Describing motion

Displacement, velocity and acceleration are vectors - direction matters, so always define a positive direction first.

• Speed is scalar (distance/time); velocity is vector (displacement/time).
• Acceleration is the rate of change of velocity, measured in m/s^2.
• On a displacement-time graph, gradient = velocity. On a velocity-time graph, gradient = acceleration and area under the graph = displacement.

The SUVAT equations

These only apply when acceleration is constant.

• v = u + at
• s = ut + 0.5at^2
• v^2 = u^2 + 2as
• s = 0.5(u+v)t

Here s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time. Always write down what each symbol equals before substituting numbers, and watch your signs - a deceleration is a negative acceleration in your chosen direction.

Projectile motion

Split motion into horizontal and vertical components - they are independent of each other.

• Horizontal velocity is constant (no horizontal force if air resistance is ignored).
• Vertical motion has constant acceleration g, taken as 9.81 m/s^2 on Earth.
• Time of flight is found from the vertical motion only, then used to find horizontal distance (range = horizontal velocity x time).
• For a projectile launched at an angle, resolve the initial velocity into horizontal (u cos theta) and vertical (u sin theta) components first.

Forces and Newton's laws

• Newton's First Law: an object stays at rest or constant velocity unless acted on by a resultant force.
• Newton's Second Law: F = ma (resultant force equals mass times acceleration); more generally F = rate of change of momentum.
• Newton's Third Law: forces come in equal and opposite pairs acting on DIFFERENT objects, of the same type.
• Weight = mg, always acts vertically downward, and is distinct from mass (a scalar measured in kg that doesn't change with location).

Momentum and impulse

• Momentum p = mv, measured in kg m/s.
• Impulse = change in momentum = F x t, also equal to the area under a force-time graph.
• In a closed system, total momentum is conserved in all collisions (elastic and inelastic).
• Elastic collisions conserve kinetic energy too; inelastic collisions do not (some energy converts to heat/sound/deformation).

Common mistakes to avoid

• Mixing up scalar speed with vector velocity in exam answers - state direction where relevant.
• Forgetting that suvat equations need constant acceleration - not valid for terminal velocity or projectile problems including air resistance.
• Using g = 9.81 inconsistently - check what your exam paper specifies (usually 9.81 m/s^2 for OCR).
• Forgetting Newton's third law pairs must be the SAME TYPE of force and act on two different bodies, never the same one.

Density and Hooke's Law

• Density = mass / volume, units kg m^-3.
• Hooke's law: F = kx, where k is the spring constant in N m^-1, valid only up to the limit of proportionality.
• Beyond the elastic limit the material no longer returns to its original shape when unloaded.

Stress, Strain and the Young Modulus

• Tensile stress = F / A (Pa), tensile strain = extension / original length (no units).
• Young modulus E = stress / strain (Pa), found from the gradient of the straight-line region of a stress-strain graph.
• Common mistake: using the extended length instead of the ORIGINAL length in the strain calculation.
• Elastic strain energy stored in a stretched wire or spring = area under the force-extension graph = 0.5 F x (only valid up to the limit of proportionality, since this is 0.5 x F x for a straight line).

Stress-Strain Graph Features

• Elastic region: obeys Hooke's law, straight line through the origin.
• Yield point: material suddenly extends with little extra load, marks start of plastic deformation.
• Ultimate tensile stress (UTS): the maximum stress the material can withstand.
• Breaking stress: the stress at which the material fractures.
• Brittle materials (e.g. glass) snap with almost no plastic deformation; ductile materials (e.g. copper) show a long plastic region before breaking.

Internal Energy, Specific Heat Capacity and Latent Heat

• Internal energy = sum of the random kinetic and potential energies of all particles in a system.
• Absolute zero (0 K = -273.15 degrees C) is the temperature at which particles have minimum kinetic energy.
• Q = mcDeltaT for a temperature change, where c is specific heat capacity in J kg^-1 K^-1.
• Q = mL for a change of state, where L is specific latent heat in J kg^-1; NO temperature change occurs during a phase change, so all the energy changes potential energy between particles.
• Common mistake: forgetting that specific latent heat of vaporisation is always LARGER than specific latent heat of fusion for the same substance, because more bonds must be fully broken.

Gas Laws and Molecular Kinetic Theory

• Ideal gas equation: pV = nRT, with R = 8.31 J mol^-1 K^-1 and T in kelvin, always.
• Also pV = NkT, where k is the Boltzmann constant, 1.38 x 10^-23 J K^-1, and N is the number of molecules.
• Kinetic theory assumptions: molecules are point-like, in random motion, collisions are perfectly elastic, negligible time in collisions, negligible forces between molecules except during collision.
• Mean kinetic energy of a molecule = (3/2)kT, showing kinetic energy depends only on absolute temperature, not on the type of gas.
• pV = (1/3)Nm(c-squared mean), the kinetic theory equation, where c-squared mean is the mean square speed.

Current, charge and potential difference

Current I is the rate of flow of charge: I = Q / t, measured in amperes (A), where 1 A = 1 coulomb per second.

Potential difference (p.d.) is the energy transferred per unit charge: V = W / Q, measured in volts (V), where 1 V = 1 joule per coulomb.

Electromotive force (e.m.f.) is the energy given to each coulomb of charge by a source (like a cell), also measured in volts. E.m.f. and p.d. are easily confused: e.m.f. is the total energy supplied per coulomb; terminal p.d. is what is left after energy is lost to internal resistance.

Resistance and Ohm's law

Resistance R = V / I, measured in ohms (ohm symbol). Ohm's law states current is directly proportional to p.d. for a component at constant temperature, giving a straight line I-V graph through the origin.

Resistance depends on resistivity: R = (rho L) / A, where rho is resistivity (ohm metres), L is length and A is cross-sectional area. Common mistake: forgetting area must be in square metres, not mm squared, in calculations.

• Filament lamp: I-V curve bends over as it heats up (resistance increases with temperature).
• Diode: almost zero current below the threshold voltage (about 0.6 V for silicon), then rises steeply.
• Thermistor (NTC): resistance falls as temperature rises.
• LDR: resistance falls as light intensity rises.

Circuit rules

Kirchhoff's first law: current into a junction equals current out (conservation of charge).

Kirchhoff's second law: the sum of e.m.f.s around a closed loop equals the sum of p.d.s (conservation of energy).

Series circuits: same current everywhere; resistances add directly, R_total = R1 + R2 + ...; p.d.s share in proportion to resistance.

Parallel circuits: same p.d. across each branch; 1/R_total = 1/R1 + 1/R2 + ...; current splits, with more current through the smaller resistance. A very common mistake is adding parallel resistances directly instead of using the reciprocal formula.

Power and internal resistance

Electrical power: P = VI = I squared R = V squared / R.

Energy transferred: E = VIt.

For a real cell, e.m.f. E = I(R + r), where r is internal resistance. Terminal p.d. V = E - Ir, so terminal p.d. drops as current drawn increases. Maximum power is transferred to the external circuit when external resistance equals internal resistance.

Potential dividers

A potential divider splits e.m.f. between two resistors in series: Vout = V_in x (R2 / (R1 + R2)). Used with thermistors or LDRs to create sensor circuits producing a variable output voltage. Common mistake: mixing up which resistor's voltage is being asked for - always check which resistor Vout is measured across.

Wave basics

A wave transfers energy without transferring matter. Transverse waves (light, all EM waves, water surface waves, s-waves) oscillate perpendicular to the direction of travel and can be polarised. Longitudinal waves (sound, p-waves) oscillate parallel to the direction of travel and cannot be polarised.

Key equation: wave speed v = f x lambda, where f is frequency in Hz and lambda is wavelength in m. Also v = distance / time for a wavefront, and T = 1/f for the period.

Polarisation

Only transverse waves polarise. A polarising filter only lets through the component of oscillation aligned with its transmission axis. Two filters at 90 degrees to each other (crossed polarisers) block all light. Malus's law: I = I0 cos^2(theta), where theta is the angle between the light's polarisation and the filter axis. Polaroid sunglasses cut glare because reflected light is partially polarised horizontally.

Superposition and standing waves

When two waves meet, their displacements add (principle of superposition). Coherent sources (same frequency, constant phase difference) produce a stable interference pattern: constructive interference where path difference = n x lambda (whole number of wavelengths), destructive where path difference = (n + 1/2) x lambda.

Standing (stationary) waves form when two waves of equal frequency and amplitude travel in opposite directions and superpose, e.g. a wave and its reflection. Nodes are points of zero amplitude (always zero displacement); antinodes are points of maximum amplitude. Adjacent nodes are lambda/2 apart. A stretched string fixed at both ends: the fundamental (first harmonic) has length L = lambda/2. Unlike progressive waves, points between adjacent nodes are in phase; points either side of a node are in antiphase.

Diffraction and interference experiments

Diffraction is the spreading of waves through a gap or around an obstacle; it's most noticeable when the gap width is similar to the wavelength.

Double-slit equation: lambda = ax/D, where a is slit separation, x is fringe spacing, D is slit-to-screen distance (D must be much greater than a).

Diffraction grating equation: d sin(theta) = n x lambda, where d = 1/N is the slit spacing (N = lines per metre), n is the order (integer), theta is the angle to the normal. Maximum order occurs when sin(theta) cannot exceed 1.

Refraction

Refractive index n = c / v = sin(theta1) / sin(theta2) (Snell's law). Total internal reflection happens only when light travels from a denser to a less dense medium and the angle of incidence exceeds the critical angle C, where sin(C) = 1/n. Optical fibres use TIR (with cladding of lower refractive index) to carry signals; modal dispersion limits bandwidth over long distances.

Common mistakes

• Forgetting D >> a is required for the double-slit formula to be valid.
• Mixing up node/antinode spacing (lambda/2, not lambda).
• Applying Malus's law with angle in degrees but forgetting to convert if using a calculator in radians.
• Confusing constructive interference condition (n x lambda) with destructive (n + 1/2) x lambda).

Gravitational fields

Mass creates a gravitational field. Field strength g = F/m (N/kg). For a point mass, g = GM/r^2, where G = 6.67 x 10^-11 N m^2 kg^-2. Field lines point radially inward towards the mass, showing the field is always attractive.

Gravitational potential V at a point is the work done per unit mass bringing a small test mass from infinity to that point: V = -GM/r. It is always negative, and zero at infinity (the natural reference point). Potential difference gives escape energy: energy needed to escape = mass x change in V.

Orbits: for a circular orbit, gravitational force provides centripetal force, so GMm/r^2 = mv^2/r = m(4 pi^2/T^2)r. This gives T^2 proportional to r^3 - Kepler's third law. Geostationary orbits have T = 24 hours (23 h 56 min sidereal, but OCR accepts 24 h), sit above the equator, and orbit west to east.

Electric fields

Electric field strength E = F/Q (N/C or V/m). For a point charge, E = Q/(4 pi epsilon_0 r^2), where epsilon_0 = 8.85 x 10^-12 F/m. Field lines point away from positive charges, towards negative ones - opposite to gravity's always-attractive rule, since like charges repel.

Electric potential V = Q/(4 pi epsilon_0 r). Potential difference between two points equals work done per unit charge. Uniform fields (parallel plates) have E = V/d, constant field strength, and straight parallel field lines.

Coulomb's law: F = Qq/(4 pi epsilon_0 r^2). Compare directly with Newton's law of gravitation - both are inverse square laws, both compare via F, E/g, and V, but gravity is always attractive while electric force can attract or repel.

Magnetic fields

A moving charge or current in a magnetic field feels a force. For a wire: F = BIL sin(theta), where B is magnetic flux density (tesla, T), I is current, L is length, theta is the angle between wire and field. Maximum force when the wire is perpendicular to B (theta = 90 degrees), zero when parallel.

For a moving charge: F = BQv sin(theta). Use Fleming's left-hand rule for force direction on a current or positive charge (thuMb = motion/force, First finger = field, seCond finger = current).

A charged particle moving perpendicular to B undergoes circular motion because the force is always perpendicular to velocity: BQv = mv^2/r, so r = mv/(BQ).

Common mistakes

• Mixing up g/V (gravity, always negative potential) with E/V (electric, sign depends on charge).
• Forgetting sin(theta) in BIL and BQv when the field isn't perpendicular.
• Using the wrong hand rule (right hand is for the motor/generator only in some textbooks - OCR uses Fleming's left-hand for force).
• Forgetting potential is a scalar (add directly) but field strength is a vector.

Atomic structure and scattering

The nuclear model comes from the Geiger-Marsden alpha scattering experiment: most alpha particles passed straight through gold foil, but a small fraction bounced back at large angles.

• This proved the atom is mostly empty space with a tiny, dense, positively charged nucleus (radius around 1 x 10^-15 m, versus atom radius around 1 x 10^-10 m).
• Closest approach calculations use energy conservation: kinetic energy of the alpha equals its electric potential energy at closest approach, giving an estimate of nuclear radius.
• Nuclear radius follows R = r0 x A^(1/3), where r0 is about 1.2 x 10^-15 m and A is the nucleon number. This shows nuclear density is roughly constant for all nuclei.

Radioactive decay

Decay is random and spontaneous, unaffected by temperature, pressure or chemical state.

• Activity A = lambda x N, where lambda is the decay constant (per second) and N is the number of undecayed nuclei.
• N = N0 x e^(-lambda t) and A = A0 x e^(-lambda t) are exponential decay equations.
• Half-life T(1/2) = ln(2) / lambda. Common mistake: forgetting ln(2) (0.693) and just writing 1/lambda.
• Alpha (helium nucleus, stopped by paper/a few cm of air), beta-minus (fast electron, stopped by a few mm of aluminium), and gamma (EM radiation, needs thick lead or concrete) each have different penetrating power and ionising ability, roughly inversely related.
• In beta-minus decay a neutron becomes a proton, electron and antineutrino; in beta-plus decay a proton becomes a neutron, positron and neutrino. Neutrinos are needed to conserve energy and momentum, explaining the continuous beta energy spectrum.

Nuclear energetics

Mass-energy equivalence: E = mc^2. Mass defect is the difference between the mass of separate nucleons and the actual nucleus mass; this missing mass converts to binding energy that holds the nucleus together.

• Binding energy per nucleon peaks around iron-56 (about 8.8 MeV per nucleon), the most stable nucleus.
• Fusion of light nuclei and fission of heavy nuclei both release energy because the products have higher binding energy per nucleon than the reactants. Common mistake: thinking binding energy release means mass increases; total mass actually decreases.

Particle physics

Hadrons (protons, neutrons) are made of quarks; leptons (electrons, neutrinos, muons) are fundamental.

• Baryons contain three quarks (proton = uud, neutron = udd); mesons contain a quark-antiquark pair.
• Each particle has an antiparticle with equal mass and opposite charge; particle-antiparticle pairs can annihilate producing photons.
• Conservation laws apply to every interaction: charge, baryon number, lepton number, and strangeness (strangeness can only change in weak interactions).
• The four fundamental forces are gravity, electromagnetic, strong nuclear, and weak nuclear, mediated by exchange particles (gluons, photons, W/Z bosons).