CG PET cheat sheet

Kinematics: v = u + at; s = ut + ½at²; v² = u² + 2as. For free fall take a = g ≈ 9.8 m/s² downward and fix a sign convention before substituting.

Newton's second law: ΣF = ma. On an incline without friction, a = g sin θ; with kinetic friction, a = g(sin θ − μk cos θ).

Work-energy: W = Fd cos θ; kinetic energy KE = ½mv²; gravitational PE = mgh; work done by net force equals change in KE.

Momentum: p = mv, conserved in collisions with no external net force. Elastic collisions also conserve KE; perfectly inelastic collisions share a common final velocity.

Circular motion: centripetal acceleration a = v²/r = ω²r, force directed toward the centre. Angular motion: ω = 2π/T, v = ωr.

Electricity: Ohm's law V = IR; power P = VI = I²R = V²/R. Resistors in series add; in parallel 1/R = Σ(1/Ri).

Waves and optics: v = fλ; for a wave crossing a boundary frequency stays constant while speed and wavelength change. Snell's law: n₁ sin θ₁ = n₂ sin θ₂.

Thermodynamics: Q = mcΔT for heating without phase change; Q = mL at a phase change. Ideal gas: PV = nRT with T in kelvin.

Mole relations: n = m/M = N/N_A = V/22.4 L at STP for gases. Molar mass in g/mol converts mass to moles directly.

Stoichiometry: balance the equation first, then use mole ratios as conversion factors. Identify the limiting reagent by dividing moles by the coefficient.

Concentration: molarity M = moles solute / litres solution. Dilution obeys M₁V₁ = M₂V₂ because moles of solute are unchanged.

Acid-base: pH = −log[H⁺]; pOH = −log[OH⁻]; pH + pOH = 14 at 25 °C. Strong acid/base dissociates completely; weak ones need Ka or Kb.

Gas laws: Boyle PV = constant at fixed T; Charles V/T = constant at fixed P; combined PV/T = constant; ideal PV = nRT with T in kelvin.

Periodic trends across a period: atomic radius decreases, ionisation energy and electronegativity increase. Down a group the reverse holds because shells are added.

Bonding: ionic between metal and non-metal with large electronegativity difference; covalent between non-metals by sharing; metallic involves delocalised electrons.

Redox: oxidation is loss of electrons (increase in oxidation number), reduction is gain. In a redox reaction the oxidising agent is itself reduced.

Quadratic formula: for ax² + bx + c = 0, x = [−b ± √(b² − 4ac)] / 2a. The discriminant b² − 4ac gives two, one, or no real roots when positive, zero, or negative.

Logarithm rules: log(xy) = log x + log y; log(x/y) = log x − log y; log xⁿ = n log x; log_b b = 1 and log_b 1 = 0. Change of base: log_b x = ln x / ln b.

Differentiation: d/dx xⁿ = nxⁿ⁻¹; d/dx sin x = cos x; d/dx cos x = −sin x; d/dx eˣ = eˣ; d/dx ln x = 1/x. Product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C; ∫eˣ dx = eˣ + C. A definite integral gives signed area under the curve.

Trigonometry: sin²θ + cos²θ = 1; tan θ = sin θ / cos θ; sine rule a/sin A = b/sin B = c/sin C; cosine rule a² = b² + c² − 2bc cos A.

Arithmetic progression: nth term a + (n−1)d, sum n/2[2a + (n−1)d]. Geometric progression: nth term arⁿ⁻¹, sum a(1 − rⁿ)/(1 − r) for r ≠ 1.

Straight line: y = mx + c with slope m = (y₂ − y₁)/(x₂ − x₁); perpendicular slopes multiply to −1. Distance between points = √[(x₂−x₁)² + (y₂−y₁)²].

Probability: P(A∪B) = P(A) + P(B) − P(A∩B); for independent events P(A∩B) = P(A)P(B); conditional P(A|B) = P(A∩B)/P(B).