Equations

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

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[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)
• 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, \(F_{e}=qE\)
• Drag or viscous force,  \(D \propto v^2\)
• 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, \(F_{up}= \rho V g\) 

 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 \(E_p = \frac{1}{2} Fx = \frac{1}{2} kx^2\)
• 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 \(\dfrac{\Delta t}{T} = \dfrac{\Delta x}{\lambda} = \dfrac{\Delta \phi}{2\pi} \)
  
• Wave intensity, \( I \propto \textrm{Amplitude}^2 \)
• Doppler effect ratios, \( \dfrac{f_0}{f_s} = \dfrac{v}{v \pm v_s}   \) (positive away, negative towards)
• Malus' Law of Polarized intensity, \(I = I_o \cos^2 \theta\)

AS Ch8 Superposition of Waves

• Path difference, \(\Delta L = L_2 - L_1 \)
• 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 in conductor (C) = \(Q = nALq \) 
• 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 ratio of segments, \( \dfrac{V_1}{V_2} = \left(\dfrac{x_1}{x_2} \right)  \)
• Potentiometre ratio to total, \( \dfrac{V}{V_\text{total}} = \left(\dfrac{x}{L} \right)  \)

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, \(a_c = \dfrac{v^2}{r} = r \omega^2  \)
• Centripetal force, \(F_c = ma_c = mr \omega^2 = \dfrac{mv^2}{r} \)
  

 A2 Ch13 Gravitation / Gravitational Fields (G-field)

• Gravitational force of attraction, \( F = \dfrac{GMm}{r^2} \)
• G-field strength, \(g = \dfrac{F}{g} = \dfrac{GM}{r^2} \)
• Gravitational potential, \( \phi = - \dfrac{GM}{r} \) 
• G-potential energy, \(E_p = m \phi = -\dfrac{G Mm}{r} \)
  
• Resultant potential, \(\phi = \phi_1 + \phi_2 \)
• Resultant field strength, \(g = g_1 + g_2  \)
• Orbital velocity, \( v = \sqrt{ \frac{GM}{r} } \) (derive from \(F_c = F_g\) )
• Escape velocity, \(v = \sqrt{\frac{2GM}{R}} \) (derive from ΔKE = ΔGPE)

A2 Ch14 Temperature & Thermal Properties

• Conversion: T/K = T/°C + 273.15 
• Thermal energy to raise temperature, \(Q = mc \Delta \theta + Q_{lost}\)
• Energy supplied by heater, \(Q = VIt \)
  
• Thermal energy to change state, \(Q = mL\)

 A2 Ch15 Ideal Gases & Kinetic Theory

• Ideal gas law \(PV = nRT = NkT\)
• Amount of gas (mol), \(n = \frac{\text{Number of Particles}, N}{\text{Avogadro Constant}, N_{A}} \) 
• Boltzmann constant, \(k = \frac{\text{Molar gas constant}, R}{\text{Avogadro constant}, N_A}\)
• 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!
• Root-mean-square speed, \(c_{\text{rms}} = \sqrt{\langle c^2 \rangle} \)
• Kinetic energy of molecule \(\langle E_k \rangle = \frac{1}{2} m \langle c^2 \rangle = \frac{3}{2} kT \)
• Kinetic energy of gas, \(\langle E_k \rangle =\frac{3}{2} NkT \)

 A2 Ch16 Thermodynamics

• Internal energy, \(U = \text{K.E.} + \text{P.E}\)
• Change in internal energy, \(\Delta U = q + W \)
  where +q is heat supplied to system, +W is work done on system (V decrease).
• Work done expanding gas, \( W = P \Delta V \)
• Internal energy link to temperature, \(\Delta U = \Delta E_k = \frac{3}{2} Nk\Delta T \) (assuming no potential energy)

 A2 Ch17 Oscillations / Simple Harmonic Motion (SHM)

• SHM acceleration, \(a = -\omega^2 x \)
• Max acc. at max displacement,  \(a_{\text{max}} = -\omega^2 A \)
• Angular frequency, \( \omega = \dfrac{2 \pi}{T} = 2 \pi f \)
• Displacement, \(x = A \sin \omega t \)
• Velocity (in time), \(v = v_0 \cos \omega t \)
• Velocity (at position), \(v = \pm \omega \sqrt{A^2 - x^2} \) 
• Max speed at equilibrium (x=0), \(v_{\text{max}} = \omega A\)
• Total energy \(E_T = \text{KE}_{\text{max}} = \dfrac{1}{2} m  v_{max}^2 = \dfrac{1}{2} m  \omega^2 A^2 \)

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 \(F_e = qE = \dfrac{kQq}{r^2} \) Coulomb's Law)
• 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 charge, \(Q = Q_o e^{-\frac{t}{RC}}  \)
• Discharging current, \(I = I_o e^{-\frac{t}{RC}}  \)
• Discharging voltage, \(V = V_o e^{-\frac{t}{RC}}  \)
• Time constant, \( \tau = RC \)

A2 Ch20 Magnetic Fields & Induction

• Field from a straight wire, \(B \propto \dfrac{I}{r} \) 
• Force on wire in field, \(F = BIL \sin \theta\)
• Force per unit length between two wires, \(\dfrac{F}{L} \propto \dfrac{I_1 I_2}{x} \)
  
• Force on charged particle, \(F = Bqv \sin \theta \)
• Hall voltage, \(V_H = \dfrac{BI}{ntq} \)
• Current (A), \(I = \frac{total \; charge \;}{time \; taken}\ = \dfrac{Q}{t} = nAvq \) 
• Magnetic flux, \( \phi = BA\)
  
• Magnetic flux Linkage, \( \Phi = N \phi = NBA\)
• Induced emf, \(E = \dfrac{d\Phi}{d t} = \dfrac{\Delta N \phi}{\Delta t} \) (Faraday's Law)

A2 Ch21 Alternating Currents (AC)

• Oscillating current. \(I = I_o \sin \omega t \)
• Oscillating voltage, \(V =  V_o \sin \omega t \)
• Root-mean-square current, \(I_{rms} = \frac{I_o}{\sqrt{2}} \) (sinusoidal only)
  
• Root-mean-square voltage , \(V_{rms} = \frac{V_o}{\sqrt{2}} \) (sinusoidal only)
• Oscillating Power, \(P = I^2 R =  P_o \sin^2 \omega t \)
• Max power, \(P_o = I_o^2 R \)
• Mean power, \(\langle P \rangle = I_{rms}^2 R \) 
• Mean value of non-sinusoidal graph = \( \frac{\text{Area under graph}}{\text{total time}} \)

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, \(\phi = hf_o = \dfrac{hc}{\lambda_o} \)
• De Broglie wavelength, \(\lambda = \dfrac{h}{p} \)
• Momentum of particle, \(p=mv \)
• Momentum of photon, \(p =  \dfrac{E}{c} \)
• Diffraction grating exp, \(n \lambda = d \sin \theta \)
• Emitted photon energy, \(\Delta E_{\text{level}} = E_1 - E_2 = hf \)

 A2 Ch23 Nuclear Physics

• Energy released. \( \Delta E = (\Delta m) c^2 \)
• Mass excess = mass (in u) - nucleon number
• Number of nuclei, \(N = N_o e^{\lambda t} \)
• Activity, \(A = A_o e^{-\lambda t} = \frac{dN}{dt} = - \lambda N \)
• Decay constant, \( \lambda = \dfrac{\ln 2}{t_{\frac{1}{2}}} =  \dfrac{0.693}{t_{\frac{1}{2}}} \)
• 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, \(I = I_0 e^{-\mu x} \)

A2 Ch24 Medical: X-Ray

• X-ray photon energy from accelerating pd of electron: \(hf_{max} = eV \)
• Attenuation of intensity \(I = I_o e^{-\mu x} \)
• Decibel (dB) = \(10 \log_{10} \left( \dfrac{I_2}{I_1 } \right) \)

A2 Ch24 Medical: PET Scan

• Total energy of gamma photons from electron-positron annihilation \( E = 2 (m_e c^2 ) \)

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 \( \dfrac{\Delta \lambda}{\lambda_o} = \dfrac{v}{c} \)
• 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), \( \dfrac{V_{out}}{V_{in}} = 1 + \dfrac{R_f}{R_{in} }\)