Displacement, velocity and acceleration are vectors - direction matters, so use +/- signs on a defined axis.
Speed is distance/time (scalar); velocity is displacement/time (vector).
Acceleration is the rate of change of velocity, measured in m/s^2. Constant gradient on a velocity-time graph means constant (uniform) acceleration.
These only work for constant (uniform) acceleration:
Here s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time. Always write down what each symbol equals before substituting numbers - this catches sign errors early.
On a displacement-time graph, gradient = velocity. On a velocity-time graph, gradient = acceleration and the area under the graph = displacement. A curved displacement-time graph means changing velocity (acceleration).
Split motion into horizontal and vertical components - they are independent of each other. Horizontal velocity stays constant (no air resistance assumed) while vertical motion has constant acceleration g = 9.81 m/s^2 downward. Time of flight is found from the vertical SUVAT equations, then used to find horizontal range.
Newton's First Law: an object stays at rest or constant velocity unless a resultant (net) force acts on it.
Newton's Second Law: F = ma, where F is the resultant force in newtons, m is mass in kg, a is acceleration in m/s^2.
Newton's Third Law: every action has an equal and opposite reaction force, acting on the OTHER object, same type of force, same line of action.
As speed increases, resistive forces (drag) increase until they equal the driving force. At this point resultant force = 0, so acceleration = 0 and velocity becomes constant - this is terminal velocity.
Density is mass per unit volume: rho = m / V, measured in kg/m3. For a wire under tension, stress = Force / cross-sectional Area (Pa), and strain = extension / original length (no units, or a ratio).
The Young modulus E = stress / strain = (F/A) / (x/L), measured in Pa. It only applies within the region where stress is proportional to strain, up to the limit of proportionality on a stress-strain graph. Beyond the elastic limit, the material stops returning to its original shape when unloaded - that is plastic deformation.
Brittle materials (like glass) snap suddenly with almost no plastic region. Ductile materials (like copper) show a long plastic region before breaking.
Work done stretching a wire = area under the force-extension graph. If the wire obeys Hooke's law (F = kx), elastic strain energy = 1/2 F x = 1/2 k x^2. This equals the elastic potential energy stored, released fully if the wire stays within its elastic limit.
F = kx, where k is the spring constant (N/m) - the force needed per unit extension. Springs in series: 1/k_total = 1/k1 + 1/k2. Springs in parallel: k_total = k1 + k2. This is the reverse pattern to resistors in circuits, a very common exam trap.
You should be able to describe the Searle's apparatus method for measuring the Young modulus of a wire, including use of a Vernier scale, control of a reference wire to cancel out temperature effects and sag, and taking repeat readings to reduce random error.
Charge Q = It, measured in coulombs (C). One coulomb is the charge passing a point when a current of 1 A flows for 1 second.
Potential difference V = W/Q, the energy transferred per coulomb of charge (volts, V).
Current is the rate of flow of charge: I = ΔQ/Δt.
Resistance R = V/I, measured in ohms (Ω).
Ohm's law: current through a metallic conductor is proportional to pd if temperature stays constant, giving a straight line I-V graph through the origin.
A filament lamp curves as temperature rises (resistance increases). A diode conducts one way only, with near-zero current in reverse until breakdown.
R = ρL/A, where ρ is resistivity (Ω m), L is length and A is cross-sectional area.
Common mistake: forgetting to convert diameter to radius and to square it correctly when finding A = πr².
P = IV = I²R = V²/R.
Energy transferred W = Pt = IVt.
Series: same current everywhere, pds add up (V = V1 + V2 + ...), total resistance R = R1 + R2 + ...
Parallel: same pd across each branch, currents add up, 1/R = 1/R1 + 1/R2 + ...
Common mistake: mixing up which quantity is shared in series vs parallel.
EMF ε is the energy given per coulomb by the source, including energy wasted inside it.
ε = I(R + r), where r is internal resistance and R is external (load) resistance.
Terminal pd V = ε - Ir. Lost volts = Ir.
To find ε and r experimentally, plot V against I; the y-intercept is ε and the gradient is -r.
Output voltage from a potential divider: Vout = (R2 / (R1 + R2)) × Vin.
Sensors (LDR, thermistor) used in one arm let the output vary with light or temperature.
Kirchhoff's first law: current into a junction equals current out (conservation of charge).
Kirchhoff's second law: the sum of emfs around a loop equals the sum of pd drops (conservation of energy).
A gravitational field is a region where a mass feels a force. Field strength g = F/m, measured in N/kg. For a point (or spherical) mass, g = GM/r^2, where G = 6.67 x 10^-11 N m^2 kg^-2. This is an inverse-square law, so doubling r cuts g to a quarter.
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, because gravity is always attractive and does positive work as mass falls inwards. The gradient of a V against r graph gives -g.
Orbits: for a satellite in a circular orbit, gravitational force provides centripetal force, so GMm/r^2 = mv^2/r = m(4 pi^2/T^2)r. This rearranges to T^2 = (4 pi^2/GM) r^3, Kepler's third law. Geostationary orbits have T = 24 hours (23 h 56 min sidereal) and sit above the equator at radius about 4.2 x 10^7 m.
Electric field strength E = F/Q, in N/C or V/m. For a point charge, E = Q/(4 pi epsilon0 r^2), with epsilon0 = 8.85 x 10^-12 F/m. Coulomb's law gives the force F = Qq/(4 pi epsilon0 r^2). Electric potential V = Q/(4 pi epsilon0 r), zero at infinity, and can be positive or negative depending on the sign of the source charge.
Uniform fields exist between parallel plates: E = V/d, pointing from positive to negative plate. Work done moving charge q through potential difference V is W = qV. This underpins the electronvolt: 1 eV = 1.6 x 10^-19 J.
Both follow inverse-square force laws and have potentials that go as 1/r, so the equations are structurally identical (swap G and M for 1/(4 pi epsilon0) and Q). The key difference: gravity is always attractive, electric force can attract or repel depending on charge signs.
A moving charge in a magnetic field feels F = BQv sin(theta), maximum when velocity is perpendicular to B. A current-carrying wire feels F = BIL sin(theta). Flux density B is measured in tesla (T). Use Fleming's left-hand rule for force direction on a positive charge or conventional current.
A charged particle moving perpendicular to B follows a circular path, since the magnetic force is always perpendicular to velocity and does no work, so speed (and kinetic energy) stays constant. This is the basis of the mass spectrometer and cyclotron.
An atom has a tiny, dense, positively-charged nucleus (protons + neutrons) surrounded by orbiting electrons. Nuclide notation: A over Z, X, where Z = proton number (atomic number) and A = nucleon number (mass number, protons + neutrons). Isotopes have the same Z but different A (different neutron number).
Alpha particles fired at thin gold foil mostly passed straight through, but a small fraction deflected through large angles, some straight back. This proved the atom is mostly empty space with a small, dense, positive nucleus (the 'plum pudding' model was disproved). Closest approach distance gives an estimate of nuclear radius.
Nuclear radius follows R = r0 A^(1/3), where r0 is about 1.2 fm (1.2 x 10^-15 m). This means nuclear density is roughly constant for all nuclei, around 10^17 kg per cubic metre, far denser than ordinary matter.
Alpha decay emits a helium nucleus (2 protons, 2 neutrons), reducing A by 4 and Z by 2. Beta-minus decay emits an electron and an antineutrino as a neutron converts to a proton, so Z increases by 1, A unchanged. Beta-plus decay emits a positron and a neutrino as a proton converts to a neutron. Gamma emission releases energy only, no change to A or Z.
Activity A = lambda N, where lambda is the decay constant and N is the number of undecayed nuclei. Decay is exponential: N = N0 e^(-lambda t). Half-life T(1/2) = ln2 / lambda. Common mistake: forgetting activity and count rate are proportional to N, not to A (mass number).
Matter is built from quarks (up, down, etc.) and leptons (electron, neutrino). Hadrons are made of quarks: baryons (3 quarks, e.g. proton = uud, neutron = udd) and mesons (quark-antiquark pair). Every particle has an antiparticle with opposite charge but equal mass.
In all particle interactions, charge, baryon number, lepton number, and strangeness (in strong/EM interactions only) must be conserved. Common mistake: assuming strangeness is conserved in weak interactions - it isn't, which explains slow kaon decay.
E = mc^2 links mass and energy. Binding energy is the energy needed to split a nucleus into separate nucleons; it corresponds to a mass defect (the nucleus has less mass than its separate parts). Binding energy per nucleon peaks around iron-56, which is why fusion releases energy for light nuclei and fission releases energy for heavy nuclei.
Don't confuse mass number (A) with atomic mass in kg. Always balance both A and Z in decay equations. Remember antineutrinos/neutrinos carry away energy and momentum in beta decay, explaining the continuous energy spectrum of beta particles.