BCECE flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Projectile range formula

For launch speed u at angle θ on level ground, the horizontal range is R = u² sin 2θ / g. Range is maximum at θ = 45° and equal for complementary angles.

Coefficient of restitution

The ratio e = (relative velocity of separation)/(relative velocity of approach) for two colliding bodies. e = 1 means perfectly elastic, e = 0 means perfectly inelastic sticking together.

Rolling without slipping condition

A body rolling without slipping on a surface satisfies v = ωR, where v is centre-of-mass speed, ω is angular speed and R is radius. Total kinetic energy then equals ½mv² + ½Iω².

Simple pendulum period

For small oscillations, the period of a simple pendulum is T = 2π√(L/g), independent of mass and amplitude. Period increases with length and decreases with stronger gravity.

RMS speed of gas molecules

The root-mean-square speed is v_rms = √(3RT/M), where R is the gas constant, T the absolute temperature and M the molar mass. It rises with temperature and falls for heavier molecules.

Mirror and lens formulas

Spherical mirror: 1/v + 1/u = 1/f with f = R/2. Thin lens: 1/v − 1/u = 1/f. Consistent sign convention must be applied before substituting values.

Photoelectric stopping potential

The stopping potential V₀ satisfies eV₀ = hν − φ, where φ is the work function. It depends on light frequency but not intensity; intensity changes photocurrent only.

Kirchhoff circuit rules

Junction rule: algebraic sum of currents at a node is zero. Loop rule: algebraic sum of potential differences around any closed loop is zero. Both follow from charge and energy conservation.

Magnetic force on charge

A charge q moving with velocity v in field B feels F = qv × B, magnitude qvB sinθ. The force is perpendicular to velocity, so it does no work and cannot change speed.

Escape velocity

The minimum speed to escape a body's gravity is v_e = √(2GM/R) = √(2gR). For Earth it is about 11.2 km/s and is independent of the projectile's mass.

Mole concept

One mole contains N_A = 6.022 × 10²³ particles and equals molar mass in grams. At STP an ideal gas mole occupies 22.4 L; molarity equals moles of solute per litre.

Quantum number constraints

Principal n ≥ 1; azimuthal l = 0 to n−1; magnetic m_l = −l to +l; spin m_s = ±½. A subshell holds 2(2l+1) electrons and a shell holds 2n².

Le Chatelier pressure rule

Raising pressure on a gaseous equilibrium shifts it toward the side with fewer moles of gas. If gaseous mole counts are equal on both sides, pressure change causes no shift.

Relation between Kp and Kc

For a gaseous reaction, K_p = K_c(RT)^Δn, where Δn is moles of gaseous products minus gaseous reactants. K_p equals K_c when Δn = 0.

Buffer pH equation

For an acidic buffer, pH = pK_a + log([salt]/[acid]). Maximum buffer capacity occurs when salt and acid concentrations are equal, giving pH = pK_a.

Nernst equation

Cell potential at non-standard conditions is E = E° − (RT/nF)ln Q, which becomes E° − (0.059/n)log Q at 298 K. Q is the reaction quotient.

Gibbs energy and spontaneity

ΔG = ΔH − TΔS. A reaction is spontaneous at constant temperature and pressure when ΔG < 0; at equilibrium ΔG = 0 and ΔG° = −RT ln K.

Order of a reaction

The order is the sum of experimental concentration exponents in the rate law, not the stoichiometric coefficients. First-order half-life is 0.693/k, independent of initial concentration.

Hybridisation and geometry

Steric number (sigma bonds plus lone pairs) sets hybridisation: 2 is sp linear, 3 is sp² trigonal planar, 4 is sp³ tetrahedral. Lone pairs compress bond angles below ideal values.

Markovnikov addition rule

When HX adds to an unsymmetrical alkene, hydrogen attaches to the carbon with more hydrogens, giving the more stable carbocation intermediate. Peroxide conditions reverse this for HBr.

Faraday electrolysis relation

Mass deposited m = (M I t)/(n F), where M is molar mass, I current, t time, n electrons per ion and F = 96500 C/mol. Charge Q = It.

Quadratic roots and discriminant

For ax² + bx + c = 0, roots sum to −b/a and multiply to c/a. Discriminant D = b² − 4ac: positive gives distinct real roots, zero equal roots, negative complex roots.

Geometric progression sum

For first term a and ratio r, S_n = a(rⁿ − 1)/(r − 1) when r ≠ 1. The infinite sum a/(1 − r) converges only when |r| < 1.

Derivative of common functions

d/dx(xⁿ) = nx^(n−1), d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x. Chain rule multiplies by the inner derivative.

Definite integral symmetry

∫₀^a f(x)dx = ∫₀^a f(a − x)dx. Also ∫₋ₐ^a f(x)dx = 0 if f is odd and 2∫₀^a f(x)dx if f is even.

Perpendicular line slopes

Two non-vertical lines are perpendicular exactly when the product of their slopes is −1. Parallel lines have equal slopes and zero angle between them.

Point-to-line distance

The perpendicular distance from point (x₁, y₁) to line ax + by + c = 0 is |ax₁ + by₁ + c|/√(a² + b²). The denominator must be positive.

Parabola standard elements

For y² = 4ax, the focus is (a, 0), the directrix is x = −a, the vertex is the origin and the latus rectum length is 4a.

Ellipse eccentricity

For x²/a² + y²/b² = 1 with a > b, eccentricity e = √(1 − b²/a²), so 0 ≤ e < 1. Foci lie at (±ae, 0) and the sum of focal distances is 2a.

Conditional probability

P(A|B) = P(A ∩ B)/P(B) provided P(B) > 0. Events are independent when P(A ∩ B) = P(A)P(B), which makes P(A|B) equal P(A).

Binomial distribution mean

For n independent trials with success probability p, P(X = r) = nCr pʳ (1 − p)^(n−r). The mean is np and the variance is np(1 − p).

Dot and cross product

a·b = |a||b|cosθ is zero for perpendicular vectors. |a × b| = |a||b|sinθ equals the parallelogram area; the cross product is perpendicular to both vectors.