Formula Reference

The complete formula bank for Physics, Chemistry & Mathematics — everything needed for UPSC, Banking, SSC, Railway, State PSC and other competitive/Govt job exams, plus higher-studies foundations.

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Motion & Laws Projectile & Rotation Work, Energy & Power SHM & Elasticity Fluid Mechanics Heat & Thermodynamics Electricity Magnetism Optics Waves & Sound Modern Physics

Motion & Newton's Laws

Equation of Motion (Velocity)

v = u + at

u = initial velocity, a = acceleration, t = time

Equation of Motion (Distance)

s = ut + ½at²

Distance covered under uniform acceleration

Equation of Motion (No time)

v² = u² + 2as

Relates velocity to distance without time

Distance in nth Second

sₙ = u + a(2n−1)/2

Distance covered in the nth second alone

Newton's First Law

F = 0 ⟹ v = constant

Law of inertia

Newton's Second Law

F = ma

Force = mass × acceleration

Newton's Third Law

F₁₂ = −F₂₁

Action and reaction are equal and opposite

Momentum

p = mv

Momentum = mass × velocity

Impulse

J = F × t = Δp

Equals change in momentum

Law of Conservation of Momentum

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

For an isolated system, total momentum is conserved

Weight

W = mg

g ≈ 9.8 m/s² on Earth

Friction Force

f = μN

μ = coefficient of friction, N = normal force

Density

ρ = m / V

Mass per unit volume

Pressure

P = F / A

Force per unit area

Projectile, Circular & Rotational Motion

Time of Flight (Projectile)

T = 2u sinθ / g

θ = angle of projection

Maximum Height (Projectile)

H = u²sin²θ / 2g

Horizontal Range (Projectile)

R = u²sin2θ / g

Maximum range occurs at θ = 45°

Centripetal Force

F = mv² / r

r = radius of circular path

Centripetal Acceleration

a = v² / r

Directed toward the centre of the circle

Angular Velocity

ω = 2π / T = 2πf

T = time period, f = frequency

Torque

τ = r × F sinθ

Turning effect of a force

Moment of Inertia (Point Mass)

I = mr²

Angular Momentum

L = Iω

Rotational analogue of linear momentum

Universal Gravitation

F = Gm₁m₂ / r²

G = 6.674 × 10⁻¹¹ N·m²/kg²

Work, Energy & Power

Work Done

W = F × d × cosθ

Force applied over a distance

Kinetic Energy

KE = ½mv²

Energy due to motion

Potential Energy

PE = mgh

Energy due to height above ground

Work-Energy Theorem

W = ΔKE

Net work equals change in kinetic energy

Law of Conservation of Energy

KE + PE = constant

In the absence of friction/external forces

Power

P = W / t

Rate of doing work

Power (in terms of velocity)

P = F × v

SHM, Elasticity & Springs

Hooke's Law

F = kx

k = spring constant, x = extension/compression

Elastic Potential Energy

PE = ½kx²

Energy stored in a stretched/compressed spring

Time Period — Spring-Mass System

T = 2π√(m/k)

Simple Pendulum Period

T = 2π√(L/g)

L = length of pendulum

Young's Modulus

Y = Stress / Strain

Measure of a material's stiffness

Fluid Mechanics

Archimedes' Principle

Fᵦ = ρ_fluid × V_displaced × g

Buoyant force equals weight of fluid displaced

Pascal's Law

P = F / A (transmitted equally)

Pressure applied to an enclosed fluid transmits undiminished

Bernoulli's Equation

P + ½ρv² + ρgh = constant

Along a streamline in an ideal fluid

Equation of Continuity

A₁v₁ = A₂v₂

Conservation of mass in fluid flow

Surface Tension

T = F / L

Force per unit length along a surface

Heat & Thermodynamics

Specific Heat

Q = mcΔT

Heat needed to change temperature

Latent Heat

Q = mL

Heat absorbed/released during a phase change (no temperature change)

First Law of Thermodynamics

ΔU = Q − W

Change in internal energy = heat added − work done by system

Carnot Engine Efficiency

η = 1 − (T_cold / T_hot)

Temperatures in Kelvin; theoretical maximum efficiency

Linear Thermal Expansion

ΔL = L₀αΔT

α = coefficient of linear expansion

Celsius to Fahrenheit

F = (9/5)C + 32

Celsius to Kelvin

K = C + 273.15

Electricity

Ohm's Law

V = IR

Voltage = Current × Resistance

Electrical Power

P = VI = I²R = V²/R

Three equivalent forms

Electrical Energy

E = P × t

Measured in watt-hours or joules

Resistors in Series

R = R₁ + R₂ + …

Total resistance adds directly

Resistors in Parallel

1/R = 1/R₁ + 1/R₂ + …

Reciprocals add

Resistance from Resistivity

R = ρL / A

ρ = resistivity, L = length, A = cross-section area

Joule's Law of Heating

H = I²Rt

Heat generated in a resistor

Capacitance

Q = CV

Charge stored = Capacitance × Voltage

Capacitors in Parallel

C = C₁ + C₂ + …

Capacitors in Series

1/C = 1/C₁ + 1/C₂ + …

Coulomb's Law

F = kq₁q₂ / r²

k = 9 × 10⁹ N·m²/C² (Coulomb's constant)

Magnetism

Force on Current-Carrying Conductor

F = BIL sinθ

B = magnetic field, I = current, L = length

Force on a Moving Charge

F = qvB sinθ

Lorentz force (magnetic component)

Magnetic Field — Straight Wire

B = μ₀I / 2πr

μ₀ = permeability of free space

Faraday's Law of Induction

EMF = −ΔΦ / Δt

Induced EMF equals rate of change of magnetic flux

Optics

Mirror Formula

1/v + 1/u = 1/f

u = object distance, v = image distance, f = focal length

Lens Formula

1/v − 1/u = 1/f

Sign convention: distances measured from optical centre

Magnification

m = v/u = h′/h

Ratio of image to object distance/height

Lens Power

P = 1/f (metres)

Measured in dioptres (D)

Refractive Index

n = c / v

c = speed of light in vacuum (3 × 10⁸ m/s)

Snell's Law

n₁ sinθ₁ = n₂ sinθ₂

Relates angles of incidence and refraction

Waves & Sound

Wave Equation

v = f × λ

Speed = Frequency × Wavelength

Time Period of a Wave

T = 1 / f

Doppler Effect

f′ = f × (v ± v₀) / (v ∓ vₛ)

v₀ = observer speed, vₛ = source speed

Frequency को हराओ

f_beat = |f₁ − f₂|

From two waves of slightly different frequency

Modern Physics

Photon Energy

E = hf

h = Planck's constant (6.626 × 10⁻³⁴ J·s)

Mass-Energy Equivalence

E = mc²

Einstein's famous relation

de Broglie Wavelength

λ = h / p = h / mv

Wave nature of matter

Radioactive Decay Law

N = N₀e^(−λt)

N₀ = initial nuclei count, λ = decay constant

Half-Life

t½ = 0.693 / λ

Time for half the radioactive nuclei to decay

Mass Number

A = Z + N

Z = protons (atomic number), N = neutrons

Mole Concept Solutions Acids & Bases Gas Laws Atomic Structure Thermochemistry Electrochemistry Kinetics & Equilibrium

Mole Concept & Stoichiometry

Number of Moles

n = m / M

mass ÷ molar mass

Number of Particles

N = n × Nₐ

Nₐ = Avogadro's number = 6.022 × 10²³ /mol

Moles of Gas at STP

n = V / 22.4 L

22.4 L = molar volume of a gas at STP

Percentage Composition

% element = (mass of element / molar mass) × 100

Per element within a compound

Empirical Formula Mass Ratio

Simplest whole-number mole ratio

Reduce mole ratios of elements to lowest terms

Molecular Formula

(Empirical Formula)ₙ, n = M / Empirical Formula Mass

Solutions & Concentration

Molarity

M = n / V (litres)

Moles of solute per litre of solution

Molality

m = n(solute) / mass(solvent, kg)

Moles of solute per kg of solvent

Mole Fraction

x_A = n_A / (n_A + n_B)

Ratio of moles of one component to total moles

Dilution Law

M₁V₁ = M₂V₂

Used when diluting a solution

Normality

N = M × n-factor

n-factor = basicity/acidity of the species

Equivalent Weight

E = Molar Mass / n-factor

Mass per gram-equivalent

Normality Dilution

N₁V₁ = N₂V₂

Equivalent form of the dilution law

Acids, Bases & pH

pH = −log[H⁺]

Measure of acidity

pOH

pOH = −log[OH⁻]

Measure of alkalinity

pH–pOH Relation

pH + pOH = 14

At 25°C, in aqueous solution

Ionic Product of Water

Kw = [H⁺][OH⁻] = 10⁻¹⁴

At 25°C

Gas Laws

Boyle's Law

P₁V₁ = P₂V₂

At constant temperature

Charles's Law

V₁/T₁ = V₂/T₂

At constant pressure (T in Kelvin)

Gay-Lussac's Law

P₁/T₁ = P₂/T₂

At constant volume

Combined Gas Law

P₁V₁/T₁ = P₂V₂/T₂

Combines Boyle's and Charles's laws

Ideal Gas Equation

PV = nRT

R = 0.0821 L·atm/mol·K

Graham's Law of Diffusion

r₁/r₂ = √(M₂/M₁)

Rate of diffusion is inversely proportional to √(molar mass)

Atomic Structure

Mass Number

A = Z + N

Protons (Z) + Neutrons (N)

Bohr Orbit Energy (H-like atom)

Eₙ = −13.6 Z² / n² eV

Z = atomic number, n = orbit number

Bohr Orbit Radius (H-like atom)

rₙ = 0.529 n² / Z Å

Maximum Electrons in a Shell

2n²

n = shell/orbit number

Thermochemistry

Hess's Law

ΔH_reaction = ΣΔH_products − ΣΔH_reactants

Enthalpy change is path-independent

Enthalpy & Internal Energy

ΔH = ΔU + ΔnRT

For reactions involving gases

Electrochemistry

Faraday's First Law

m = ZIt

Z = electrochemical equivalent, I = current, t = time

Faraday's Constant

F = 96,500 C/mol

Charge carried by one mole of electrons

Nernst Equation

E = E° − (0.0591/n) log Q

At 25°C; relates cell potential to concentration

Chemical Kinetics & Equilibrium

Rate Law

Rate = k[A]ᵐ[B]ⁿ

k = rate constant; m, n = reaction orders

Arrhenius Equation

k = Ae^(−Ea/RT)

Ea = activation energy

Equilibrium Constant (Kp–Kc)

Kp = Kc(RT)^Δn

Δn = moles of gaseous products − reactants

Number System Algebra — Identities Progressions & Quadratics Percentage Profit, Loss & Discount Simple & Compound Interest Ratio, Mixture & Partnership Time, Speed & Distance Time & Work Averages Mensuration — 2D Mensuration — 3D Coordinate Geometry Geometry Trigonometry Permutation, Combination & Probability Statistics Logarithms

Number System

Divisible by 2

Last digit is even

Divisible by 3

Sum of digits divisible by 3

Divisible by 4

Last two digits divisible by 4

Divisible by 5

Last digit is 0 or 5

Divisible by 8

Last three digits divisible by 8

Divisible by 9

Sum of digits divisible by 9

Divisible by 11

(Sum of digits at odd places − sum at even places) divisible by 11

HCF × LCM

HCF × LCM = Product of the two numbers

HCF of Fractions

HCF(numerators) / LCM(denominators)

LCM of Fractions

LCM(numerators) / HCF(denominators)

Sum of First n Natural Numbers

n(n+1) / 2

Sum of Squares of First n Numbers

n(n+1)(2n+1) / 6

Sum of Cubes of First n Numbers

[n(n+1)/2]²

Algebra — Identities

Square of Sum

(a+b)² = a² + 2ab + b²

Square of Difference

(a−b)² = a² − 2ab + b²

Difference of Squares

a² − b² = (a+b)(a−b)

Square of a Trinomial

(a+b+c)² = a²+b²+c²+2(ab+bc+ca)

Cube of Sum

(a+b)³ = a³ + 3a²b + 3ab² + b³

Cube of Difference

(a−b)³ = a³ − 3a²b + 3ab² − b³

Sum of Cubes

a³ + b³ = (a+b)(a² − ab + b²)

Difference of Cubes

a³ − b³ = (a−b)(a² + ab + b²)

Sum of Three Cubes

a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca)

Special Case: a+b+c = 0

a³ + b³ + c³ = 3abc

Progressions & Quadratic Equations

Quadratic Formula

x = [−b ± √(b²−4ac)] / 2a

For ax² + bx + c = 0

Sum of Roots

α + β = −b/a

Product of Roots

α × β = c/a

AP — nth Term

aₙ = a + (n−1)d

a = first term, d = common difference

AP — Sum of n Terms

Sₙ = n/2 [2a + (n−1)d]

GP — nth Term

aₙ = a·rⁿ⁻¹

r = common ratio

GP — Sum of n Terms

Sₙ = a(rⁿ−1) / (r−1)

For r ≠ 1

GP — Sum to Infinity

S∞ = a / (1−r)

Valid only when |r| < 1

Percentage

% = (Part / Whole) × 100

Successive Percentage Change

Net% = x + y + xy/100

Use negative values for decreases

Reduction to Restore (after increase)

Decrease% = [x / (100+x)] × 100

Increase to Restore (after decrease)

Increase% = [x / (100−x)] × 100

Profit, Loss & Discount

Profit Percentage

Profit% = (SP − CP) / CP × 100

SP = Selling Price, CP = Cost Price

Loss Percentage

Loss% = (CP − SP) / CP × 100

Selling Price

SP = CP × (1 ± %/100)

+ for profit, − for loss

Cost Price

CP = SP × 100 / (100 ± %)

Discount Percentage

Discount% = (MP − SP) / MP × 100

MP = Marked Price

Selling Price (after discount)

SP = MP × (1 − Discount%/100)

Successive Discounts

Net Discount% = d₁ + d₂ − d₁d₂/100

Simple & Compound Interest

Simple Interest

SI = (P × R × T) / 100

P = Principal, R = Rate %, T = Time (years)

Compound Interest — Amount

A = P(1 + R/100)ᵀ

CI = A − P

CI — Compounded Half-Yearly

A = P(1 + (R/2)/100)^(2T)

CI — Compounded Quarterly

A = P(1 + (R/4)/100)^(4T)

CI − SI Difference (2 years)

Diff = P(R/100)²

CI − SI Difference (3 years)

Diff = P(R/100)² × [3 + R/100]

Ratio, Proportion, Mixture & Partnership

Proportion

a:b = c:d ⟹ ad = bc

Partnership Profit Share

Profit ratio = (Investment × Time) ratio

Mixture & Alligation

(Cheaper qty)/(Dearer qty) = (CP_dearer − Mean)/(Mean − CP_cheaper)

Time, Speed & Distance

Speed

Speed = Distance / Time

km/h to m/s

× 5/18

m/s to km/h

× 18/5

Relative Speed (same direction)

|S₁ − S₂|

Relative Speed (opposite direction)

S₁ + S₂

Average Speed (two equal distances)

2S₁S₂ / (S₁+S₂)

Boats & Streams — Downstream

Speed = B + S

B = boat speed, S = stream speed

Boats & Streams — Upstream

Speed = B − S

Boat/Stream Speed from Down & Up

B = (Down+Up)/2, S = (Down−Up)/2

Train Crossing a Pole

Time = Length of train / Speed

Train Crossing a Platform

Time = (Train length + Platform length) / Speed

Time & Work

Work Rate

1 day's work = 1 / (days to finish alone)

Combined Work Rate

1/A + 1/B = 1/Together

A, B = days each takes alone

Pipes & Cisterns — Net Rate

1/Fill − 1/Empty

Inlet fills, outlet/leak empties

Averages

औसत

Average = Sum of terms / Number of terms

Weighted Average

Σ(wᵢxᵢ) / Σwᵢ

Mensuration — 2D

Rectangle — Area

A = l × b

Rectangle — Perimeter

P = 2(l + b)

Square — Area

A = a²

Square — Perimeter

P = 4a

Triangle — Area

A = ½ × base × height

Triangle — Heron's Formula

A = √[s(s−a)(s−b)(s−c)]

s = (a+b+c)/2 (semi-perimeter)

Equilateral Triangle — Area

A = (√3/4)a²

Parallelogram — Area

A = base × height

Rhombus — Area

A = ½ × d₁ × d₂

d₁, d₂ = diagonals

Trapezium — Area

A = ½ × (sum of parallel sides) × height

Circle — Area

A = πr²

Circle — Circumference

C = 2πr

Sector — Area

A = (θ/360) × πr²

Arc Length

L = (θ/360) × 2πr

Mensuration — 3D

Cuboid — Volume

V = l × b × h

Cuboid — Surface Area

SA = 2(lb + bh + hl)

Cube — Volume

V = a³

Cube — Surface Area

SA = 6a²

Cylinder — Volume

V = πr²h

Cylinder — Curved Surface Area

CSA = 2πrh

Cylinder — Total Surface Area

TSA = 2πr(r + h)

Cone — Volume

V = ⅓πr²h

Cone — Curved Surface Area

CSA = πrl

l = slant height = √(r²+h²)

Sphere — Volume

V = (4/3)πr³

Sphere — Surface Area

SA = 4πr²

Hemisphere — Volume

V = (2/3)πr³

Hemisphere — Total Surface Area

TSA = 3πr²

Coordinate Geometry

Distance Formula

d = √[(x₂−x₁)² + (y₂−y₁)²]

Midpoint Formula

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Section Formula (internal)

P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))

Slope of a Line

m = (y₂−y₁) / (x₂−x₁)

Slope-Intercept Form

y = mx + c

Geometry

Pythagorean Theorem

a² + b² = c²

c = hypotenuse of a right triangle

Sum of Interior Angles (polygon)

(n−2) × 180°

n = number of sides

Each Interior Angle (regular polygon)

[(n−2) × 180°] / n

Angle in a Semicircle

= 90°

Angle subtended by a diameter

Angle at Centre vs Circumference

Angle at centre = 2 × angle at circumference

For the same arc

Trigonometry

Sine

sinθ = Opposite / Hypotenuse

Cosine

cosθ = Adjacent / Hypotenuse

Tangent

tanθ = Opposite / Adjacent

Cosecant, Secant, Cotangent

cosecθ=1/sinθ, secθ=1/cosθ, cotθ=1/tanθ

Pythagorean Identity

sin²θ + cos²θ = 1

Identity — Tangent/Secant

1 + tan²θ = sec²θ

Identity — Cotangent/Cosecant

1 + cot²θ = cosec²θ

Height & Distance

Height = Distance × tanθ

θ = angle of elevation/depression

Permutation, Combination & Probability

Permutation

ⁿPᵣ = n! / (n−r)!

Combination

ⁿCᵣ = n! / (r!(n−r)!)

Combination Complement

ⁿCᵣ = ⁿCₙ₋ᵣ

Circular Permutation

(n−1)!

Ways to arrange n items in a circle

Probability

P(E) = Favourable outcomes / Total outcomes

Addition Rule (Probability)

P(A∪B) = P(A) + P(B) − P(A∩B)

Statistics

Mean

Mean = Σx / n

Median (odd n)

Value at position (n+1)/2

After arranging data in order

Mode

Most frequently occurring value

Range

Range = Maximum − Minimum

Variance

σ² = Σ(x − mean)² / n

Standard Deviation

σ = √Variance

Logarithms

Product Rule

log(xy) = log x + log y

Quotient Rule

log(x/y) = log x − log y

Power Rule

log(xⁿ) = n log x

Log of Base to Itself

log_a(a) = 1, log_a(1) = 0

Change of Base

log_a x = log_b x / log_b a