Physics Formula Reference

1.1 Linear Motion

v = v₀ + at

Final velocity from initial velocity v₀, acceleration a, and time t

s = v₀t + ½at²

Displacement with constant acceleration

v² = v₀² + 2as

Velocity-displacement relationship

s = vt

Displacement at constant velocity

v_avg = (v₀ + v)/2

Average velocity for constant acceleration

1.2 Circular Motion

v = ωr

Tangential velocity, where ω is angular velocity and r is radius

ω = 2π/T = 2πf

Angular velocity from period T or frequency f

a_c = v²/r = ω²r

Centripetal acceleration

F_c = mv²/r = mω²r

Centripetal force

1.3 Projectile Motion

h = v₀²sin²θ/(2g)

Maximum height for projectile launched at angle θ

R = v₀²sin(2θ)/g

Range of projectile

t_flight = 2v₀sinθ/g

Total flight time

2.1 Newton's Laws

F = ma

Newton's second law

F_g = mg

Gravitational force (weight)

F_friction = μN

Friction force, where μ is coefficient of friction and N is normal force

2.2 Gravitation

F = GMm/r²

Universal gravitation, G = 6.67×10⁻¹¹ N·m²/kg²

g = GM/R²

Gravitational acceleration at surface

U = -GMm/r

Gravitational potential energy

v_escape = √(2GM/R)

Escape velocity

v_orbital = √(GM/r)

Orbital velocity for circular orbit

3.1 Work and Power

W = F·s = Fs cosθ

Work done by force F over displacement s

P = W/t = F·v

Power as work per unit time

P = τω

Rotational power, where τ is torque

3.2 Mechanical Energy

KE = ½mv²

Kinetic energy (translational)

PE = mgh

Gravitational potential energy (near surface)

PE_elastic = ½kx²

Elastic potential energy in spring with stiffness k

E_total = KE + PE = constant

Conservation of mechanical energy

p = mv

Linear momentum

F = Δp/Δt

Newton's second law (momentum form)

m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

Conservation of momentum

J = FΔt = Δp

Impulse

5.1 Rotational Kinematics

ω = ω₀ + αt

Angular velocity with angular acceleration α

θ = ω₀t + ½αt²

Angular displacement

ω² = ω₀² + 2αθ

Angular velocity-displacement relation

5.2 Rotational Dynamics

τ = r × F = rF sinθ

Torque (perpendicular distance × force)

τ = Iα

Rotational analog of F = ma

L = Iω

Angular momentum

KE_rot = ½Iω²

Rotational kinetic energy

5.3 Moments of Inertia

I = Σmᵢrᵢ² (discrete) or I = ∫r²dm (continuous)

General definition

I_rod(center) = (1/12)ML²

Rod of length L rotating about center

I_rod(end) = (1/3)ML²

Rod rotating about end

I_disk = ½MR²

Solid disk or cylinder about axis

I_sphere = (2/5)MR²

Solid sphere about diameter

I_hoop = MR²

Thin hoop or cylindrical shell

I = I_cm + Md²

Parallel axis theorem: d is distance between axes

6.1 Heat and Temperature

Q = mcΔT

Heat transfer, where c is specific heat capacity

Q = mL

Heat for phase change, where L is latent heat

L = L_f (fusion) or L_v (vaporization)

L_f ≈ 334 kJ/kg (water), L_v ≈ 2260 kJ/kg (water)

ΔL = αL₀ΔT

Linear thermal expansion, α is coefficient

ΔV = βV₀ΔT

Volumetric expansion, β ≈ 3α for solids

6.2 Heat Transfer

P = kA(ΔT)/d

Heat conduction, k is thermal conductivity

P = σεAT⁴

Stefan-Boltzmann law, σ = 5.67×10⁻⁸ W/(m²·K⁴)

6.3 Ideal Gas Law

PV = nRT

n is moles, R = 8.314 J/(mol·K)

PV = NkT

N is number of molecules, k = 1.38×10⁻²³ J/K

ρ = PM/(RT)

Density from ideal gas law, M is molar mass

6.4 Thermodynamic Processes

ΔU = Q - W

First law of thermodynamics

U = (f/2)nRT

Internal energy, f = degrees of freedom (3 mono, 5 dia, 6 poly)

W = ∫PdV = P(V₂ - V₁) (isobaric)

Work done by gas

PVᵞ = constant (adiabatic)

γ = C_p/C_v = (f+2)/f

W_adiabatic = (P₁V₁ - P₂V₂)/(γ-1)

Work in adiabatic process

η = W/Q_in = 1 - Q_out/Q_in

Efficiency of heat engine

7.1 Pressure and Buoyancy

P = F/A

Pressure definition

P = P₀ + ρgh

Hydrostatic pressure at depth h

F_buoyant = ρ_fluid V g

Archimedes' principle, V is displaced volume

P₁ + ρgh₁ + ½ρv₁² = P₂ + ρgh₂ + ½ρv₂²

Bernoulli's equation

A₁v₁ = A₂v₂

Continuity equation (incompressible flow)

8.1 Electric Force and Field

F = kq₁q₂/r²

Coulomb's law, k = 8.99×10⁹ N·m²/C²

k = 1/(4πε₀)

ε₀ = 8.85×10⁻¹² C²/(N·m²)

E = F/q = kQ/r²

Electric field

V = kQ/r

Electric potential

U = qV = kq₁q₂/r

Electric potential energy

8.2 Circuits

V = IR

Ohm's law

P = IV = I²R = V²/R

Electric power

R = ρL/A

Resistance, ρ is resistivity

R_series = R₁ + R₂ + R₃ + ...

Resistors in series

1/R_parallel = 1/R₁ + 1/R₂ + 1/R₃ + ...

Resistors in parallel

Q = CV

Capacitor charge

U_C = ½CV² = ½Q²/C

Energy stored in capacitor

8.3 Magnetism

F = qvB sinθ

Magnetic force on moving charge

F = BIL sinθ

Magnetic force on current-carrying wire

r = mv/(qB)

Radius of charged particle in magnetic field (Larmor radius)

T = 2πm/(qB)

Period of circular motion in magnetic field

Φ = BA cosθ

Magnetic flux

ε = -dΦ/dt = -N(dΦ/dt)

Faraday's law of induction

9.1 Wave Properties

v = fλ

Wave speed, frequency, wavelength relation

v_sound ≈ 343 m/s (air at 20°C)

Speed of sound

c = 3.00×10⁸ m/s

Speed of light in vacuum

v = c/n

Speed of light in medium with index n

9.2 Optics

n₁sinθ₁ = n₂sinθ₂

Snell's law of refraction

θ_c = arcsin(n₂/n₁)

Critical angle for total internal reflection (n₁ > n₂)

1/f = 1/d_o + 1/d_i

Thin lens equation

m = -d_i/d_o = h_i/h_o

Magnification

f = R/2

Focal length of spherical mirror

10.1 Special Relativity

γ = 1/√(1 - v²/c²)

Lorentz factor

L = L₀/γ

Length contraction

Δt = γΔt₀

Time dilation

E = γmc²

Relativistic energy

E² = (pc)² + (mc²)²

Energy-momentum relation

E = mc²

Rest energy

p = γmv

Relativistic momentum

10.2 Photons

E = hf = hc/λ

Photon energy, h = 6.626×10⁻³⁴ J·s

p = h/λ = E/c

Photon momentum

F = -kx

Hooke's law for springs

ω = √(k/m)

Angular frequency of mass-spring system

T = 2π√(m/k) = 2π/ω

Period of mass-spring oscillator

T = 2π√(L/g)

Period of simple pendulum

x(t) = A cos(ωt + φ)

Simple harmonic motion

v_max = Aω

Maximum velocity in SHM

a_max = Aω²

Maximum acceleration in SHM

g = 9.8 m/s² ≈ 10 m/s²

Gravitational acceleration (Earth surface)

G = 6.67×10⁻¹¹ N·m²/kg²

Universal gravitational constant

k = 8.99×10⁹ N·m²/C²

Coulomb's constant

e = 1.602×10⁻¹⁹ C

Elementary charge

m_e = 9.109×10⁻³¹ kg

Electron mass

m_p = 1.673×10⁻²⁷ kg

Proton mass

c = 3.00×10⁸ m/s

Speed of light

h = 6.626×10⁻³⁴ J·s

Planck's constant

k_B = 1.38×10⁻²³ J/K

Boltzmann constant

R = 8.314 J/(mol·K)

Universal gas constant

N_A = 6.022×10²³ mol⁻¹

Avogadro's number

σ = 5.67×10⁻⁸ W/(m²·K⁴)

Stefan-Boltzmann constant

ρ_water = 1000 kg/m³

Density of water

c_water = 4186 J/(kg·K)

Specific heat capacity of water

P_atm = 101,325 Pa ≈ 1.01×10⁵ Pa

Standard atmospheric pressure

sin(x) ≈ x (for small x in radians)

Valid for |x| < 0.3 rad ≈ 17°

cos(x) ≈ 1 - x²/2 (for small x)

Valid for small angles

tan(x) ≈ x (for small x)

Valid for small angles

(1 + x)ⁿ ≈ 1 + nx (for |x| << 1)

Binomial approximation

√(1 + x) ≈ 1 + x/2 (for |x| << 1)

Useful approximation

1/(1 + x) ≈ 1 - x (for |x| << 1)

Useful approximation

A_circle = πr²

Area of circle

C_circle = 2πr

Circumference of circle

V_sphere = (4/3)πr³

Volume of sphere

A_sphere = 4πr²

Surface area of sphere

V_cylinder = πr²h

Volume of cylinder

A_triangle = ½bh

Area of triangle

c² = a² + b² - 2ab cos(C)

Law of cosines

sin(A)/a = sin(B)/b = sin(C)/c

Law of sines

a² + b² = c²

Pythagorean theorem

Physics Formula Reference Guide

1. Kinematics & Motion

1.1 Linear Motion

1.2 Circular Motion

1.3 Projectile Motion

2. Forces & Dynamics

2.1 Newton's Laws

2.2 Gravitation

3. Energy & Work

3.1 Work and Power

3.2 Mechanical Energy

4. Momentum

5. Rotational Motion

5.1 Rotational Kinematics

5.2 Rotational Dynamics

5.3 Moments of Inertia

6. Thermodynamics

6.1 Heat and Temperature

6.2 Heat Transfer

6.3 Ideal Gas Law

6.4 Thermodynamic Processes

7. Fluid Mechanics

7.1 Pressure and Buoyancy

8. Electricity & Magnetism

8.1 Electric Force and Field

8.2 Circuits

8.3 Magnetism

9. Waves & Optics

9.1 Wave Properties

9.2 Optics

10. Modern Physics

10.1 Special Relativity

10.2 Photons

11. Oscillations

12. Useful Constants

13. Useful Approximations

14. Geometry & Trigonometry