PIEAS Test 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 v at angle θ on level ground with gravity g, range R = v²sin(2θ)/g. Range is maximum at 45° and equal for complementary angles.

Work–energy theorem

The net work done on a body equals its change in kinetic energy: W_net = ½mv₂² − ½mv₁². This holds even when forces vary along the path.

Ohm's law and power

For a resistor, V = IR. Electrical power dissipated is P = VI = I²R = V²/R, so doubling the current through a fixed resistor quadruples the power.

Mole and molarity

One mole contains 6.022 × 10²³ particles and has mass equal to the molar mass in grams. Molarity is moles of solute per litre of solution.

Quadratic discriminant rule

For ax² + bx + c = 0, the discriminant b² − 4ac decides the roots: positive gives two distinct real roots, zero one repeated root, negative two complex roots.

Logarithm laws

Products become sums, quotients become differences, and powers become multipliers: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a.

Snell's law and critical angle

Refraction obeys n₁sinθ₁ = n₂sinθ₂. Total internal reflection needs travel from denser to rarer medium beyond the critical angle sinθ_c = n₂/n₁.

Photoelectric equation

Maximum photoelectron kinetic energy is K_max = hf − φ, where φ is the metal's work function. No emission occurs below the threshold frequency φ/h.

Half-life decay fraction

Radioactive decay follows N = N₀e^{−λt} with λ = ln2/t½. After n half-lives the remaining fraction is (1/2)ⁿ, independent of the starting amount.

Le Chatelier's principle

A system at equilibrium shifts to oppose a disturbance: adding reactant shifts forward, removing product shifts forward, and heating favours the endothermic direction.

Thin lens equation

For a thin lens or spherical mirror, 1/f = 1/v + 1/u with magnification m = −v/u. A converging lens has positive focal length.

Kirchhoff circuit rules

The junction rule states currents into a node sum to zero, conserving charge. The loop rule states voltages around any closed loop sum to zero, conserving energy.

Calorimetry relations

Sensible heat for a temperature change is Q = mcΔT. At a phase change Q = mL, where L is latent heat, and temperature stays constant during melting or boiling.

Stoichiometry procedure

Balance the equation, convert given masses to moles, apply the mole ratio from coefficients, then convert moles of the wanted substance to mass or volume.

Escape velocity

The minimum speed to leave a body's gravity without further propulsion is v = √(2GM/R), about 11.2 km/s for Earth. It is independent of projectile mass.

Doppler effect for sound

Observed frequency increases when source and observer approach and decreases when they recede. The shift depends on both speeds relative to the sound medium.

Eigenvalue equation

A nonzero vector v is an eigenvector of A with eigenvalue λ when Av = λv. Eigenvalues satisfy det(A − λI) = 0 and sum to the trace.

Divergence and curl meaning

Divergence measures net outflow per volume and is zero for incompressible fields. Curl measures local rotation and is zero for conservative fields.

Carnot efficiency

The maximum efficiency of any heat engine between hot and cold reservoirs is η = 1 − T_c/T_h using absolute temperatures. No real engine exceeds this limit.

Reynolds number regimes

Re = ρVD/μ compares inertia to viscous forces in pipe flow. Flow is typically laminar below about 2300 and turbulent above about 4000.

Bernoulli equation

Along a streamline in steady, inviscid, incompressible flow, P + ½ρV² + ρgz is constant. It expresses energy conservation per unit volume.

Fourier conduction law

Heat rate through a plane wall is q = kAΔT/L, with k the thermal conductivity. In series layers, thermal resistances L/(kA) add like electrical resistances.

First law for control mass

For a closed system, Q − W = ΔU, where W is work done by the system. For an ideal gas ΔU = nC_vΔT, independent of the process path.

Axial deformation formula

A prismatic bar under axial load P stretches by δ = PL/(AE), where L is length, A cross-sectional area, and E Young's modulus.

Euler buckling load

For a pinned–pinned slender column, the critical buckling load is P_cr = π²EI/L². End conditions change the effective length and hence the load.

Mohr's circle principal stresses

Principal stresses are σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]. The circle radius equals the maximum in-plane shear stress.

Nyquist sampling criterion

To avoid aliasing, the sampling rate must exceed twice the highest frequency present in the signal. An anti-aliasing filter precedes the sampler.

Newton–Raphson iteration

The root is refined by x_{n+1} = x_n − f(x_n)/f′(x_n). Convergence is quadratic near a simple root but fails where the derivative is zero.

Second-order system parameters

Natural frequency is ω_n = √(k/m) and damping ratio ζ = c/(2√(km)). ζ below one gives oscillation; ζ equal to one gives critical damping.

Bayes' theorem

The posterior probability is P(A|B) = P(B|A)P(A)/P(B), where P(B) is the total probability of the evidence across all possible causes.

Error propagation rules

For sums and differences, absolute uncertainties add. For products and quotients, fractional uncertainties add. These rules assume independent random errors.

Buckingham pi theorem

The number of independent dimensionless groups describing a physical problem equals the number of variables minus the rank of the dimensional matrix.