ground.

• Using energy: m g h = ⁄₂ m v^2 → v = sqrt(2 g h) = sqrt(2·9.81·5) ≈ 9.90 m/s

Momentum and Collisions

• Momentum: p = m v
• Impulse: J = Δp = F_avg Δt
• Elastic collision (1D) — relative speed reverses; formulas depend on masses. For two masses m1 and m2 with initial velocities u1, u2: v1 = (u1 (m1 − m2) + 2 m2 u2) / (m1 + m2)
  v2 = (u2 (m2 − m1) + 2 m1 u1) / (m1 + m2)

Worked example: m1 = 1 kg at 3 m/s collides elastically with m2 = 2 kg at rest. Find speeds after collision.

• v1 = (3(1−2) + 2·2·0)/(1+2) = (−3)/3 = −1 m/s
• v2 = (0(2−1) + 2·1·3)/3 = ⁄₃ = 2 m/s

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Rotational Motion

Important formulas

• Angular displacement: θ (radians)
• Angular velocity: ω = dθ/dt
• Angular acceleration: α = dω/dt
• Relation to linear: v = ω r, a_tangential = α r, a_centripetal = ω^2 r
• Rotational kinetic energy: K_rot = ⁄₂ I ω^2
• Torque: τ = I α = r × F
• Moment of inertia: depends on shape (e.g., solid disk I = ⁄₂ m R^2)

Worked example: A solid disk (m = 4 kg, R = 0.5 m) spins at 10 rad/s. Find its rotational kinetic energy.

• I = ⁄₂ m R^2 = 0.5·4·0.5^2 = 1·0.25 = 0.5 kg·m^2
• K_rot = ⁄₂ I ω^2 = 0.5·0.5·10^2 = 0.25·100 = 25 J

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Gravitation & Orbits

Formulas

• Newton’s universal gravitation: F = G m1 m2 / r^2 (G = 6.674×10^−11 N·m^2/kg^2)
• Gravitational potential energy (near Earth): U = m g h (approximate)
• Orbital speed (circular): v = sqrt(G M / r)
• Period (circular orbit): T = 2π r / v = 2π sqrt(r^3 / (G M))

Worked example: Find orbital speed for a satellite 300 km above Earth’s surface (R_earth = 6371 km). r = 6671 km = 6.671×10^6 m, M = 5.972×10^24 kg.

• v ≈ sqrt(G M / r) ≈ sqrt(6.674e-11·5.972e24 / 6.671e6) ≈ 7.73×10^3 m/s

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Thermodynamics

Key relations

• Ideal gas law: PV = n R T (R = 8.314 J/mol·K)
• First law: ΔU = Q − W (sign convention: W = work done by system)
• Heat capacities: Q = m c ΔT
• Heat transfer (conduction): Q/t = k A ΔT / L

Worked example: How much heat to raise 0.5 kg of water (c = 4184 J/kg·K) from 20°C to 80°C?

• Q = m c ΔT = 0.5·4184·60 = 125,520 J ≈ 1.26×10^5 J

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Waves & Oscillations

Formulas

• Wave speed: v = f λ
• Period and frequency: f = 1/T
• Simple harmonic motion (spring): x(t) = A cos(ω t + φ), ω = sqrt(k/m)
• Energy in SHM (spring): E = ⁄₂ k A^2

Worked example: Mass m = 0.2 kg on spring k = 50 N/m. Find ω and period.

• ω = sqrt(k/m) = sqrt(⁄₀.2) = sqrt(250) ≈ 15.81 rad/s
• T = 2π / ω ≈ 2π / 15.81 ≈ 0.397 s

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Electricity & Magnetism

Electrostatics

• Coulomb’s law: F = k q1 q2 / r^2 (k = 1/(4π ε0) ≈ 8.988×10^9 N·m^2/C^2)
• Electric field: E = F / q = k Q / r^2
• Electric potential (point charge): V = k Q / r

Worked example: Charge 1 μC at origin, find E at 0.1 m on x-axis.

• E = k Q / r^2 = 8.99e9·1e-6 / 0.1^2 = 8.99e9·1e-6 / 0.01 = 8.99e5 N/C

Circuits

• Ohm’s law: V = I R
• Power: P = V I = I^2 R = V^2 / R
• Series resistors: R_eq = Σ R_i
• Parallel resistors: 1/R_eq = Σ 1/R_i

Worked example: Two resistors 4 Ω and 6 Ω in series with 12 V source. Current?

• R_eq = 10 Ω, I = V / R_eq = 12 / 10 = 1.2 A

Magnetism & Induction

• Magnetic force on moving charge: F = q v × B
• EMF (Faraday): ε = −dΦ_B/dt, Φ_B = ∫ B · dA

Worked example: A loop with area 0.02 m^2 in a uniform B changing from 0.1 T to 0.01 T in 0.2 s. EMF magnitude:

• ΔΦ = A ΔB = 0.02·(0.01−0.1) = −0.0018 Wb; |ε| = |ΔΦ/Δt| = 0.0018/0.2 = 0.009 V

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Optics

Geometrical optics

• Mirror/lens equation: 1/f = 1/do + 1/di
• Magnification: m = −di/do = h_image / h_object

Wave optics

• Single-slit diffraction (small angle): a sinθ ≈ m λ (m = ±1, ±2, …)
• Double-slit interference: d sinθ = m λ

Worked example: Young’s double-slit with d = 0.5 mm, λ = 500 nm, first-order fringe angle:

• sinθ = λ / d = 5e-7 / 5e-4 = 1e-3 → θ ≈ 0.0573°

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Modern Physics

Relativity (special)

• Time dilation: Δt = γ Δt0, γ = 1 / sqrt(1 − v^2/c^2)
• Length contraction: L = L0 / γ
• Energy–mass: E = γ m c^2, rest energy E0 = m c^2

Quantum (basic)

• Photon energy: E = h f = h c / λ (h = 6.626×10^−34 J·s)
• de Broglie wavelength: λ = h / p

Worked example: Photon wavelength 500 nm energy:

• E = h c / λ = 6.626e-34·3e8 / 5e-7 ≈ 3.976e-19 J ≈ 2.48 eV

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Tips for Using Formulas

• Always check units; use SI units for consistency.
• Sketch the problem and list knowns/unknowns.
• Identify whether energy, kinematics, forces, or conservation laws best simplify the problem.
• For multi-step problems, solve symbolically first, then plug numbers to reduce algebra errors.

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Quick Reference: Selected Formulas

• v = v0 + a t
• v^2 = v0^2 + 2 a Δx
• F = m a
• W = F d cosθ
• KE = ⁄₂ m v^2
• p = m v
• τ = I α
• K_rot = ⁄₂ I ω^2
• v_orbit = sqrt(G M / r)
• PV = n R T
• v_wave = f λ
• V = I R
• E_photon = h c / λ

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If you want, I can: provide this as a printable PDF, add practice problems with solutions, or create topic-specific cheat sheets (mechanics, E&M, or waves).