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Cambridge A-Level Mathematics & Physics Formulas

Essential pure mathematics calculus rules, mechanics equations, and electric field relationships for A-Level exams.

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SUVAT Kinematic Equations (Constant Acceleration)

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SUVAT Kinematic Equations (Constant Acceleration)

v = u + a t, s = u t + \frac{1}{2} a t^{2}, v^{2} = u^{2} + 2 a s, s = \frac{( u + v )}{2} t

Friction Maximum Limit Formula

F_{max} = μ R

w h er e

\mu$ is coefficient of friction and $R$ is normal contact reaction force.

Centripetal Acceleration and Force

a_{c} = \frac{v ^{2}}{r} = ω^{2} r, F_{c} = \frac{m v ^{2}}{r} = m ω^{2} r

Newton's Law of Universal Gravitation

F = G \frac{M m}{r ^{2}}, g = \frac{GM}{r ^{2}}

$ with gravitational constant $G = 6.67 \times 10^{-11}\text{ N m}^2\text{ kg}^{-2}$.

Gravitational Potential in a Radial Field

V = - \frac{GM}{r}

$ Potential is zero at infinity and negative everywhere in the field.

Coulomb's Law of Electrostatic Force

F = \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r ^{2}}

w h er e

\varepsilon_0$ is permittivity of free space.

Electric Field Strength and Potential

E = \frac{V}{d} (uniform), E = - \frac{d V}{d r} (general)

Capacitance and Stored Energy

C = \frac{Q}{V}, E = \frac{1}{2} Q V = \frac{1}{2} C V^{2} = \frac{Q ^{2}}{2 C}

Magnetic Force on a Moving Charge

F = B q v sin θ

$ with direction given by Fleming's left-hand rule.

Magnetic Force on a Current-Carrying Wire

F = B I L sin θ

Derivative of

e^{k x}

and

ln (k x)

\frac{d}{d x} e^{k x} = k e^{k x}, \frac{d}{d x} ln (k x) = \frac{1}{x}

Derivative of

tan (k x)

\frac{d}{d x} tan (k x) = k sec^{2} (k x)

Integration of

\frac{f ^{'} ( x )}{f ( x )}

\int \frac{f ^{'} ( x )}{f ( x )} d x = ln ∣ f (x) ∣ + c

Integration of

f^{'} (x) [f (x)]^{n}

\int f^{'} (x) [f (x)]^{n} d x = \frac{[ f ( x ) ] ^{n + 1}}{n + 1} + c (n \neq = - 1)

Double Angle Formulas for

cos (2 A)

cos (2 A) = cos^{2} A - sin^{2} A = 2 cos^{2} A - 1 = 1 - 2 sin^{2} A

Double Angle Formula for

sin (2 A)

sin (2 A) = 2 sin A cos A

Trigonometric Identity

1 + tan^{2} θ

1 + tan^{2} θ = sec^{2} θ, 1 + cot^{2} θ = csc^{2} θ

Binomial Expansion for Any Rational

n

(1 + x)^{n} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \dots valid for ∣ x ∣ < 1

Parametric Differentiation Formula

\frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}}, \frac{d ^{2} y}{d x ^{2}} = \frac{d}{d t} (\frac{d y}{d x}) \div \frac{d x}{d t}

Separable Differential Equations

To solve

\frac{d y}{d x} = f (x) g (y)

, rewrite as $

\int \frac{1}{g ( y )} d y = \int f (x) d x

Standard Normal Distribution Z-Score

Z = \frac{X - μ}{σ} \sim N (0, 1)

Hypothesis Testing: Type I and Type II Errors

Type I: Rejecting null hypothesis

H_{0}

when it is true (significance level

α

). Type II: Failing to reject

H_{0}

when alternative

H_{1}

is true.

Moment of a Force

Moment = Force \times perpendicular distance to pivot

$ For equilibrium: $\sum \text{clockwise moments} = \sum \text{anticlockwise moments}$.

Photon Energy and Planck's Constant

E = h f = \frac{h c}{λ} (h = 6.63 \times 1 0^{- 34} J s)

Photoelectric Effect Equation

h f = Φ + E_{k, max} = Φ + \frac{1}{2} m v_{max}^{2}

w h er e

\Phi$ is the metal's work function.

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