Lower Saxony Zentralabitur Physics cheat sheet

Uniform acceleration: v = v₀ + a·t, s = v₀·t + ½a·t², v² = v₀² + 2a·s; free fall uses a = g ≈ 9.81 m/s².

Newton's second law in vector form: F_net = m·a; in the Zentralabitur always name the system and draw the force diagram before resolving components.

Inclined plane without friction: a = g·sin α; with kinetic friction: a = g·(sin α − μ·cos α).

Momentum p = m·v is conserved in every collision and explosion; kinetic energy is conserved only in elastic collisions.

Perfectly inelastic collision: v' = (m₁v₁ + m₂v₂)/(m₁ + m₂); energy loss ΔE = ½·μ·(v₁ − v₂)² with reduced mass μ = m₁m₂/(m₁+m₂).

Circular motion: a_z = v²/r = ω²·r, centripetal force F_z = m·v²/r directed toward the centre; ω = 2π/T.

Gravitation: F = G·m₁m₂/r² with G = 6.674·10⁻¹¹ N·m²/kg²; gravitational field strength g = G·M/r².

Kepler's third law: T²/r³ = 4π²/(G·M) is constant for all orbits around the same central body; geostationary orbit requires T = 24 h and equatorial plane.

Work-energy theorem: W = ∫F·ds = ΔE_kin; for a spring W = ½D·s² and E_pot = ½D·s².

Coulomb's law: F = (1/4πε₀)·q₁q₂/r² with 1/4πε₀ = 8.99·10⁹ N·m²/C²; the force acts along the connecting line and can attract or repel.

Uniform field between parallel plates: E = U/d; force on charge F = q·E; work W = q·U independent of path.

Potential and field: E = −ΔV/Δs; potential of a point charge V = (1/4πε₀)·q/r; potential is a scalar and superposes algebraically.

Ohm's law U = R·I for ohmic conductors at constant temperature; series R = ΣR_i, parallel 1/R = Σ1/R_i.

Kirchhoff's rules: node rule ΣI = 0 (charge conservation), mesh rule ΣU = 0 (energy conservation); include source EMFs with sign.

Magnetic flux density of a long straight wire: B = μ₀·I/(2π·r); inside a solenoid: B = μ₀·n·I with n = N/l.

Lorentz force F = q·v×B acts perpendicular to v, so it does no work; radius of a circular path r = m·v/(q·B).

Induction: U_ind = −N·dΦ/dt with Φ = B·A·cos φ; the minus sign (Lenz) means the induced current opposes the flux change.

Self-induction: U = −L·dI/dt; energy stored in an inductor E = ½L·I²; in a coil the current decays as I = I₀·e^(−t/τ) with τ = L/R.

Harmonic oscillator: restoring force F = −D·s, so a = −(D/m)·s; solution s(t) = ŝ·sin(ωt + φ₀) with ω = √(D/m).

Pendulum for small angles: T = 2π·√(l/g); spring-mass: T = 2π·√(m/D); period is independent of amplitude only in the harmonic limit.

Energy in a harmonic oscillator: E_total = ½D·ŝ²; E_kin and E_pot exchange with a phase shift of π/2 and both oscillate at 2ω.

Wave equation: c = λ·f = λ/T; a wave transports energy and phase, not matter.

Double slit: Δs = n·λ gives constructive interference, Δs = (n + ½)·λ destructive; for small angles sin α ≈ tan α ≈ d_n/a.

Grating: d·sin α_n = n·λ, so λ = d·sin α_n/n; the maximum observable order satisfies n_max = ⌊d/λ⌋.

Single slit minimum: a·sin α_n = n·λ (n ≥ 1); the central maximum is twice as wide as the side maxima.

Sound: c ≈ 343 m/s in air at 20 °C; Doppler for a moving source f' = f·c/(c ± v), with the sign giving a higher pitch on approach.

Standing waves on a string fixed at both ends: l = n·λ/2; open pipe: l = n·λ/2; closed pipe: l = (2n−1)·λ/4.

Photon energy E = h·f = h·c/λ with h = 6.626·10⁻³⁴ J·s and h·c = 1240 eV·nm; 1 eV = 1.602·10⁻¹⁹ J.

Photoelectric effect: E_kin,max = h·f − W_A; the work function W_A is material-specific and the stopping potential gives e·U_G = E_kin,max.

Threshold frequency f_G = W_A/h; below it no emission occurs regardless of intensity, while above it intensity only sets the photocurrent.

de Broglie wavelength λ = h/p = h/(m·v); for electrons accelerated through U, λ = h/√(2·m_e·e·U).

Bohr model: E_n = −13.6 eV/n² for hydrogen; transition energy h·f = E_n − E_m; r_n = n²·r₁ with r₁ = 5.29·10⁻¹¹ m.

Heisenberg uncertainty: Δx·Δp ≥ h/(4π); it limits simultaneous knowledge, not measurement precision of a single quantity.

Wave-particle duality: interference and diffraction demonstrate wave behaviour, the photoelectric effect and Compton scattering demonstrate particle behaviour.

Notation ᴬ_Z X: Z protons, A − Z neutrons; isotopes share Z but differ in A; nuclide charts plot N against Z with the stability valley.

Alpha decay: A decreases by 4, Z by 2; beta-minus: Z increases by 1, A unchanged (n → p + e⁻ + ν̄_e); gamma emission changes neither A nor Z.

Radioactive decay law: N(t) = N₀·e^(−λt); activity A = λ·N with unit becquerel; half-life T₁/₂ = ln2/λ ≈ 0.693/λ.

After n half-lives the remaining fraction is (½)ⁿ; the decay constant λ has unit s⁻¹ and is independent of temperature and pressure.

Mass defect: Δm = Z·m_p + (A−Z)·m_n − m_nucleus; binding energy E_B = Δm·c², often quoted in MeV with 1 u = 931.5 MeV/c².

Mass-energy equivalence E = m·c²; in nuclear reactions the Q value equals the kinetic energy released and follows from the mass difference of reactants and products.

Conservation laws in nuclear reactions: charge Z, nucleon number A, total energy and momentum must each balance on both sides of the equation.

Temperature conversion: T/K = θ/°C + 273.15; absolute zero is 0 K = −273.15 °C and cannot be reached.

Ideal gas law p·V = n·R·T with R = 8.314 J/(mol·K); alternatively p·V = N·k_B·T with k_B = 1.381·10⁻²³ J/K.

First law: ΔU = Q + W, with W the work done on the gas; for work done by the gas write ΔU = Q − p·ΔV.

Isothermal: p·V = const, ΔU = 0, Q = W. Isochoric: V constant, W = 0, Q = ΔU. Isobaric: W = p·ΔV. Adiabatic: Q = 0, so ΔU = W.

Internal energy of an ideal gas depends only on temperature: U = (f/2)·n·R·T, with f = 3 for monatomic and f = 5 for diatomic gases.

Efficiency of a heat engine: η = W/Q_in = 1 − Q_out/Q_in; Carnot limit η_C = 1 − T_cold/T_hot with temperatures in kelvin.

Entropy change for a reversible process: ΔS = Q_rev/T; the entropy of an isolated system never decreases (second law).

Mean kinetic energy per particle: Ē_kin = (3/2)·k_B·T; root-mean-square speed v_rms = √(3·R·T/M).

Lorentz factor γ = 1/√(1 − v²/c²) ≥ 1; it approaches 1 for v ≪ c and diverges as v → c.

Time dilation: Δt = γ·Δt₀, where Δt₀ is the proper time measured on the clock at rest relative to the events.

Length contraction: l = l₀/γ, where l₀ is the proper length measured in the rest frame of the object; contraction occurs only along the direction of motion.

Relativistic momentum p = γ·m·v; total energy E = γ·m·c²; rest energy E₀ = m·c²; kinetic energy E_kin = (γ − 1)·m·c².

Energy-momentum relation E² = (p·c)² + (m·c²)²; for photons m = 0, so E = p·c.

Velocity addition: u = (u' + v)/(1 + u'·v/c²); the result never exceeds c, unlike the Galilean sum.

Relativity principle: the speed of light in vacuum is the same in all inertial frames; simultaneity is frame-dependent for spatially separated events.

Distinguish systematic errors (offset, wrong calibration) from random errors (scatter); averaging reduces random but not systematic error.

Absolute error of a sum or difference: Δ(a ± b) = Δa + Δb; relative errors of products add: Δ(ab)/(ab) = Δa/a + Δb/b.

For a power law y = k·xⁿ the relative error is Δy/y = n·Δx/x; for log or exponential forms linearise before fitting.

Linearisation rule: transform the equation so a straight line results, e.g. T² against l for a pendulum; the slope then yields the physical constant.

Significant figures: the result carries no more significant figures than the least precise input; state measurement uncertainty with one significant figure.

A hypothesis is testable only if it predicts a measurable outcome; the control experiment differs from the test in exactly one variable.

Graphical evaluation: slope and intercept carry units; check whether the intercept is physically expected to be zero before interpreting it.

Model critique: state the assumptions (frictionless, point mass, ideal gas), name the regime where the model fails, and suggest an improvement.