Slovak Physics Olympiad Cat A cheat sheet

Uniformly accelerated motion: v = v0 + at, s = v0·t + ½at², v² = v0² + 2as; these hold only for constant acceleration.

Projectile motion decomposes into independent horizontal (constant v) and vertical (a = −g) components; range R = v0²·sin2α / g over level ground.

Newton's second law in vector form: ΣF = ma; in non-inertial frames add fictitious force −ma_frame.

Work–energy theorem: W_net = ΔEk = ½mv² − ½mv0²; conservative forces admit potential energy with F = −dU/dx.

Momentum conservation holds when net external force is zero; perfectly inelastic collision loses maximum kinetic energy while conserving momentum.

Static friction satisfies f ≤ μs·N; kinetic friction is f = μk·N and opposes relative sliding.

Kepler's third law: T² = (4π²/GM)·a³; Newtonian gravitation F = GMm/r² with potential U = −GMm/r.

Rigid body: torque τ = r × F, moment of inertia I = Σmr², rotational form τ = Iα and Ek_rot = ½Iω².

Fluids: continuity A1v1 = A2v2, Bernoulli p + ½ρv² + ρgh = const, buoyancy F = ρ_fluid·V_submerged·g.

Ideal gas law: pV = nRT = NkT; average translational kinetic energy per molecule = (3/2)kT.

Internal energy of monatomic ideal gas U = (3/2)nRT; diatomic adds rotational degrees of freedom (5/2 nRT at moderate T).

First law: ΔU = Q − W, with W = ∫p dV the work done by the gas; sign conventions must be stated explicitly.

Isothermal work W = nRT·ln(V2/V1); adiabatic process obeys pV^κ = const with κ = Cp/Cv.

Efficiency of a heat engine η = W/Q_in ≤ 1 − T_cold/T_hot (Carnot bound, absolute temperatures).

Mean free path λ = kT/(√2·πd²p); rms speed v_rms = √(3RT/M).

Phase change: latent heat Q = mL at constant temperature; Clausius–Clapeyron relates saturation pressure to temperature.

Entropy change for reversible process ΔS = ∫dQ_rev/T; for an isolated system ΔS ≥ 0.

Simple harmonic motion: x = A·sin(ωt + φ), a = −ω²x; for a spring ω = √(k/m), for a pendulum ω = √(g/l).

Total energy of a harmonic oscillator E = ½kA² = ½mω²A²; period is independent of amplitude for ideal SHM.

Damped oscillator amplitude decays as e^(−bt/2m); resonance occurs near the natural frequency.

Wave equation: v = λf; for a string v = √(F/μ), for sound in gas v = √(κRT/M).

Superposition yields beats at frequency |f1 − f2|; standing waves have nodes spaced λ/2 apart.

Doppler effect for a moving source/observer: f' = f·(v ± v_o)/(v ∓ v_s), signs depending on approach or recession.

Sound intensity level β = 10·log(I/I0) dB with I0 = 10⁻¹² W/m²; intensity falls as 1/r² from a point source.

Closed pipe supports odd harmonics only (f_n = nv/4L, n odd); open pipe supports all harmonics (f_n = nv/2L).

Coulomb's law F = kq1q2/r² with k = 1/(4πε0); electric field E = F/q and potential V = kq/r with E = −dV/dx.

Gauss's law ∮E·dA = Q_enclosed/ε0; field inside a conductor in equilibrium is zero.

Capacitance C = Q/U; parallel plate C = ε0εr·A/d; energy stored W = ½CU².

DC circuits: Kirchhoff's loop rule ΣU = 0 and node rule ΣI = 0; series R adds, parallel conductance adds.

Magnetic force on moving charge F = qv × B; on a straight wire F = BIL·sinθ; field of a long wire B = μ0I/(2πr).

Faraday's law ε = −dΦ/dt; motional emf ε = Bℓv for a rod moving perpendicular to B.

AC circuits: impedance Z = √(R² + (X_L − X_C)²), resonance at f = 1/(2π√(LC)).

Transformer relation U1/U2 = N1/N2 and I1/I2 = N2/N1 for an ideal transformer.

Snell's law n1·sinθ1 = n2·sinθ2; total internal reflection when sinθ_c = n2/n1 for n1 > n2.

Thin lens and mirror equation 1/f = 1/s + 1/s'; magnification m = −s'/s; sign conventions must be fixed before use.

Young's double slit: fringe spacing Δy = λL/d; constructive when path difference = mλ.

Single-slit diffraction minima at a·sinθ = mλ; diffraction grating maxima at d·sinθ = mλ.

Thin-film interference depends on both path difference 2nd·cosθ and phase shifts on reflection (0 or π).

Brewster angle satisfies tanθ_B = n2/n1, where reflected light is fully polarized perpendicular to the plane of incidence.

Light of wavelength λ in vacuum has λ/n inside a medium of refractive index n; frequency is unchanged.

Rayleigh criterion for resolution: θ_min ≈ 1.22·λ/D for a circular aperture.

Lorentz factor γ = 1/√(1 − v²/c²); time dilation Δt = γ·Δt0 and length contraction L = L0/γ.

Relativistic energy E = γmc²; rest energy E0 = mc²; total energy–momentum relation E² = (pc)² + (mc²)².

Photon energy E = hf = hc/λ; photoelectric equation hf = W + Ek_max with threshold f0 = W/h.

de Broglie wavelength λ = h/p; for electrons accelerated by voltage U, λ ≈ h/√(2meU).

Bohr model: En = −13.6 eV/n² for hydrogen; spectral lines from hf = Em − En.

Radioactive decay N = N0·e^(−λt); half-life T½ = ln2/λ; activity A = λN.

Mass defect and binding energy: E_b = Δm·c²; 1 u corresponds to about 931.5 MeV.

Heisenberg uncertainty: Δx·Δp ≥ ħ/2; nuclear radius approximately r = r0·A^(1/3) with r0 ≈ 1.2 fm.

Report a measurement as (x̄ ± σ) with stated confidence; standard uncertainty of the mean is σ/√N for N readings.

Type A uncertainty from repeated readings uses the sample standard deviation; Type B uses instrument resolution or manufacturer specification.

Error propagation: for z = x + y, σz = √(σx² + σy²); for z = x·y, relative errors add in quadrature.

For z = x^a·y^b, relative uncertainty is √((a·σx/x)² + (b·σy/y)²).

Linearize data before fitting (e.g., T² vs l for a pendulum) and extract the constant from the slope.

Least-squares slope uncertainty should be compared with the scatter of residuals to judge fit quality.

Systematic errors shift all readings in the same direction and are not reduced by averaging; random errors are.

Significant figures of a result should match the precision of its uncertainty, typically one or two digits in the uncertainty.

Uniformly accelerated motion: v = v0 + at, s = v0·t + ½at², v² = v0² + 2as; these hold only for constant acceleration.

Projectile motion decomposes into independent horizontal (constant v) and vertical (a = −g) components; range R = v0²·sin2α / g over level ground.

Newton's second law in vector form: ΣF = ma; in non-inertial frames add fictitious force −ma_frame.

Work–energy theorem: W_net = ΔEk = ½mv² − ½mv0²; conservative forces admit potential energy with F = −dU/dx.

Momentum conservation holds when net external force is zero; perfectly inelastic collision loses maximum kinetic energy while conserving momentum.

Static friction satisfies f ≤ μs·N; kinetic friction is f = μk·N and opposes relative sliding.

Kepler's third law: T² = (4π²/GM)·a³; Newtonian gravitation F = GMm/r² with potential U = −GMm/r.

Rigid body: torque τ = r × F, moment of inertia I = Σmr², rotational form τ = Iα and Ek_rot = ½Iω².

Fluids: continuity A1v1 = A2v2, Bernoulli p + ½ρv² + ρgh = const, buoyancy F = ρ_fluid·V_submerged·g.

Ideal gas law: pV = nRT = NkT; average translational kinetic energy per molecule = (3/2)kT.

Internal energy of monatomic ideal gas U = (3/2)nRT; diatomic adds rotational degrees of freedom (5/2 nRT at moderate T).

First law: ΔU = Q − W, with W = ∫p dV the work done by the gas; sign conventions must be stated explicitly.

Isothermal work W = nRT·ln(V2/V1); adiabatic process obeys pV^κ = const with κ = Cp/Cv.

Efficiency of a heat engine η = W/Q_in ≤ 1 − T_cold/T_hot (Carnot bound, absolute temperatures).

Mean free path λ = kT/(√2·πd²p); rms speed v_rms = √(3RT/M).

Phase change: latent heat Q = mL at constant temperature; Clausius–Clapeyron relates saturation pressure to temperature.

Entropy change for reversible process ΔS = ∫dQ_rev/T; for an isolated system ΔS ≥ 0.

Simple harmonic motion: x = A·sin(ωt + φ), a = −ω²x; for a spring ω = √(k/m), for a pendulum ω = √(g/l).

Total energy of a harmonic oscillator E = ½kA² = ½mω²A²; period is independent of amplitude for ideal SHM.

Damped oscillator amplitude decays as e^(−bt/2m); resonance occurs near the natural frequency.

Wave equation: v = λf; for a string v = √(F/μ), for sound in gas v = √(κRT/M).

Superposition yields beats at frequency |f1 − f2|; standing waves have nodes spaced λ/2 apart.

Doppler effect for a moving source/observer: f' = f·(v ± v_o)/(v ∓ v_s), signs depending on approach or recession.

Sound intensity level β = 10·log(I/I0) dB with I0 = 10⁻¹² W/m²; intensity falls as 1/r² from a point source.

Closed pipe supports odd harmonics only (f_n = nv/4L, n odd); open pipe supports all harmonics (f_n = nv/2L).

Coulomb's law F = kq1q2/r² with k = 1/(4πε0); electric field E = F/q and potential V = kq/r with E = −dV/dx.

Gauss's law ∮E·dA = Q_enclosed/ε0; field inside a conductor in equilibrium is zero.

Capacitance C = Q/U; parallel plate C = ε0εr·A/d; energy stored W = ½CU².

DC circuits: Kirchhoff's loop rule ΣU = 0 and node rule ΣI = 0; series R adds, parallel conductance adds.

Magnetic force on moving charge F = qv × B; on a straight wire F = BIL·sinθ; field of a long wire B = μ0I/(2πr).

Faraday's law ε = −dΦ/dt; motional emf ε = Bℓv for a rod moving perpendicular to B.

AC circuits: impedance Z = √(R² + (X_L − X_C)²), resonance at f = 1/(2π√(LC)).

Transformer relation U1/U2 = N1/N2 and I1/I2 = N2/N1 for an ideal transformer.

Snell's law n1·sinθ1 = n2·sinθ2; total internal reflection when sinθ_c = n2/n1 for n1 > n2.

Thin lens and mirror equation 1/f = 1/s + 1/s'; magnification m = −s'/s; sign conventions must be fixed before use.

Young's double slit: fringe spacing Δy = λL/d; constructive when path difference = mλ.

Single-slit diffraction minima at a·sinθ = mλ; diffraction grating maxima at d·sinθ = mλ.

Thin-film interference depends on both path difference 2nd·cosθ and phase shifts on reflection (0 or π).

Brewster angle satisfies tanθ_B = n2/n1, where reflected light is fully polarized perpendicular to the plane of incidence.

Light of wavelength λ in vacuum has λ/n inside a medium of refractive index n; frequency is unchanged.

Rayleigh criterion for resolution: θ_min ≈ 1.22·λ/D for a circular aperture.

Lorentz factor γ = 1/√(1 − v²/c²); time dilation Δt = γ·Δt0 and length contraction L = L0/γ.

Relativistic energy E = γmc²; rest energy E0 = mc²; total energy–momentum relation E² = (pc)² + (mc²)².

Photon energy E = hf = hc/λ; photoelectric equation hf = W + Ek_max with threshold f0 = W/h.

de Broglie wavelength λ = h/p; for electrons accelerated by voltage U, λ ≈ h/√(2meU).

Bohr model: En = −13.6 eV/n² for hydrogen; spectral lines from hf = Em − En.

Radioactive decay N = N0·e^(−λt); half-life T½ = ln2/λ; activity A = λN.

Mass defect and binding energy: E_b = Δm·c²; 1 u corresponds to about 931.5 MeV.

Heisenberg uncertainty: Δx·Δp ≥ ħ/2; nuclear radius approximately r = r0·A^(1/3) with r0 ≈ 1.2 fm.

Report a measurement as (x̄ ± σ) with stated confidence; standard uncertainty of the mean is σ/√N for N readings.

Type A uncertainty from repeated readings uses the sample standard deviation; Type B uses instrument resolution or manufacturer specification.

Error propagation: for z = x + y, σz = √(σx² + σy²); for z = x·y, relative errors add in quadrature.

For z = x^a·y^b, relative uncertainty is √((a·σx/x)² + (b·σy/y)²).

Linearize data before fitting (e.g., T² vs l for a pendulum) and extract the constant from the slope.

Least-squares slope uncertainty should be compared with the scatter of residuals to judge fit quality.

Systematic errors shift all readings in the same direction and are not reduced by averaging; random errors are.

Significant figures of a result should match the precision of its uncertainty, typically one or two digits in the uncertainty.

Uniformly accelerated motion: v = v0 + at, s = v0·t + ½at², v² = v0² + 2as; these hold only for constant acceleration.

Projectile motion decomposes into independent horizontal (constant v) and vertical (a = −g) components; range R = v0²·sin2α / g over level ground.

Newton's second law in vector form: ΣF = ma; in non-inertial frames add fictitious force −ma_frame.

Work–energy theorem: W_net = ΔEk = ½mv² − ½mv0²; conservative forces admit potential energy with F = −dU/dx.

Momentum conservation holds when net external force is zero; perfectly inelastic collision loses maximum kinetic energy while conserving momentum.

Static friction satisfies f ≤ μs·N; kinetic friction is f = μk·N and opposes relative sliding.

Kepler's third law: T² = (4π²/GM)·a³; Newtonian gravitation F = GMm/r² with potential U = −GMm/r.

Rigid body: torque τ = r × F, moment of inertia I = Σmr², rotational form τ = Iα and Ek_rot = ½Iω².

Fluids: continuity A1v1 = A2v2, Bernoulli p + ½ρv² + ρgh = const, buoyancy F = ρ_fluid·V_submerged·g.

Ideal gas law: pV = nRT = NkT; average translational kinetic energy per molecule = (3/2)kT.

Internal energy of monatomic ideal gas U = (3/2)nRT; diatomic adds rotational degrees of freedom (5/2 nRT at moderate T).

First law: ΔU = Q − W, with W = ∫p dV the work done by the gas; sign conventions must be stated explicitly.

Isothermal work W = nRT·ln(V2/V1); adiabatic process obeys pV^κ = const with κ = Cp/Cv.

Efficiency of a heat engine η = W/Q_in ≤ 1 − T_cold/T_hot (Carnot bound, absolute temperatures).

Mean free path λ = kT/(√2·πd²p); rms speed v_rms = √(3RT/M).

Phase change: latent heat Q = mL at constant temperature; Clausius–Clapeyron relates saturation pressure to temperature.

Entropy change for reversible process ΔS = ∫dQ_rev/T; for an isolated system ΔS ≥ 0.

Simple harmonic motion: x = A·sin(ωt + φ), a = −ω²x; for a spring ω = √(k/m), for a pendulum ω = √(g/l).

Total energy of a harmonic oscillator E = ½kA² = ½mω²A²; period is independent of amplitude for ideal SHM.

Damped oscillator amplitude decays as e^(−bt/2m); resonance occurs near the natural frequency.

Wave equation: v = λf; for a string v = √(F/μ), for sound in gas v = √(κRT/M).

Superposition yields beats at frequency |f1 − f2|; standing waves have nodes spaced λ/2 apart.

Doppler effect for a moving source/observer: f' = f·(v ± v_o)/(v ∓ v_s), signs depending on approach or recession.

Sound intensity level β = 10·log(I/I0) dB with I0 = 10⁻¹² W/m²; intensity falls as 1/r² from a point source.

Closed pipe supports odd harmonics only (f_n = nv/4L, n odd); open pipe supports all harmonics (f_n = nv/2L).

Coulomb's law F = kq1q2/r² with k = 1/(4πε0); electric field E = F/q and potential V = kq/r with E = −dV/dx.

Gauss's law ∮E·dA = Q_enclosed/ε0; field inside a conductor in equilibrium is zero.

Capacitance C = Q/U; parallel plate C = ε0εr·A/d; energy stored W = ½CU².

DC circuits: Kirchhoff's loop rule ΣU = 0 and node rule ΣI = 0; series R adds, parallel conductance adds.

Magnetic force on moving charge F = qv × B; on a straight wire F = BIL·sinθ; field of a long wire B = μ0I/(2πr).

Faraday's law ε = −dΦ/dt; motional emf ε = Bℓv for a rod moving perpendicular to B.

AC circuits: impedance Z = √(R² + (X_L − X_C)²), resonance at f = 1/(2π√(LC)).

Transformer relation U1/U2 = N1/N2 and I1/I2 = N2/N1 for an ideal transformer.

Snell's law n1·sinθ1 = n2·sinθ2; total internal reflection when sinθ_c = n2/n1 for n1 > n2.

Thin lens and mirror equation 1/f = 1/s + 1/s'; magnification m = −s'/s; sign conventions must be fixed before use.

Young's double slit: fringe spacing Δy = λL/d; constructive when path difference = mλ.

Single-slit diffraction minima at a·sinθ = mλ; diffraction grating maxima at d·sinθ = mλ.

Thin-film interference depends on both path difference 2nd·cosθ and phase shifts on reflection (0 or π).

Brewster angle satisfies tanθ_B = n2/n1, where reflected light is fully polarized perpendicular to the plane of incidence.

Light of wavelength λ in vacuum has λ/n inside a medium of refractive index n; frequency is unchanged.

Rayleigh criterion for resolution: θ_min ≈ 1.22·λ/D for a circular aperture.

Lorentz factor γ = 1/√(1 − v²/c²); time dilation Δt = γ·Δt0 and length contraction L = L0/γ.

Relativistic energy E = γmc²; rest energy E0 = mc²; total energy–momentum relation E² = (pc)² + (mc²)².

Photon energy E = hf = hc/λ; photoelectric equation hf = W + Ek_max with threshold f0 = W/h.

de Broglie wavelength λ = h/p; for electrons accelerated by voltage U, λ ≈ h/√(2meU).

Bohr model: En = −13.6 eV/n² for hydrogen; spectral lines from hf = Em − En.

Radioactive decay N = N0·e^(−λt); half-life T½ = ln2/λ; activity A = λN.

Mass defect and binding energy: E_b = Δm·c²; 1 u corresponds to about 931.5 MeV.

Heisenberg uncertainty: Δx·Δp ≥ ħ/2; nuclear radius approximately r = r0·A^(1/3) with r0 ≈ 1.2 fm.

Report a measurement as (x̄ ± σ) with stated confidence; standard uncertainty of the mean is σ/√N for N readings.

Type A uncertainty from repeated readings uses the sample standard deviation; Type B uses instrument resolution or manufacturer specification.

Error propagation: for z = x + y, σz = √(σx² + σy²); for z = x·y, relative errors add in quadrature.

For z = x^a·y^b, relative uncertainty is √((a·σx/x)² + (b·σy/y)²).

Linearize data before fitting (e.g., T² vs l for a pendulum) and extract the constant from the slope.

Least-squares slope uncertainty should be compared with the scatter of residuals to judge fit quality.

Systematic errors shift all readings in the same direction and are not reduced by averaging; random errors are.

Significant figures of a result should match the precision of its uncertainty, typically one or two digits in the uncertainty.

Uniformly accelerated motion: v = v0 + at, s = v0·t + ½at², v² = v0² + 2as; these hold only for constant acceleration.

Projectile motion decomposes into independent horizontal (constant v) and vertical (a = −g) components; range R = v0²·sin2α / g over level ground.

Newton's second law in vector form: ΣF = ma; in non-inertial frames add fictitious force −ma_frame.

Work–energy theorem: W_net = ΔEk = ½mv² − ½mv0²; conservative forces admit potential energy with F = −dU/dx.

Momentum conservation holds when net external force is zero; perfectly inelastic collision loses maximum kinetic energy while conserving momentum.

Static friction satisfies f ≤ μs·N; kinetic friction is f = μk·N and opposes relative sliding.

Kepler's third law: T² = (4π²/GM)·a³; Newtonian gravitation F = GMm/r² with potential U = −GMm/r.

Rigid body: torque τ = r × F, moment of inertia I = Σmr², rotational form τ = Iα and Ek_rot = ½Iω².

Fluids: continuity A1v1 = A2v2, Bernoulli p + ½ρv² + ρgh = const, buoyancy F = ρ_fluid·V_submerged·g.

Ideal gas law: pV = nRT = NkT; average translational kinetic energy per molecule = (3/2)kT.

Internal energy of monatomic ideal gas U = (3/2)nRT; diatomic adds rotational degrees of freedom (5/2 nRT at moderate T).

First law: ΔU = Q − W, with W = ∫p dV the work done by the gas; sign conventions must be stated explicitly.

Isothermal work W = nRT·ln(V2/V1); adiabatic process obeys pV^κ = const with κ = Cp/Cv.

Efficiency of a heat engine η = W/Q_in ≤ 1 − T_cold/T_hot (Carnot bound, absolute temperatures).

Mean free path λ = kT/(√2·πd²p); rms speed v_rms = √(3RT/M).

Phase change: latent heat Q = mL at constant temperature; Clausius–Clapeyron relates saturation pressure to temperature.

Entropy change for reversible process ΔS = ∫dQ_rev/T; for an isolated system ΔS ≥ 0.

Simple harmonic motion: x = A·sin(ωt + φ), a = −ω²x; for a spring ω = √(k/m), for a pendulum ω = √(g/l).

Total energy of a harmonic oscillator E = ½kA² = ½mω²A²; period is independent of amplitude for ideal SHM.

Damped oscillator amplitude decays as e^(−bt/2m); resonance occurs near the natural frequency.

Wave equation: v = λf; for a string v = √(F/μ), for sound in gas v = √(κRT/M).

Superposition yields beats at frequency |f1 − f2|; standing waves have nodes spaced λ/2 apart.

Doppler effect for a moving source/observer: f' = f·(v ± v_o)/(v ∓ v_s), signs depending on approach or recession.

Sound intensity level β = 10·log(I/I0) dB with I0 = 10⁻¹² W/m²; intensity falls as 1/r² from a point source.

Closed pipe supports odd harmonics only (f_n = nv/4L, n odd); open pipe supports all harmonics (f_n = nv/2L).

Coulomb's law F = kq1q2/r² with k = 1/(4πε0); electric field E = F/q and potential V = kq/r with E = −dV/dx.

Gauss's law ∮E·dA = Q_enclosed/ε0; field inside a conductor in equilibrium is zero.

Capacitance C = Q/U; parallel plate C = ε0εr·A/d; energy stored W = ½CU².

DC circuits: Kirchhoff's loop rule ΣU = 0 and node rule ΣI = 0; series R adds, parallel conductance adds.

Magnetic force on moving charge F = qv × B; on a straight wire F = BIL·sinθ; field of a long wire B = μ0I/(2πr).

Faraday's law ε = −dΦ/dt; motional emf ε = Bℓv for a rod moving perpendicular to B.

AC circuits: impedance Z = √(R² + (X_L − X_C)²), resonance at f = 1/(2π√(LC)).

Transformer relation U1/U2 = N1/N2 and I1/I2 = N2/N1 for an ideal transformer.

Snell's law n1·sinθ1 = n2·sinθ2; total internal reflection when sinθ_c = n2/n1 for n1 > n2.

Thin lens and mirror equation 1/f = 1/s + 1/s'; magnification m = −s'/s; sign conventions must be fixed before use.

Young's double slit: fringe spacing Δy = λL/d; constructive when path difference = mλ.

Single-slit diffraction minima at a·sinθ = mλ; diffraction grating maxima at d·sinθ = mλ.

Thin-film interference depends on both path difference 2nd·cosθ and phase shifts on reflection (0 or π).

Brewster angle satisfies tanθ_B = n2/n1, where reflected light is fully polarized perpendicular to the plane of incidence.

Light of wavelength λ in vacuum has λ/n inside a medium of refractive index n; frequency is unchanged.

Rayleigh criterion for resolution: θ_min ≈ 1.22·λ/D for a circular aperture.

Lorentz factor γ = 1/√(1 − v²/c²); time dilation Δt = γ·Δt0 and length contraction L = L0/γ.

Relativistic energy E = γmc²; rest energy E0 = mc²; total energy–momentum relation E² = (pc)² + (mc²)².

Photon energy E = hf = hc/λ; photoelectric equation hf = W + Ek_max with threshold f0 = W/h.

de Broglie wavelength λ = h/p; for electrons accelerated by voltage U, λ ≈ h/√(2meU).

Bohr model: En = −13.6 eV/n² for hydrogen; spectral lines from hf = Em − En.

Radioactive decay N = N0·e^(−λt); half-life T½ = ln2/λ; activity A = λN.

Mass defect and binding energy: E_b = Δm·c²; 1 u corresponds to about 931.5 MeV.

Heisenberg uncertainty: Δx·Δp ≥ ħ/2; nuclear radius approximately r = r0·A^(1/3) with r0 ≈ 1.2 fm.

Report a measurement as (x̄ ± σ) with stated confidence; standard uncertainty of the mean is σ/√N for N readings.

Type A uncertainty from repeated readings uses the sample standard deviation; Type B uses instrument resolution or manufacturer specification.

Error propagation: for z = x + y, σz = √(σx² + σy²); for z = x·y, relative errors add in quadrature.

For z = x^a·y^b, relative uncertainty is √((a·σx/x)² + (b·σy/y)²).

Linearize data before fitting (e.g., T² vs l for a pendulum) and extract the constant from the slope.

Least-squares slope uncertainty should be compared with the scatter of residuals to judge fit quality.

Systematic errors shift all readings in the same direction and are not reduced by averaging; random errors are.

Significant figures of a result should match the precision of its uncertainty, typically one or two digits in the uncertainty.

Uniformly accelerated motion: v = v0 + at, s = v0·t + ½at², v² = v0² + 2as; these hold only for constant acceleration.

Projectile motion decomposes into independent horizontal (constant v) and vertical (a = −g) components; range R = v0²·sin2α / g over level ground.

Newton's second law in vector form: ΣF = ma; in non-inertial frames add fictitious force −ma_frame.

Work–energy theorem: W_net = ΔEk = ½mv² − ½mv0²; conservative forces admit potential energy with F = −dU/dx.

Momentum conservation holds when net external force is zero; perfectly inelastic collision loses maximum kinetic energy while conserving momentum.

Static friction satisfies f ≤ μs·N; kinetic friction is f = μk·N and opposes relative sliding.

Kepler's third law: T² = (4π²/GM)·a³; Newtonian gravitation F = GMm/r² with potential U = −GMm/r.

Rigid body: torque τ = r × F, moment of inertia I = Σmr², rotational form τ = Iα and Ek_rot = ½Iω².

Fluids: continuity A1v1 = A2v2, Bernoulli p + ½ρv² + ρgh = const, buoyancy F = ρ_fluid·V_submerged·g.

Ideal gas law: pV = nRT = NkT; average translational kinetic energy per molecule = (3/2)kT.

Internal energy of monatomic ideal gas U = (3/2)nRT; diatomic adds rotational degrees of freedom (5/2 nRT at moderate T).

First law: ΔU = Q − W, with W = ∫p dV the work done by the gas; sign conventions must be stated explicitly.

Isothermal work W = nRT·ln(V2/V1); adiabatic process obeys pV^κ = const with κ = Cp/Cv.

Efficiency of a heat engine η = W/Q_in ≤ 1 − T_cold/T_hot (Carnot bound, absolute temperatures).

Mean free path λ = kT/(√2·πd²p); rms speed v_rms = √(3RT/M).

Phase change: latent heat Q = mL at constant temperature; Clausius–Clapeyron relates saturation pressure to temperature.

Entropy change for reversible process ΔS = ∫dQ_rev/T; for an isolated system ΔS ≥ 0.

Simple harmonic motion: x = A·sin(ωt + φ), a = −ω²x; for a spring ω = √(k/m), for a pendulum ω = √(g/l).

Total energy of a harmonic oscillator E = ½kA² = ½mω²A²; period is independent of amplitude for ideal SHM.

Damped oscillator amplitude decays as e^(−bt/2m); resonance occurs near the natural frequency.

Wave equation: v = λf; for a string v = √(F/μ), for sound in gas v = √(κRT/M).

Superposition yields beats at frequency |f1 − f2|; standing waves have nodes spaced λ/2 apart.

Doppler effect for a moving source/observer: f' = f·(v ± v_o)/(v ∓ v_s), signs depending on approach or recession.

Sound intensity level β = 10·log(I/I0) dB with I0 = 10⁻¹² W/m²; intensity falls as 1/r² from a point source.

Closed pipe supports odd harmonics only (f_n = nv/4L, n odd); open pipe supports all harmonics (f_n = nv/2L).

Coulomb's law F = kq1q2/r² with k = 1/(4πε0); electric field E = F/q and potential V = kq/r with E = −dV/dx.

Gauss's law ∮E·dA = Q_enclosed/ε0; field inside a conductor in equilibrium is zero.

Capacitance C = Q/U; parallel plate C = ε0εr·A/d; energy stored W = ½CU².

DC circuits: Kirchhoff's loop rule ΣU = 0 and node rule ΣI = 0; series R adds, parallel conductance adds.

Magnetic force on moving charge F = qv × B; on a straight wire F = BIL·sinθ; field of a long wire B = μ0I/(2πr).

Faraday's law ε = −dΦ/dt; motional emf ε = Bℓv for a rod moving perpendicular to B.

AC circuits: impedance Z = √(R² + (X_L − X_C)²), resonance at f = 1/(2π√(LC)).

Transformer relation U1/U2 = N1/N2 and I1/I2 = N2/N1 for an ideal transformer.

Snell's law n1·sinθ1 = n2·sinθ2; total internal reflection when sinθ_c = n2/n1 for n1 > n2.

Thin lens and mirror equation 1/f = 1/s + 1/s'; magnification m = −s'/s; sign conventions must be fixed before use.

Young's double slit: fringe spacing Δy = λL/d; constructive when path difference = mλ.

Single-slit diffraction minima at a·sinθ = mλ; diffraction grating maxima at d·sinθ = mλ.

Thin-film interference depends on both path difference 2nd·cosθ and phase shifts on reflection (0 or π).

Brewster angle satisfies tanθ_B = n2/n1, where reflected light is fully polarized perpendicular to the plane of incidence.

Light of wavelength λ in vacuum has λ/n inside a medium of refractive index n; frequency is unchanged.

Rayleigh criterion for resolution: θ_min ≈ 1.22·λ/D for a circular aperture.

Lorentz factor γ = 1/√(1 − v²/c²); time dilation Δt = γ·Δt0 and length contraction L = L0/γ.

Relativistic energy E = γmc²; rest energy E0 = mc²; total energy–momentum relation E² = (pc)² + (mc²)².

Photon energy E = hf = hc/λ; photoelectric equation hf = W + Ek_max with threshold f0 = W/h.

de Broglie wavelength λ = h/p; for electrons accelerated by voltage U, λ ≈ h/√(2meU).

Bohr model: En = −13.6 eV/n² for hydrogen; spectral lines from hf = Em − En.

Radioactive decay N = N0·e^(−λt); half-life T½ = ln2/λ; activity A = λN.

Mass defect and binding energy: E_b = Δm·c²; 1 u corresponds to about 931.5 MeV.

Heisenberg uncertainty: Δx·Δp ≥ ħ/2; nuclear radius approximately r = r0·A^(1/3) with r0 ≈ 1.2 fm.

Report a measurement as (x̄ ± σ) with stated confidence; standard uncertainty of the mean is σ/√N for N readings.

Type A uncertainty from repeated readings uses the sample standard deviation; Type B uses instrument resolution or manufacturer specification.

Error propagation: for z = x + y, σz = √(σx² + σy²); for z = x·y, relative errors add in quadrature.

For z = x^a·y^b, relative uncertainty is √((a·σx/x)² + (b·σy/y)²).

Linearize data before fitting (e.g., T² vs l for a pendulum) and extract the constant from the slope.

Least-squares slope uncertainty should be compared with the scatter of residuals to judge fit quality.

Systematic errors shift all readings in the same direction and are not reduced by averaging; random errors are.

Significant figures of a result should match the precision of its uncertainty, typically one or two digits in the uncertainty.

Uniformly accelerated motion: v = v0 + at, s = v0·t + ½at², v² = v0² + 2as; these hold only for constant acceleration.

Projectile motion decomposes into independent horizontal (constant v) and vertical (a = −g) components; range R = v0²·sin2α / g over level ground.

Newton's second law in vector form: ΣF = ma; in non-inertial frames add fictitious force −ma_frame.

Work–energy theorem: W_net = ΔEk = ½mv² − ½mv0²; conservative forces admit potential energy with F = −dU/dx.

Momentum conservation holds when net external force is zero; perfectly inelastic collision loses maximum kinetic energy while conserving momentum.

Static friction satisfies f ≤ μs·N; kinetic friction is f = μk·N and opposes relative sliding.

Kepler's third law: T² = (4π²/GM)·a³; Newtonian gravitation F = GMm/r² with potential U = −GMm/r.

Rigid body: torque τ = r × F, moment of inertia I = Σmr², rotational form τ = Iα and Ek_rot = ½Iω².

Fluids: continuity A1v1 = A2v2, Bernoulli p + ½ρv² + ρgh = const, buoyancy F = ρ_fluid·V_submerged·g.

Ideal gas law: pV = nRT = NkT; average translational kinetic energy per molecule = (3/2)kT.

Internal energy of monatomic ideal gas U = (3/2)nRT; diatomic adds rotational degrees of freedom (5/2 nRT at moderate T).

First law: ΔU = Q − W, with W = ∫p dV the work done by the gas; sign conventions must be stated explicitly.

Isothermal work W = nRT·ln(V2/V1); adiabatic process obeys pV^κ = const with κ = Cp/Cv.

Efficiency of a heat engine η = W/Q_in ≤ 1 − T_cold/T_hot (Carnot bound, absolute temperatures).

Mean free path λ = kT/(√2·πd²p); rms speed v_rms = √(3RT/M).

Phase change: latent heat Q = mL at constant temperature; Clausius–Clapeyron relates saturation pressure to temperature.

Entropy change for reversible process ΔS = ∫dQ_rev/T; for an isolated system ΔS ≥ 0.

Simple harmonic motion: x = A·sin(ωt + φ), a = −ω²x; for a spring ω = √(k/m), for a pendulum ω = √(g/l).

Total energy of a harmonic oscillator E = ½kA² = ½mω²A²; period is independent of amplitude for ideal SHM.

Damped oscillator amplitude decays as e^(−bt/2m); resonance occurs near the natural frequency.

Wave equation: v = λf; for a string v = √(F/μ), for sound in gas v = √(κRT/M).

Superposition yields beats at frequency |f1 − f2|; standing waves have nodes spaced λ/2 apart.

Doppler effect for a moving source/observer: f' = f·(v ± v_o)/(v ∓ v_s), signs depending on approach or recession.

Sound intensity level β = 10·log(I/I0) dB with I0 = 10⁻¹² W/m²; intensity falls as 1/r² from a point source.

Closed pipe supports odd harmonics only (f_n = nv/4L, n odd); open pipe supports all harmonics (f_n = nv/2L).

Coulomb's law F = kq1q2/r² with k = 1/(4πε0); electric field E = F/q and potential V = kq/r with E = −dV/dx.

Gauss's law ∮E·dA = Q_enclosed/ε0; field inside a conductor in equilibrium is zero.

Capacitance C = Q/U; parallel plate C = ε0εr·A/d; energy stored W = ½CU².

DC circuits: Kirchhoff's loop rule ΣU = 0 and node rule ΣI = 0; series R adds, parallel conductance adds.

Magnetic force on moving charge F = qv × B; on a straight wire F = BIL·sinθ; field of a long wire B = μ0I/(2πr).

Faraday's law ε = −dΦ/dt; motional emf ε = Bℓv for a rod moving perpendicular to B.

AC circuits: impedance Z = √(R² + (X_L − X_C)²), resonance at f = 1/(2π√(LC)).

Transformer relation U1/U2 = N1/N2 and I1/I2 = N2/N1 for an ideal transformer.

Snell's law n1·sinθ1 = n2·sinθ2; total internal reflection when sinθ_c = n2/n1 for n1 > n2.

Thin lens and mirror equation 1/f = 1/s + 1/s'; magnification m = −s'/s; sign conventions must be fixed before use.

Young's double slit: fringe spacing Δy = λL/d; constructive when path difference = mλ.

Single-slit diffraction minima at a·sinθ = mλ; diffraction grating maxima at d·sinθ = mλ.

Thin-film interference depends on both path difference 2nd·cosθ and phase shifts on reflection (0 or π).

Brewster angle satisfies tanθ_B = n2/n1, where reflected light is fully polarized perpendicular to the plane of incidence.

Light of wavelength λ in vacuum has λ/n inside a medium of refractive index n; frequency is unchanged.

Rayleigh criterion for resolution: θ_min ≈ 1.22·λ/D for a circular aperture.

Lorentz factor γ = 1/√(1 − v²/c²); time dilation Δt = γ·Δt0 and length contraction L = L0/γ.

Relativistic energy E = γmc²; rest energy E0 = mc²; total energy–momentum relation E² = (pc)² + (mc²)².

Photon energy E = hf = hc/λ; photoelectric equation hf = W + Ek_max with threshold f0 = W/h.

de Broglie wavelength λ = h/p; for electrons accelerated by voltage U, λ ≈ h/√(2meU).

Bohr model: En = −13.6 eV/n² for hydrogen; spectral lines from hf = Em − En.

Radioactive decay N = N0·e^(−λt); half-life T½ = ln2/λ; activity A = λN.

Mass defect and binding energy: E_b = Δm·c²; 1 u corresponds to about 931.5 MeV.

Heisenberg uncertainty: Δx·Δp ≥ ħ/2; nuclear radius approximately r = r0·A^(1/3) with r0 ≈ 1.2 fm.

Report a measurement as (x̄ ± σ) with stated confidence; standard uncertainty of the mean is σ/√N for N readings.

Type A uncertainty from repeated readings uses the sample standard deviation; Type B uses instrument resolution or manufacturer specification.

Error propagation: for z = x + y, σz = √(σx² + σy²); for z = x·y, relative errors add in quadrature.

For z = x^a·y^b, relative uncertainty is √((a·σx/x)² + (b·σy/y)²).

Linearize data before fitting (e.g., T² vs l for a pendulum) and extract the constant from the slope.

Least-squares slope uncertainty should be compared with the scatter of residuals to judge fit quality.

Systematic errors shift all readings in the same direction and are not reduced by averaging; random errors are.

Significant figures of a result should match the precision of its uncertainty, typically one or two digits in the uncertainty.