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IB Physics SL Data Booklet

Search the key equations and constants used in IB Physics SL, with variables defined and each formula connected to revision notes.

These formulas are rendered as real mathematics rather than images. Read the definition, check every variable, and follow the topic link to see the idea in context. Always confirm the permitted official booklet or sheet for your examination session.

Reference section

A · Space, Time & Motion

Kinematics (constant acceleration)

v = u + at \qquad s = ut + \frac{1}{2}at^2 \qquad v^2 = u^2 + 2as \qquad s=\frac{1}{2}(u+v)t

Used for any object with constant (uniform) acceleration — free fall, braking vehicles, projectile components. List the five variables s, u, v, a, t; pick the equation that omits the one you don't need.

$s$: displacement (m)
$u$: initial velocity (m s⁻¹)
$v$: final velocity (m s⁻¹)
$a$: acceleration (m s⁻²)
$t$: time (s)

Projectile motion

T=\frac{2u\sin\theta}{g} \qquad R=\frac{u^2\sin2\theta}{g} \qquad H_{max}=\frac{u^2\sin^2\theta}{2g}

Valid only when launch and landing heights are equal and air resistance is neglected. For unequal heights, resolve into components and use SUVAT on each axis independently with time as the link.

$T$: total time of flight (s)
$R$: horizontal range (m)
$H_max$: maximum height (m)
$u$: launch speed (m s⁻¹)
$θ$: launch angle above horizontal
$g$: gravitational field strength (m s⁻²)

Newton's second law & momentum

F_{net}=ma \qquad p=mv \qquad F=\frac{\Delta p}{\Delta t}

F = Δp/Δt is the most general form; F = ma follows when mass is constant. Use impulse-momentum (J = FΔt = Δp) for collisions and impact problems.

$F_net$: net (resultant) force (N)
$m$: mass (kg)
$a$: acceleration (m s⁻²)
$p$: momentum (kg m s⁻¹)
$v$: velocity (m s⁻¹)
$F$: force (N)
$Δp$: change in momentum — impulse (N s)
$Δt$: time interval (s)

Work, energy, power

W = Fs\cos\theta \qquad E_k = \frac{1}{2}mv^2 \qquad E_p=mgh \qquad P = \frac{W}{t}=Fv

Use the work-energy theorem (W_net = ΔE_k) to avoid finding acceleration. P = Fv applies when a vehicle travels at constant speed (driving force = resistive force).

$W$: work done (J)
$F$: applied force (N)
$s$: displacement (m)
$θ$: angle between force and displacement
$E_k$: kinetic energy (J)
$E_p$: gravitational potential energy (J)
$h$: height (m)
$P$: power (W)
$t$: time (s)
$v$: velocity (m s⁻¹)

Circular motion

a = \frac{v^2}{r}=\omega^2r \qquad F=\frac{mv^2}{r} \qquad \omega=\frac{2\pi}{T}=2\pi f

Centripetal acceleration always points toward the centre; the centripetal force is the net inward force (which might be tension, gravity, friction, or a combination). Speed is constant but velocity is not.

$a$: centripetal acceleration (m s⁻²)
$F$: centripetal force (N)
$v$: speed (m s⁻¹)
$r$: radius of circular path (m)
$ω$: angular velocity (rad s⁻¹)
$m$: mass (kg)
$T$: period of revolution (s)
$f$: frequency of revolution (Hz)

Reference section

B · Particulate Nature of Matter

Specific heat capacity & latent heat

Q = mc\Delta T \qquad Q=mL

Use Q = mcΔT when temperature changes (no phase change). Switch to Q = mL during a phase change (melting/boiling) — temperature stays constant while bonds break or form. A single heating process may require both equations applied in sequence.

$Q$: thermal energy transferred (J)
$m$: mass (kg)
$c$: specific heat capacity (J kg⁻¹ K⁻¹)
$ΔT$: change in temperature (K or °C — the same size either way)
$L$: specific latent heat — L_f (fusion/melting) or L_v (vaporisation/boiling) (J kg⁻¹)

Stefan–Boltzmann law

P = e\sigma A T^4

Applies to thermal radiation from any object. For stars, e ≈ 1 (black body). Combined with Wien's law, lets you compare two stars' radii from their luminosities and temperatures: L ∝ R²T⁴.

$P$: total power radiated (W)
$e$: emissivity (0 ≤ e ≤ 1; e = 1 for an ideal black body)
$σ$: Stefan–Boltzmann constant = 5.67×10⁻⁸ W m⁻²K⁻⁴
$A$: surface area of the emitter (m²)
$T$: absolute (kelvin) temperature of the surface (K)

Wien's displacement law

\lambda_{max}T = 2.90\times10^{-3}\text{ m K}

Lets you find a star's surface temperature from the colour of its peak emission, or predict the peak wavelength from its temperature. Hotter objects emit peak radiation at shorter (bluer) wavelengths.

$λ_max$: wavelength at which emission intensity is greatest (m)
$T$: absolute temperature of the surface (K)

Earth's energy balance

\frac{(1-\alpha)S}{4} = \sigma T^4

Derived by setting absorbed solar power equal to emitted thermal power. The factor of 4 arises because Earth intercepts sunlight over a disc (πR²) but radiates from its whole sphere (4πR²). Predicts ~255 K without greenhouse gases; actual ~288 K due to the greenhouse effect.

$α$: albedo — fraction of incident radiation reflected (Earth ≈ 0.30)
$S$: solar constant ≈ 1361 W m⁻²
$σ$: Stefan–Boltzmann constant = 5.67×10⁻⁸ W m⁻²K⁻⁴
$T$: predicted equilibrium surface temperature (K)

Ideal gas law

pV = nRT = NkT

Always use T in kelvin. To compare two states of the same gas, use pV/T = constant (combined gas law). The two forms are equivalent: R = N_A × k, so nR = Nk.

$p$: pressure (Pa)
$V$: volume (m³)
$n$: amount of substance (mol)
$R$: molar gas constant = 8.31 J mol⁻¹K⁻¹
$T$: absolute temperature (K)
$N$: number of molecules
$k$: Boltzmann constant = 1.38×10⁻²³ J K⁻¹

Kinetic theory of gases

p=\frac{1}{3}\frac{Nm}{V}\langle c^2\rangle \qquad \langle E_k\rangle=\frac{1}{2}m\langle c^2\rangle=\frac{3}{2}kT \qquad c_{rms}=\sqrt{\frac{3kT}{m}}

Links microscopic molecular properties to macroscopic pressure and temperature. Since ⟨E_k⟩ ∝ T, temperature is a direct measure of average molecular KE. c_rms ∝ √(T/m), so lighter molecules move faster at the same temperature.

$p$: pressure of the gas (Pa)
$N$: number of molecules
$m$: mass of one molecule (kg)
$V$: volume (m³)
$⟨c²⟩$: mean square speed of the molecules (m² s⁻²)
$⟨E_k⟩$: average translational kinetic energy per molecule (J)
$k$: Boltzmann constant = 1.38×10⁻²³ J K⁻¹
$T$: absolute temperature (K)
$c_rms$: root-mean-square speed of the molecules (m s⁻¹)

Ohm's law & resistor combinations

V=IR \qquad R_{series}=R_1+R_2+\ldots \qquad \frac{1}{R_{parallel}}=\frac{1}{R_1}+\frac{1}{R_2}+\ldots

For series circuits: same current everywhere, voltages add. For parallel: same voltage across each branch, currents add. R_parallel is always less than the smallest individual resistor.

$V$: potential difference across the component (V)
$I$: current (A)
$R$: resistance (Ω)
$R_series$: total resistance of resistors in series (Ω)
$R_parallel$: total resistance of resistors in parallel (Ω)

EMF, internal resistance & power

\varepsilon=I(R+r) \qquad P = IV = I^2R = \frac{V^2}{R}

Terminal voltage V = ε − Ir drops below EMF when current flows. Use I²R for finding heat dissipated when R is known; V²/R when V is known. Maximum power transfer to external load occurs when R = r.

$ε$: electromotive force (EMF) of the source (V)
$I$: current (A)
$R$: external (load) resistance (Ω)
$r$: internal resistance of the source (Ω)
$P$: electrical power dissipated (W)
$V$: potential difference (V)

Capacitance and stored energy

C=\frac{Q}{V} \qquad E=\frac{1}{2}CV^2=\frac{Q^2}{2C} \qquad Q=Q_0e^{-t/RC}

After one time constant τ = RC, charge falls to Q₀/e ≈ 37%. After 5τ, capacitor is considered fully discharged (~1% remaining). Capacitors in series combine like parallel resistors and vice versa.

$C$: capacitance (F)
$Q$: charge stored on the capacitor (C)
$V$: potential difference across the capacitor (V)
$E$: energy stored in the capacitor (J)
$Q_0$: initial charge (C)
$t$: time since discharge began (s)
$R$: resistance in the discharge circuit (Ω)
$RC$: time constant τ — time for charge to fall to 1/e of its initial value (s)

Reference section

C · Wave Behaviour

Wave equation

v = f\lambda \qquad T=\frac{1}{f}

Wave speed depends on the medium, not the source. Frequency is set by the source and is unchanged as a wave passes between media; wavelength changes instead (λ = v/f in each medium).

$v$: wave speed (m s⁻¹)
$f$: frequency (Hz)
$λ$: wavelength (m)
$T$: period (s)

SHM defining equation & solutions

a=-\omega^2x \qquad x=x_0\cos(\omega t) \qquad v=\pm\omega\sqrt{x_0^2-x^2}

The defining condition of SHM is a ∝ −x (acceleration always opposes displacement). Use x = x₀cos(ωt) when displacement is maximum at t = 0; use sin if starting from equilibrium. v = ±ω√(x₀²−x²) gives speed at any displacement without needing time.

$a$: acceleration (m s⁻²)
$ω$: angular frequency (rad s⁻¹) = 2πf
$x$: displacement from equilibrium at time t (m)
$x_0$: amplitude — maximum displacement (m)
$v$: velocity at displacement x (m s⁻¹)

SHM energy

E_k=\frac{1}{2}m\omega^2(x_0^2-x^2) \qquad E_p=\frac{1}{2}m\omega^2x^2 \qquad E_{total}=\frac{1}{2}m\omega^2x_0^2

KE is maximum at equilibrium (x = 0) and zero at the extremes (x = ±x₀). PE is the reverse. Total energy is constant (no damping) and proportional to amplitude squared — doubling amplitude quadruples energy.

$E_k$: kinetic energy at displacement x (J)
$E_p$: potential energy at displacement x (J)
$E_total$: total mechanical energy (constant) (J)
$m$: mass of the oscillator (kg)
$ω$: angular frequency (rad s⁻¹)
$x$: displacement (m)
$x_0$: amplitude (m)

Pendulum & mass-spring periods

T = 2\pi\sqrt{\frac{L}{g}} \qquad T = 2\pi\sqrt{\frac{m}{k}}

Pendulum period is independent of mass and amplitude (for small angles < ~10°). Mass-spring period increases with heavier mass and weaker spring. These can be rearranged to measure g (pendulum experiment) or k (spring experiment).

$T$: period of oscillation (s)
$L$: length of the pendulum (m)
$g$: gravitational field strength (m s⁻²)
$m$: mass on the spring (kg)
$k$: spring constant (N m⁻¹)

Snell's law, refractive index & critical angle

n_1\sin\theta_1 = n_2\sin\theta_2 \qquad n=\frac{c}{v} \qquad \sin\theta_c=\frac{n_2}{n_1}

All angles measured from the normal to the boundary. TIR only occurs when going from a denser medium to a less dense one (n₁ > n₂) and the angle of incidence exceeds θ_c. Used for optical fibres, diamonds, and prism design.

$n₁, n₂$: refractive indices of medium 1 and medium 2
$θ₁, θ₂$: angles from the normal in medium 1 and medium 2
$c$: speed of light in vacuum (m s⁻¹)
$v$: speed of light in the medium (m s⁻¹)
$θ_c$: critical angle (only defined when going from a denser medium n₁ to a less dense medium n₂)

Double-slit interference & diffraction grating

s = \frac{\lambda D}{d} \qquad d\sin\theta=n\lambda

For Young's double-slit use s = λD/d (small angles). For a diffraction grating use d sinθ = nλ; gratings give sharper, brighter maxima than double slits. Maximum order n_max = d/λ (since sinθ ≤ 1).

$s$: fringe spacing on the screen (m)
$λ$: wavelength of light (m)
$D$: distance from slits/grating to screen (m)
$d$: slit separation, or grating spacing = 1/(lines per metre) (m)
$θ$: angle to the nth order maximum
$n$: order number (0, 1, 2, …)

Single-slit diffraction

b\sin\theta=m\lambda \qquad \theta\approx\frac{\lambda}{b}

Gives the positions of minima (dark fringes) — note the central maximum is twice as wide as the secondary maxima. Diffraction is most pronounced when b ≈ λ. The central maximum angular half-width θ ≈ λ/b (small angle approximation).

$b$: slit width (m)
$θ$: angle to the mth minimum (or angular half-width of central maximum, small-angle case)
$m$: minimum order number (±1, ±2, …)
$λ$: wavelength (m)

Standing waves — strings & pipes

f_n=\frac{nv}{2L}\ (\text{string, both ends fixed; pipe, both ends open}) \qquad f_n=\frac{nv}{4L}\ (\text{pipe, one end closed})

Standing waves require nodes at fixed ends and antinodes at open ends. A closed-end pipe only supports odd harmonics (n = 1, 3, 5…), giving it a different timbre from an open pipe.

$f_n$: frequency of the nth harmonic (Hz)
$n$: harmonic number: 1,2,3,… for open systems; 1,3,5,… (odd only) for a pipe closed at one end
$v$: wave speed in the string or air column (m s⁻¹)
$L$: length of the string or air column (m)

Doppler effect (moving source/observer)

f' = f\left(\frac{v \pm v_o}{v \mp v_s}\right) \qquad \frac{\Delta\lambda}{\lambda}\approx\frac{v}{c}\ \text{(light, } v\ll c\text{)}

Source approaching → higher observed frequency (blue shift); receding → lower (red shift). Use the top signs when source/observer approach; bottom when receding. The light version (Δλ/λ ≈ v/c) is used to determine recession speeds of galaxies.

$f'$: observed frequency (Hz)
$f$: emitted (source) frequency (Hz)
$v$: speed of the wave in the medium (m s⁻¹)
$v_o$: speed of the observer relative to the medium (m s⁻¹)
$v_s$: speed of the source relative to the medium (m s⁻¹)
$Δλ/λ$: fractional wavelength shift for light
$c$: speed of light = 3.00×10⁸ m s⁻¹

Hubble's law

v = H_0 d

Hubble's Law is evidence that the universe is expanding — more distant galaxies recede faster. The Hubble constant H₀ gives the age of the universe as approximately 1/H₀. Current best estimates: H₀ ≈ 67–73 km s⁻¹ Mpc⁻¹.

$v$: recession speed of a galaxy (km s⁻¹)
$H_0$: Hubble constant (km s⁻¹ Mpc⁻¹)
$d$: distance to the galaxy (Mpc)

Reference section

D · Fields

Newton's law of gravitation & field strength

F = \frac{Gm_1m_2}{r^2} \qquad g=\frac{F}{m}=\frac{GM}{r^2}

Both masses feel equal and opposite forces (Newton's third law). g is the acceleration due to gravity AND the gravitational field strength — numerically the same (N kg⁻¹ = m s⁻²). Force follows an inverse-square law: doubling distance quarters the force.

$F$: gravitational force between the two masses (N)
$G$: gravitational constant = 6.67×10⁻¹¹ N m² kg⁻²
$m₁, m₂$: the two point (or spherical) masses (kg)
$r$: separation between their centres (m)
$g$: gravitational field strength at distance r from mass M (N kg⁻¹)
$M$: the source mass creating the field (kg)

Gravitational potential & orbital motion

V_g=-\frac{GM}{r} \qquad E_p=-\frac{GMm}{r} \qquad v=\sqrt{\frac{GM}{r}} \qquad T^2=\frac{4\pi^2}{GM}r^3

Gravitational potential is always negative (zero at infinity). Orbital speed decreases with increasing radius. Kepler's third law (T² ∝ r³) follows directly from equating gravitational and centripetal forces.

$V_g$: gravitational potential (J kg⁻¹) — always negative, zero at infinity
$E_p$: gravitational potential energy of mass m (J)
$v$: orbital speed for a stable circular orbit (m s⁻¹)
$T$: orbital period (s)
$G$: gravitational constant
$M$: mass being orbited (kg)
$r$: orbital radius (m)

Escape velocity & total orbital energy

v_{esc}=\sqrt{\frac{2GM}{R}} \qquad E_{total}=-\frac{GMm}{2r}

Escape velocity is derived by setting total mechanical energy to zero (KE + PE = 0). Total orbital energy is negative — a bound orbit. Increasing r makes E_total less negative, so energy must be added to raise an orbit.

$v_esc$: minimum speed to escape the gravitational field entirely (m s⁻¹)
$R$: radius from which the object escapes (m)
$E_total$: total mechanical energy of an orbiting mass (J) — negative, indicating a bound orbit
$m$: orbiting mass (kg)
$r$: orbital radius (m)

Coulomb's law & electric field

F = \frac{kq_1q_2}{r^2} \qquad E=\frac{F}{q}=\frac{kQ}{r^2} \qquad E=\frac{V}{d}\ \text{(uniform field)}

Same inverse-square form as gravity but can repel (same-sign charges) or attract (opposite). E = V/d applies only in a uniform field (between parallel plates). Field direction is defined as the force on a positive test charge.

$F$: electrostatic force between the charges (N)
$k$: Coulomb constant = 8.99×10⁹ N m² C⁻²
$q₁, q₂$: the two point charges (C)
$r$: separation between the charges (m)
$E$: electric field strength (N C⁻¹ = V m⁻¹)
$q$: a small positive test charge (C)
$Q$: the source charge creating the field (C)
$V$: potential difference between the plates (V)
$d$: plate separation (m)

Electric potential & potential energy

V = \frac{kQ}{r} \qquad E_p=qV=\frac{kQq}{r}

Unlike gravitational potential (always negative), electric potential can be positive or negative depending on the sign of the source charge Q. Work done moving charge q between points of potential V₁ and V₂ is W = q(V₂ − V₁).

$V$: electric potential (V) — can be positive or negative depending on the sign of Q
$k$: Coulomb constant
$Q$: source charge (C)
$r$: distance from the source charge (m)
$E_p$: electric potential energy of charge q at potential V (J)
$q$: the charge experiencing the potential energy (C)

Magnetic force on a moving charge / current

F = qvB\sin\theta \qquad F=BIL\sin\theta \qquad r=\frac{mv}{qB}

Magnetic force is always perpendicular to velocity — it does no work and cannot change a particle's speed, only its direction. For a particle moving perpendicular to B (θ = 90°), the force is centripetal → circular motion of radius r = mv/qB.

$F$: magnetic force (N)
$q$: charge of the particle (C)
$v$: speed of the charged particle (m s⁻¹)
$B$: magnetic flux density (T)
$θ$: angle between velocity (or current) and B
$I$: current in the wire (A)
$L$: length of wire in the field (m)
$r$: radius of the circular path of a charged particle in a uniform field perpendicular to v (m)
$m$: mass of the particle (kg)

Magnetic field from a current-carrying wire

B = \frac{\mu_0 I}{2\pi r}

Field decreases with distance from the wire (1/r, not 1/r²). Direction given by the right-hand grip rule: thumb along current, fingers curl in direction of field lines. Two parallel wires with currents in the same direction attract each other.

$B$: magnetic flux density at distance r from the wire (T)
$μ₀$: permeability of free space = 4π×10⁻⁷ T m A⁻¹
$I$: current in the wire (A)
$r$: perpendicular distance from the wire (m)

Hall voltage

V_H = \frac{BI}{nqt}

Used to measure magnetic field strength and to determine the sign and density of charge carriers in a semiconductor. A larger Hall voltage results from a stronger B, larger I, or lower carrier density n.

$V_H$: Hall voltage (V)
$B$: magnetic flux density (T)
$I$: current through the conductor (A)
$n$: number density of charge carriers (m⁻³)
$q$: charge of each carrier (C)
$t$: thickness of the conductor in the direction of B (m)

Reference section

E · Nuclear & Quantum

Photon energy

E = hf = \frac{hc}{\lambda}

Photons are discrete packets of electromagnetic energy. A photon of shorter wavelength (higher frequency) carries more energy. Use E = hf when frequency is given, E = hc/λ when wavelength is given. Remember: 1 eV = 1.60×10⁻¹⁹ J.

$E$: energy of a single photon (J or eV)
$h$: Planck constant = 6.63×10⁻³⁴ J s
$f$: photon frequency (Hz)
$c$: speed of light = 3.00×10⁸ m s⁻¹
$λ$: wavelength (m)

Photoelectric equation

hf = \Phi + E_{k,max}

Emission only occurs if hf ≥ Φ (threshold frequency f₀ = Φ/h). Above threshold, increasing intensity increases the number of emitted electrons, not their energy. Increasing frequency increases E_k,max. Stopping voltage V_s = E_k,max/e.

$h$: Planck constant
$f$: frequency of incident light (Hz)
$Φ$: work function of the metal — minimum energy to eject an electron (J or eV)
$E_k,max$: maximum kinetic energy of the ejected (photo)electrons (J or eV)

Nuclear radius & mass–energy equivalence

R=R_0A^{1/3} \qquad E = \Delta mc^2

R ∝ A^(1/3) implies nuclear volume ∝ A, meaning nuclear density is constant regardless of the nucleus. E = Δmc² converts the mass defect to binding energy (use Δm in kg) or energy released in reactions. 1 u = 931.5 MeV/c².

$R$: nuclear radius (m)
$R_0$: constant ≈ 1.2×10⁻¹⁵ m
$A$: mass (nucleon) number
$E$: energy equivalent of a mass defect Δm (J or MeV)
$Δm$: mass defect / mass difference (kg or u)
$c$: speed of light = 3.00×10⁸ m s⁻¹

Radioactive decay law

N = N_0e^{-\lambda t} \qquad A=\lambda N \qquad T_{1/2}=\frac{\ln2}{\lambda}

Decay is random and spontaneous — λ is a probability per nucleus per unit time, not a certainty. For integer numbers of half-lives, use N = N₀(½)ⁿ. For non-integer, use the exponential form. Always ensure λ and t use the same time units.

$N$: number of undecayed nuclei remaining at time t
$N_0$: initial number of undecayed nuclei
$λ$: decay constant (s⁻¹) — probability of decay per nucleus per unit time
$t$: elapsed time (s)
$A$: activity — decays per second (Bq)
$T_1/2$: half-life (s)

Reviewed by the Study to Learn editorial team · Updated 2026-07-24