BCECE cheat sheet

Projectile motion: range R = u²sin2θ/g on level ground, maximum at θ = 45°; maximum height H = u²sin²θ/(2g); time of flight T = 2u sinθ/g.

Newton's second law in vector form: F_net = dp/dt = ma when mass is constant; on an incline of angle θ, acceleration down a frictionless slope is g sinθ.

Work-energy theorem: W_net = ΔKE = ½mv² − ½mu²; for a spring, elastic potential energy stored is ½kx² and work by friction is −μN·d.

Momentum conservation applies when net external force is zero: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂; coefficient of restitution e = (v₂ − v₁)/(u₁ − u₂).

Rotational dynamics: torque τ = Iα, angular momentum L = Iω, and rolling without slipping requires v = ωR; moment of inertia of a solid cylinder about its axis is ½MR².

Simple harmonic motion: for a spring–mass system T = 2π√(m/k); for a simple pendulum T = 2π√(L/g); acceleration a = −ω²x and v_max = ωA.

Ideal gas and kinetic theory: PV = nRT; average translational KE per molecule = (3/2)k_BT; rms speed v_rms = √(3RT/M).

Optics decision rule: use the mirror formula 1/v + 1/u = 1/f with f = R/2 for mirrors and the lens formula 1/v − 1/u = 1/f for lenses; real-is-positive sign convention must be fixed before substituting.

Photoelectric effect: maximum kinetic energy K_max = hν − φ, threshold frequency ν₀ = φ/h; stopping potential V₀ satisfies eV₀ = K_max, independent of intensity.

Mole relations: number of moles n = mass/molar mass = N/N_A = V/22.4 L at STP for a gas; molarity M = moles of solute per litre of solution.

Ideal gas equation PV = nRT with R = 0.0821 L·atm·mol⁻¹·K⁻¹ = 8.314 J·mol⁻¹·K⁻¹; at STP one mole of ideal gas occupies 22.4 L.

Quantum numbers: n = 1,2,3…; l = 0 to n−1; m_l = −l to +l; m_s = ±½; maximum electrons in a shell = 2n² and in a subshell = 2(2l+1).

Aufbau, Pauli and Hund rules set electron configuration order: 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p; Cr and Cu show anomalous configurations due to extra stability of half-filled and fully filled d subshells.

Le Chatelier's principle: increasing pressure shifts equilibrium toward the side with fewer gaseous moles; raising temperature shifts an exothermic reaction toward reactants (K decreases).

Equilibrium and acid–base relations: K_p = K_c(RT)^Δn; pH = −log[H⁺]; for a weak acid pH = ½(pK_a − log C); buffer pH = pK_a + log([salt]/[acid]).

Thermochemistry: ΔH = ΔU + Δn_gRT; Hess's law allows summing enthalpy changes of steps; bond breaking is endothermic and bond formation exothermic.

Electrochemistry: Nernst equation E = E° − (0.059/n)log Q at 298 K; at the cathode reduction occurs, at the anode oxidation; ΔG° = −nFE°.

Quadratic equation ax² + bx + c = 0: roots α, β satisfy α + β = −b/a and αβ = c/a; discriminant D = b² − 4ac gives two distinct real roots if D > 0, equal if D = 0, complex if D < 0.

Arithmetic progression: n-th term a_n = a + (n−1)d; sum S_n = (n/2)[2a + (n−1)d]. Geometric progression: a_n = ar^(n−1); S_n = a(rⁿ − 1)/(r − 1) for r ≠ 1; infinite sum a/(1 − r) only if |r| < 1.

Limits and derivatives: d/dx(xⁿ) = nx^(n−1); d/dx(sin x) = cos x; d/dx(ln x) = 1/x; L'Hôpital's rule applies to 0/0 and ∞/∞ indeterminate forms.

Definite integrals: ∫₀^a f(x)dx = ∫₀^a f(a − x)dx is a standard symmetry substitution; ∫₋ₐ^a f(x)dx = 0 for odd f and 2∫₀^a f(x)dx for even f.

Straight lines: slope m = (y₂ − y₁)/(x₂ − x₁); point-slope form y − y₁ = m(x − x₁); perpendicular lines satisfy m₁m₂ = −1; distance from (x₁,y₁) to ax + by + c = 0 is |ax₁ + by₁ + c|/√(a² + b²).

Conic sections: circle x² + y² = a²; parabola y² = 4ax with focus (a, 0) and directrix x = −a; ellipse x²/a² + y²/b² = 1 has eccentricity e = √(1 − b²/a²) when a > b.

Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); conditional probability P(A|B) = P(A ∩ B)/P(B); binomial distribution P(X = r) = nCr pʳ q^(n−r) with mean np.

Vectors: dot product a·b = |a||b|cosθ gives perpendicularity when zero; cross product magnitude |a||b|sinθ equals the area of the parallelogram spanned by a and b.