Equations

An extended equation sheet for AS and A2 revision. Not all equations are provided in the booklet during exams. 

Jump to sections:
[AS Chapters][A2 Chapters] 

 

AS Ch1 Physical Units and Quantities

• All 2D vectors can be broken down into two components, right angles to each other.
• Horizontal vector component, $F_x = F \cos \theta$
• Vertical vector component, $F_y = F \sin \theta$
• To recombine vector components, $F = \sqrt{F_x^2 + F_y^2}$
• Fractional uncertainty = $\frac{\textrm{absolute uncertainty}}{\textrm{measurement}} = \dfrac{\Delta x}{x}$ 
• Percentage uncertainty = $\dfrac{\Delta x}{x} \times 100$ %
• If $p=\dfrac{x^{3} y^\frac{1}{2}}{7z^4}$, then percentage uncertainty p is $\pm \;  (3 \Delta x) + (\frac{1}{2}  \Delta y) + (4 \Delta z)$%
• ... and fractional uncertainty $\frac{\Delta p}{p} = 3\frac{\Delta x}{x} + \frac{1}{2}\frac{\Delta y}{y} +4 \frac{\Delta z}{z}$

AS Ch2 Kinematics

• Displacement, s = area under v-t graph 
• Velocity, $v= \dfrac{\Delta s}{\Delta t}=$ gradient of s-t graph 
• Acceleration, $a=\dfrac{\Delta v}{\Delta t}=$ gradient of v-t graph 
• $v=u+at$
• $s=\frac{1}{2}(u+v)t$ 
• $s=ut+\frac{1}{2} at^2$ 
• $v^2 = u^2 + 2as$ 
• Projectile components if $\theta$ is elevation: $V_x = v \cos\theta$ and $V_y = v \sin\theta$ 
• Projectile max vertical height, $H=\frac{u^2 \sin^2 \theta}{2g}$ 
• Projectile max horizontal range, $R=\frac{u^2 \sin 2\theta}{g}$ 

AS Ch3 Dynamics

• Momentum, $p=mv$
• Force causes acceleration, $\Sigma F= \dfrac{\Delta p}{\Delta t}=ma$ (2nd law)
• Impulse, $F \Delta t = \Delta(mv)$
• Total momentum, $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$
• Elastic collisions $u_1 - u_2 = v_2 - v_1$ (leftward is negative, rightward is positive)
• Weight, $W=mg$

AS Ch4 Forces, Density, Pressure

• Gravitational force, $F_{g}=mg$
• Electrostatic force, $F_{e}=qE$
• Friction, $F_{f}= \mu R$
• Drag or viscous force,  $F_D \propto v^2$
• Moment of force/torque $= F \, \times d_{\perp} = F_{\perp} \times d$
• Density, $\rho = \frac{\textrm{mass}}{\textrm{volume}} = \dfrac{m}{V}$
• Pressure (Pa), $P=\frac{\textrm{Force}}{\textrm{Area}}=\dfrac{F}{A}$
• Pressure in fluids, $\Delta P = \rho g \Delta h$
• Upthrust force, $F_{up}= \rho V g$ 
• Equilibrium in fluid $\rho V g = mg$

 AS Ch5 Work, Energy, Power

• Work Done (J), $W = force \times distance = Fs \cos \theta$
  
• Gravitational Potential Energy $E_p = mgh$
• Kinetic Energy $E_k = \frac{1}{2} mv^2$
• Elastic potential energy $E_p = \frac{1}{2} Fx = \frac{1}{2} kx^2$
• Electric potential energy in uniform electric field, $E_p= qEd$
• Work done expanding gas, $W = P \Delta V$
• Efficiency (%), $\eta = \frac{\textrm{power or energy output}}{\textrm{power or energy input}} \times 100$
• Power (W), $P =\frac{\textrm{work done or energy change}}{\textrm{time taken}}= \dfrac{W}{t} = Fv$

AS Ch6 Deformation of Solids

• Restoring force (Hooke's Law), $F=k \; \Delta x$
• Parallel springs effective constant $k = k_1 + k_2 + ...$
• Series springs effective constant $\frac{1}{k} = \frac{1}{k_1} + \frac{1}{k_2} + ...$
• Elastic potential or strain energy $E_s = \frac{1}{2} Fx = \frac{1}{2} kx^2$
• Work done on spring $W = \Delta E_s =$ area under F-x graph
• Stress (Pa), $\sigma = \frac{\textrm{force applied}}{\textrm{cross-section area}} = \dfrac{F}{A}$
• Strain, $\epsilon = \frac{\textrm{extension}}{\textrm{original length}} = \dfrac{x}{l_0}$
• Young's modulus (Pa), $E = \frac{\textrm{stress}}{\textrm{strain}} = \dfrac{\sigma}{\epsilon} = \dfrac{Fl_0}{Ax}$ = Gradient of linear stress-strain graph
• Strain energy per unit volume = Area under stress-strain graph

AS Ch7 Waves

• Frequency (Hz), $f=\frac{\textrm{one cycle}}{\textrm{period}} = \dfrac{1}{T}$
• Wave speed, $v=f\lambda$
• Phase difference ratios $\dfrac{\Delta t}{T} = \dfrac{\Delta x}{\lambda} = \dfrac{\Delta \theta}{2\pi}$
  
• Wave intensity, $I \propto A^2$ and $I \propto f^2$ and $I \propto \frac{1}{r^2}$
• Doppler effect ratios, $\dfrac{f_0}{f_s} = \dfrac{v}{v \pm v_s}$ (positive away, negative towards)
• Malus' Law of Polarized intensity, $I = I_o \cos^2 \theta$ [2022 syllabus]

AS Ch8 Superposition of Waves

• Path difference, $\Delta L = L_2 - L_1 = a \sin \theta$
• For constructive interference $\Delta L = 0,\; 1\lambda, \; 2\lambda,\; 3\lambda,...  n\lambda$
• For destructive interference $\Delta L = 0.5\lambda,\; 1.5\lambda,\; 2.5\lambda,...  (n+\frac{1}{2})\lambda$
• Young's double-slit fringe separation, $x=\dfrac{\lambda D}{a}$
• Multi-slit diffraction grating, $n \lambda = d \sin \theta$ (observing maxima)
• Grating slit separation, $d = \dfrac{1}{N}$ (e.g. N = 30000 lines/metre)
• To find greatest number of maxima n, use $n \lambda < d \sin 90$

AS Ch9 Current of Electricity

• Total charge (C) of free electrons = $Q = nALe$ 
• Current (A), $I = \frac{total \; charge \;}{time \; taken}\ = \dfrac{Q}{t} = \dfrac{naLq}{t} = nAvq$
• Potential difference aka voltage (V), $V = \frac{\textrm{energy}}{\textrm{charge}} = \dfrac{W}{Q}$
• Resistance ($\Omega$), $R=\dfrac{V}{I}=$ constant (if obeys Ohms's Law)
• Resistance in wire ($\Omega$ m), $R=\dfrac{\rho L}{A}$
• Power (W), $P=IV = I^2R = \dfrac{V^2}{R}$
• Max Power Dissipated, $P_{max} = \dfrac{E^2 R}{(R+r)^2}$
  

AS Ch10 DC Circuits

• e.m.f (V) from battery, $E =  IR + Ir =$ terminal p.d. + lost volts
• Combined Series Resistance, $R = R_1 + R_2 + R_3 + ...$
• Combined Parallel Resistance, $R = \left( \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3} + ...\right)^{-1}$
• Kirchoff I: At a junction, total current in = total current out
• Kirchoff II: For loop in circuit, $\sum V = 0$ or  sum of e.m.f = sum of potential drops
  
• Potential divider (ratio), $\dfrac{V_1}{V_2} = \dfrac{R_1}{R_2}$ or $\dfrac{V_1}{V_{cell}} = \dfrac{R_1}{R_{total}}$
• Potentiometre output p.d. (ratio), $\frac{V_o}{V_s} = \left(\dfrac{x}{l} \right)$
• Potentiometre to find emf (ratio), $\frac{E}{V_0}= \left(\dfrac{x}{l} \right)$
• Strain gauge ratio if no change in area $\dfrac{\Delta R}{R} = \dfrac{\Delta L}{L}$, but if changing area but constant volume then $\dfrac{\Delta R}{R} = \dfrac{2 \Delta L}{L}$

AS Ch11 Particle and Nuclear Physics

• Alpha decay ${}_{10}^{100}\text{X} \longrightarrow {}_{8}^{96}\text{Y} + {}_{2}^{4} \alpha$
• Beta-minus decay ${}_{10}^{100}\text{X} \longrightarrow {}_{11}^{100}\text{Y} + {}_{-1}^{0} \beta + \overline{\nu}_e$ 
• Beta-plus decay ${}_{10}^{100}\text{X} \longrightarrow {}_{9}^{100}\text{Y} + {}_{1}^{0} \beta + \nu_e$  
• Gamma decay ${}_{10}^{100}\text{X} \longrightarrow {}_{10}^{100}\text{Y} + \gamma$

Jump to sections:
[AS Chapters][A2 Chapters] 

  = = = = = = = = = = = = = = = = = = =

A2 Practical Paper 5

• General uncertainty, $\Delta x = \dfrac{1}{2} (x_{max} - x_{min})$
• Absolute unc. gradient, $\Delta m = | m_{best} - m_{worst} |$
• Absolute unc. y-intercept $\Delta C = | C_{best} - C_{worst} |$
• General log uncertainty, $\Delta log_a (x) = \dfrac{1}{ln(a)} \left(\dfrac{\Delta x}{x} \right)$
  
• Absolute uncertainty of lg, $\Delta lg (x) = \dfrac{1}{ln(10)} \left(\dfrac{\Delta x}{x} \right)$
• Absolute uncertainty of ln, $\Delta ln (x) = \dfrac{\Delta x}{x}$
  

A2 Ch12 Circular Motion

• Arc length, $s = r \theta$ (definition of radian)
  
• Angular velocity, $\omega = \dfrac{\Delta \theta}{\Delta t} = \dfrac{2 \pi}{t} =2 \pi f$
• Average speed or linear/tangential velocity, $v = r \omega$
• Period of revolution, $T =  \dfrac{2 \pi}{\omega} = \dfrac{2 \pi r}{v}$
  
• Centripetal acceleration, $a = vw = r \omega^2 = \dfrac{v^2}{r}$
• Centripetal force, $F_c = ma = mr \omega^2 = \dfrac{mv^2}{r}$
  

 A2 Ch13 Gravitation / Gravitational Fields (G-field)

• Gravitational force of attraction, $F = \dfrac{GMm}{r^2}$
• G-field strength, $g = \dfrac{GM}{r^2}$
• Gravitational potential, $\phi = - \dfrac{GM}{r}$ 
• G-potential energy, $E_g = m \phi = -\dfrac{G Mm}{r}$
  
• Orbital velocity, $v = \sqrt{ \frac{GM}{r} }$
  
• Orbital period, $T = \dfrac{2 \pi r}{v} = 2 \pi \sqrt{\frac{r^3}{GM}}$
• Escape velocity, $v_{escape} = \sqrt{\frac{2GM}{R}}$

A2 Ch14 Temperature & Thermal Properties

• Conversion: T/K = T/°C + 273.15 
• Heat energy to raise liquid temperature, $Q = mc \Delta \theta + Q_{lost}$
• Energy supplied by heater, $Q = VIt$
  
• Internal energy, $U = \text{sum of random distribution of K.E. and P.E of molecules}$
  
• Change in internal energy, $\Delta U = q + W$
  where +q is heat supplied to system, +W is work done on system (V decrease).
• Internal energy of ideal gas, $U = E_k = \frac{3}{2} kT$ (no potential energy)

 A2 Ch15 Ideal Gases

• State for ideal gas $PV = nRT = NkT$
• Amount of gas (mol), $n = \frac{\text{Number of Particles}, N}{\text{Avogadro Constant}, N_{A}}$ 
• Boltzmann constant, $k = \frac{\text{Molar gas constant}, R}{\text{Avogadro constant}, N_A}$
  
• Boyle's Law $P \propto \dfrac{1}{V}$ (constant T)
  
• Charles' Law $V \propto T$ (constant P)
  
• Pressure Law $P \propto T$ (constant V)
  
• Combined laws $\frac{PV}{T} = constant$

 A2 Ch16 Thermodynamics

• Pressure of gas $P = \dfrac{1}{3} \left(\dfrac{Nm}{V}\right) \langle c^2 \rangle = \dfrac{1}{3} \rho \langle c^2 \rangle$
  NOTE: m is mass of a single particle!
• Kinetic energy of gas molecule $\langle E_k \rangle = \frac{1}{2} m \langle c^2 \rangle = \frac{3}{2} kT$
• Internal energy of ideal gas, $U = E_k = \frac{3}{2} kT$ (no potential energy)

 A2 Ch17 Oscillations / Simple Harmonic Motion (SHM)

• Angular frequency, $\omega = \dfrac{2 \pi}{T} = 2 \pi f$
• Period of a spring, $T = 2\pi \sqrt{\frac{m}{k} }$
• Period of a pendulum, $T = 2 \pi \sqrt{\frac{l}{g} }$
  
• SHM acceleration, $a = -\omega^2 x$
• Max acc. at max displacement (x=A),  $a = -\omega^2 A$
• SHM velocity, $v = \pm \omega \sqrt{A^2 - x^2}$ 
• Max speed at equilibrium (x=0), $v_{max} = \omega A$
  
• E.g. if displacement is $x = A \sin \omega t$ then velocity is $v = \omega A \cos \omega t$
• Total energy $E_T = \frac{1}{2} mv^2 + \frac{1}{2} kx^2$
• Trigonometric identity $\sin^2 \omega t + \cos^2 \omega t = 1$

A2 Ch18 Electric Fields (E-field)

• Constant $k = \dfrac{1}{4 \pi \epsilon_o}$ 
• Electric potential (J/C), $V = \dfrac{kQ}{r}$  
• E-field strength, $E = - \dfrac{dV}{dr} = \dfrac{kQ}{r^2}$
• E-field parallel plates, $E=\dfrac{F}{Q} = \dfrac{V}{d}$
• Electric force $F_e = qE = \dfrac{kQq}{r^2}$ 
• E-potential energy, $U = -\int{F_{e} \; dr} = \dfrac{kQq}{r} = qV$ 
• Work done (J) on charge, $W = Fd = \Delta U = q \Delta V$
  
• Uniform E-field strength, $E = - \text{potential gradient} = - \dfrac{dV}{dr} =  - \dfrac{\Delta V}{\Delta x}$

A2 Ch19 Capacitance

• Capacitance (F), $C = \dfrac{Q}{V}$
• Combined Parallel Capacitance, $C = C_1 + C_2 + C_3 + ...$
• Combined Series Capacitance, $C = \left( \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dfrac{1}{C_3} + ...\right)^{-1}$
• Energy stored in capacitors, $E = \dfrac{1}{2} QV = \dfrac{1}{2} CV^2$ 
• Current (A), $I = \frac{total \; charge \;}{time \; taken}\ = \dfrac{Q}{t}$ 
  
• Discharging capacitor charge, $Q = Q_o e^{-\frac{t}{RC}}$
• Charging capacitor charge, $Q = Q_o (1 - e^{-\frac{t}{RC}} )$

A2 Ch20 Magnetic Fields & Induction

• Field from a straight wire, $B = \dfrac{\mu_o I}{2 \pi r}$ 
• Field inside solenoid, $B = \dfrac{\mu_o N I}{L}$
• Force on wire in field, $F = BIL \sin \theta$
• Force per unit length between two wires, $\dfrac{F}{L} = \dfrac{\mu_o I_1 I_2}{2 \pi x}$
  
• Force on charged particle, $F = Bqv \sin \theta$
• Hall voltage, $V_H = \dfrac{BId}{nAq} = \dfrac{BI}{ntq}$
• Current (A), $I = \frac{total \; charge \;}{time \; taken}\ = \dfrac{Q}{t} = nAvq$ 
• Magnetic flux (Wb), $\Phi = BA$
  
• Induced e.m.f, $E = Blv = B\dfrac{dA}{dt}$
• Faraday's Law $E = - \dfrac{d \Phi}{dt} = -\dfrac{\Delta N \Phi}{\Delta t}$
  

A2 Ch21 Alternating Currents (AC)

• Root-mean-square current, $I_{rms} = \frac{I_o}{\sqrt{2}}$
  
• Root-mean-square voltage , $V_{rms} = \frac{V_o}{\sqrt{2}}$
• Max power, $P_{max} = I_o V_o$
• Average power, $P_{avg} = I_{rms} V_{rms} = \frac{1}{2} I_o V_o$
• Oscillating current. $I = I_o \sin (2 \pi f t) = I_o \sin \omega t$
• Oscillating voltage, $V = V_o \sin (2 \pi f t) =  V_o \sin \omega t$
• Oscillating Power, $P = IV = \dfrac{V_o I_o}{2} [1 - \cos(4 \pi f t)]$
  
• Ideal transformer $\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p} = \dfrac{I_p}{I_s}$

A2 Ch22 Quantum Physics

• Energy of a photon, $E = hf = \frac{hc}{\lambda}$
  
• Max k.e. of photoelectrons $E_{K(max)} = hf - \phi$ 
• Radiation intensity $= \dfrac{hfN}{At}$ so $I \propto f_{photons} N_{photons}$
  
• Work function energy, $\phi = hf_o = \frac{hc}{\lambda_o}$
  
• Threshold frequency, $f_o = \frac{\phi}{h}$
  
• Threshold wavelength, $\lambda_o = \frac{c}{f_o}$
• De Broglie wavelength, $\lambda = \frac{h}{p} = \frac{h}{mv}$
• Emitted photon energy, $\Delta E = E_1 - E_2 = hf$

 A2 Ch23 Nuclear Physics

• Energy released. $\Delta E = (\Delta m) c^2$
• Mass excess = mass (in u) - nucleon number
• Number of nuclei, $N = N_o e^{\lambda t}$
• Radioactivity, $A = A_o e^{-\lambda t} = \frac{dN}{dt} = - \lambda N$
• Half-life duration, $t_{1/2} = \dfrac{\ln 2}{\lambda} =  \dfrac{0.693}{\lambda}$
• Mass, $m = \text{moles} \times \text{molecular weight}$
• Number of particles, $N = \text{moles} \times N_A$  

A2 Ch24 Medical: Ultrasound

• Acoustic impedance, $Z = \rho c$
• Intensity reflection coefficient $\dfrac{I_R}{I_O} = \dfrac{(Z_1 - Z_2)^2}{(Z_1+Z_2)^2}$
• Wave intensity in material, $I = I_0 e^{-\mu x}$
• Decibel (dB) = $10 \log_{10} \left( \dfrac{I_2}{I_1 } \right)$

A2 Ch24 Medical: X-Ray

• X-ray photon max frequency $hf_{max} = eV$
• X-ray attenuation $I = I_o e^{-\mu x}$
• Decibel (dB) = $10 \log_{10} \left( \dfrac{I_2}{I_1 } \right)$

A2 Ch25 Astronomy & Cosmology

• Radiant flux intensity, $F = \dfrac{L}{4 \pi d^2}$
• Wien's blackbody radiation, $\lambda_{peak} \propto \dfrac{1}{T}$
• Stefan-Boltzmann's stellar radii, $L = 4 \pi \sigma r^2 T^4$
• Doppler Shift $\dfrac{\lambda_{star}}{\lambda_{lab}} = \dfrac{v}{c}$
• Hubble's Law $v = H_0 d$

 Jump to sections:
[AS Chapters][A2 Chapters]

 =================

Discontinued Chapters:

A2 Communications

• Bandwidth $= (f_{c} + f_{s}) - (f_{c} - f_{s}) = 2 f_s$
  NOTE: Maximum frequency of signal = $f_s$
• Bitrate = sample rate × bit depth
• Attenuation (dB) = $10 \log_{10} \left( \dfrac{P_1}{P_2} \right)$
• Gain (dB) = $10 \log_{10} \left( \dfrac{P_{out}}{P_{in} } \right)$

A2 Electronics (Op-Amp)

• Gain, $G = \dfrac{V_{out}}{V_{in}}$
  
• Output voltage, $V_{out} = G (V_{+} - V_{-})$
• Voltage gain (inverting), $\dfrac{V_{out}}{V_{in}} = -\dfrac{R_F}{R_{in} }$
• Voltage gain (non-inverting), $\dfrac{V_{out}}{V_{in}} = 1 + \dfrac{R_f}{R_{in} }$