⚡ UK GCE A/L Physics - Course Notes

1. Mechanics & Kinematics

SUVAT · graphs · free fall · projectiles

Displacement (s) – vector: distance in a specific direction. Velocity (v) – rate of change of displacement. Acceleration (a) – rate of change of velocity.
v = Δs/Δt · a = Δv/Δt

SUVAT equations (uniform acceleration):
v = u + at · s = ut + ½at² · v² = u² + 2as · s = ½(u+v)t · s = vt – ½at²
Choose based on known/unknown variables.

Motion graphs: s–t gradient = velocity; v–t gradient = acceleration; area under v–t = displacement. a–t area = change in velocity.

Free fall: a = g = 9.81 m/s² downward. Terminal velocity – air resistance balances weight; net force = 0, constant speed.

Projectile motion: horizontal: constant velocity (no air resistance). Vertical: constant acceleration g. Time to max height: t = u sinθ / g. Range: R = u² sin2θ / g. Max height: H = u² sin²θ / 2g.

Relative motion: for two objects: v_rel = v₁ – v₂ (same direction) or v₁ + v₂ (opposite). Instantaneous velocity = gradient of tangent to s–t graph.

📌 Additional notes: In projectile motion, the horizontal and vertical components are independent. The trajectory is a parabola. For a projectile launched from height h, time of flight is found from s = ut + ½at² (vertical). The range can be extended by launching from a height.
🔑 v–t gradient = a 🔑 area under v–t = displacement 🔑 SUVAT only for constant a

2. Forces & Motion

Newton · equilibrium · friction · moments

Newton I: object remains at rest or constant velocity unless acted by resultant force. Newton II: F = ma (resultant force = mass × acceleration). Newton III: action & reaction are equal, opposite, on different bodies.

Weight W = mg. Normal reaction (R) – perpendicular to surface. Friction F = μR (μ = coefficient of friction). Tension – force transmitted through a rope/cable.

Equilibrium: resultant force = 0 (no acceleration) and resultant moment = 0 (no rotation). Moment = F × perpendicular distance from pivot. Clockwise moments = anticlockwise moments.

Centre of mass: point where whole mass appears to act. For uniform shapes: sphere (centre), rectangle (intersection of diagonals). Stability: if line of action of weight falls outside base → topples.

Momentum p = mv (vector). Impulse Ft = Δp = m(v – u). Conservation of momentum: total p before = total p after (no external force).

Couple: two equal, opposite, parallel forces → torque = force × perpendicular distance. Centre of gravity = point where weight acts.

📌 Additional notes: For a body in equilibrium under three forces, they must be concurrent (lines of action meet at a point). Friction always opposes relative motion (or tendency). Static friction has a maximum value (limiting friction) before motion starts.
⚡ F = ma ⚡ impulse = Ft = Δp ⚡ sum of moments = 0 for equilibrium

3. Energy & Work

work · KE · GPE · conservation · power

Work W = F·d·cosθ (scalar, J). Work done = energy transferred. Power P = W/t = Fv (for constant force in direction of motion).

Kinetic energy KE = ½mv². Gravitational potential energy GPE = mgh (near Earth). Elastic potential EPE = ½kx² (spring, Hooke’s law F = kx).

Conservation of energy: energy cannot be created/destroyed, only transformed. In a closed system, total energy is constant.

Work-energy principle: net work done = change in kinetic energy (W_net = ΔKE). For conservative forces: KE + PE = constant.

Efficiency = useful output energy / total input energy × 100%. Dissipated energy: often as heat (friction, resistance).

Power in circuits: P = IV = I²R = V²/R. Energy in circuits: E = Pt = VIt = I²Rt.

📌 Additional notes: The work done against gravity is mgh. The work done to stretch a spring is ½kx² (area under F–x graph). Power is the rate of energy transfer – 1 W = 1 J/s.
🔋 KE = ½mv² 🔋 GPE = mgh 🔋 EPE = ½kx² 🔋 efficiency = useful/total ×100%

4. Momentum & Collisions

elastic · inelastic · restitution · explosions

Momentum p = mv (vector). Impulse = force × time = Ft = Δp. Impulse is area under force–time graph.

Conservation of momentum: for a closed system (no external forces): m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. This applies to all collisions and explosions.

Elastic collision: both momentum and kinetic energy are conserved. Inelastic: momentum conserved, KE not conserved (some energy → heat/sound). Perfectly inelastic: objects stick together after collision.

Coefficient of restitution e = (relative speed of separation) / (relative speed of approach). e = 1 (elastic), e = 0 (perfectly inelastic).

Explosions: total momentum before = 0 (if initially at rest). After: m₁v₁ + m₂v₂ = 0. Kinetic energy comes from internal energy (chemical, nuclear).

Centre of mass: moves with constant velocity if no external force. Internal forces (collisions) do not affect motion of COM.

📌 Additional notes: In 2D collisions, momentum is conserved in both x and y directions. For a perfectly elastic collision of two equal masses, they exchange velocities. The coefficient of restitution is a measure of the "bounciness" of a collision.
💥 m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ 💥 e = (v₂ – v₁)/(u₁ – u₂)

5. Circular Motion

centripetal · banked tracks · vertical circles

Angular velocity ω = θ/t = 2π/T = 2πf. Linear speed v = ωr. Angular acceleration α = Δω/Δt.

Centripetal acceleration a = v²/r = ω²r (directed towards centre). Centripetal force F = mv²/r = mω²r.

Banked track: tanθ = v²/(rg) (no friction). With friction, the horizontal component of N provides centripetal force.

Vertical circle: at top: T + mg = mv²/r; at bottom: T – mg = mv²/r. Minimum speed to maintain contact: v_min = √(rg) at top.

Conical pendulum: T sinθ = mv²/r, T cosθ = mg → tanθ = v²/(rg). Period T = 2π√(L cosθ / g).

Satellite orbits: orbital speed v = √(GM/r). Period T = 2π√(r³/GM). Total energy E = –GMm/(2r). Geostationary: T = 24h, above equator.

📌 Additional notes: Centripetal force is not a new force – it is the resultant force directed towards the centre. In a banked track, the normal reaction provides the centripetal force. For a satellite, the gravitational force provides the centripetal force.
🌀 F = mv²/r 🌀 v = ωr 🌀 T = 2π/ω

6. Gravitational Fields

Newton · Kepler · escape velocity · potential

Newton’s law of gravitation: F = Gm₁m₂/r² (G = 6.67×10⁻¹¹ N m² kg⁻²). The force is attractive and acts along the line joining the masses.

Gravitational field strength g = F/m = GM/r² (N/kg). It is a vector pointing towards the mass. On Earth’s surface, g = 9.81 N/kg.

Gravitational potential V = –GM/r (scalar, J/kg). Gravitational potential energy U = –GMm/r. V is zero at infinity.

Escape velocity v_esc = √(2GM/r). For Earth: 11.2 km/s. It is the minimum speed to escape the gravitational field.

Kepler’s laws: (1) elliptical orbits with Sun at focus, (2) equal areas in equal times, (3) T² ∝ r³ (for circular orbits: T² = (4π²/GM)r³).

Orbital energy: KE = GMm/(2r), PE = –GMm/r, total E = –GMm/(2r). A satellite in a higher orbit has greater total energy (less negative).

📌 Additional notes: Gravitational field lines point towards the mass. For a uniform solid sphere, the field outside is as if all mass is at the centre; inside the field is proportional to r. The gravitational potential is always negative.
🌍 F = Gm₁m₂/r² 🌍 v_esc = √(2GM/r) 🌍 T² ∝ r³

7. Oscillations & Waves

SHM · damping · wave properties · Doppler

Simple harmonic motion (SHM): a = –ω²x. Displacement: x = A sin(ωt + φ). Velocity: v = ±ω√(A² – x²). Acceleration: a = –ω²x.

Mass-spring system: ω = √(k/m), T = 2π√(m/k). Simple pendulum: ω = √(g/l), T = 2π√(l/g) (small angles).

Energy in SHM: E_total = ½kA² = ½mω²A². KE = ½mω²(A² – x²), PE = ½kx².

Damping: light damping (amplitude decays exponentially), critical damping (returns to equilibrium fastest), heavy damping (slow return).

Wave equation: v = fλ. Transverse (oscillations perpendicular to direction) and longitudinal (parallel). Reflection, refraction, diffraction, interference are key wave phenomena.

Doppler effect: f_obs = f_source (v ± v_obs) / (v ∓ v_source). For a moving source: f_obs = f_s v/(v ∓ v_s).

📌 Additional notes: In SHM, the acceleration is always directed towards the equilibrium position. The total energy is proportional to the square of the amplitude. For a standing wave, nodes are points of zero displacement, antinodes are points of maximum displacement.
🌊 a = –ω²x 🌊 v = fλ 🌊 T = 2π√(l/g) (pendulum)

8. Thermodynamics

gas laws · kinetic theory · entropy · engines

Ideal gas equation: PV = nRT (R = 8.31 J mol⁻¹ K⁻¹). Boyle’s law: P₁V₁ = P₂V₂ (T constant). Charles’ law: V₁/T₁ = V₂/T₂ (P constant). Pressure law: P₁/T₁ = P₂/T₂ (V constant).

Kinetic theory: pressure P = ⅓ρ⟨c²⟩. Average kinetic energy: ½m⟨c²⟩ = ³/₂kT. Temperature is a measure of average KE.

Internal energy: U = sum of kinetic and potential energies of particles. For an ideal gas, U depends only on temperature: ΔU = ³/₂nRΔT.

First law: ΔU = Q – W. Isothermal: ΔU = 0, Q = W. Adiabatic: Q = 0, ΔU = –W. Isobaric: W = PΔV. Isochoric: W = 0, Q = ΔU.

Second law: heat cannot spontaneously flow from cold to hot. Entropy S: ΔS = Q/T (reversible). Entropy of the universe increases.

Heat engine efficiency: η = (Q₁ – Q₂)/Q₁ = 1 – Q₂/Q₁. Carnot efficiency: η_max = 1 – T₂/T₁ (theoretical maximum).

📌 Additional notes: A perfect heat engine would have 100% efficiency, but this is impossible due to the second law. Refrigerators and heat pumps work by transferring heat from cold to hot (requiring work input).
🔥 PV = nRT 🔥 ΔU = Q – W 🔥 η = 1 – T₂/T₁ (Carnot)

9. Electricity & Circuits

Ohm · Kirchhoff · potential divider · bridges

Current I = ΔQ/Δt (A). Ohm’s law: V = IR (for ohmic conductors at constant temperature). Resistance R = ρL/A (ρ = resistivity).

Power P = IV = I²R = V²/R. Energy E = Pt = VIt = I²Rt = V²t/R.

Resistors in series: R_total = R₁ + R₂ + R₃ + ... Resistors in parallel: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + ...

Kirchhoff’s laws: I: ΣI_in = ΣI_out (charge conservation). II: ΣV = 0 around any closed loop (energy conservation).

EMF (ε) and internal resistance (r): terminal voltage V = ε – Ir. Maximum power transfer occurs when R = r.

Potential divider: V_out = V_in × R₂/(R₁ + R₂). Wheatstone bridge: balanced when R₁/R₂ = R₃/R₄ (no current through galvanometer).

📌 Additional notes: In a parallel circuit, the voltage across each branch is the same. The total current is the sum of branch currents. For a potential divider, a variable resistor can be used to adjust the output voltage.
⚡ V = IR ⚡ P = IV ⚡ V_out = V_in × R₂/(R₁+R₂)

10. Magnetism

B-field · force · induction · transformers

Magnetic flux density B = F/IL (T). Force on a current-carrying wire: F = BIL sinθ. Force on a moving charge: F = Bqv sinθ.

Motion of charged particle in B-field: if perpendicular → circular motion (r = mv/Bq). If parallel → straight line. Helical if at an angle.

Hall effect: V_H = BIt/(nqe). Used to measure B and determine charge carrier type.

Faraday’s law: ε = –NΔΦ/Δt. Lenz’s law: the induced current opposes the change that produced it.

Magnetic flux Φ = BA cosθ. Flux linkage = NΦ = NBA cosθ.

Transformer: V_s/V_p = N_s/N_p (ideal). AC generator: ε = NBAω sinωt. DC motor: commutator reverses current to maintain rotation.

📌 Additional notes: The direction of the force on a current-carrying wire is given by Fleming’s left-hand rule. In a transformer, power is conserved (P_p = P_s for an ideal transformer). Eddy currents are induced currents in conductors that cause energy loss.
🧲 F = BIL sinθ 🧲 ε = –NΔΦ/Δt 🧲 V_s/V_p = N_s/N_p

11. Nuclear Physics

radioactivity · fission · fusion · half-life

Alpha decay: ⁴₂He emitted, Z–2, A–4. Beta decay: neutron → proton + electron + antineutrino, Z+1. Gamma decay: high-energy photon, no change in Z or A.

Decay law: N(t) = N₀e⁻λt. Activity A = λN = –dN/dt. Half-life t₁/₂ = ln2/λ = 0.693/λ.

Nuclear fission: heavy nucleus splits (e.g. ²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n). Energy released due to mass defect. Chain reaction in reactors.

Nuclear fusion: light nuclei combine (e.g. ²H + ³H → ⁴He + n). Requires very high temperatures. Main source of energy in stars.

Mass–energy equivalence: E = mc². Mass defect = [Zm_p + (A–Z)m_n] – m_atom. Binding energy = Δmc². Iron has maximum binding energy per nucleon.

Radiation hazards: alpha (paper), beta (aluminium), gamma (lead). Ionisation can damage DNA, causing cancer. Internal exposure is more hazardous.

📌 Additional notes: Nuclear reactors use control rods (boron/cadmium) to absorb neutrons and moderate the reaction. The moderator (water/graphite) slows neutrons. Fusion is the energy source of the Sun and future energy production (ITER).
☢️ N = N₀e⁻λt ☢️ E = mc² ☢️ t₁/₂ = ln2/λ

12. Astrophysics

stars · HR diagram · cosmology · exoplanets

Stellar evolution: nebula → protostar → main sequence (H→He fusion) → red giant / supergiant → planetary nebula / supernova → white dwarf / neutron star / black hole.

Hertzsprung–Russell diagram: luminosity vs temperature. Main sequence (diagonal), red giants (cool, luminous), white dwarfs (hot, dim). Mass–luminosity: L ∝ M³·⁵.

Luminosity: L = 4πR²σT⁴ (Stefan–Boltzmann). Wien’s law: λ_max ∝ 1/T. Distance modulus: m – M = 5log₁₀(d/10).

Galaxies: spiral (Milky Way), elliptical, irregular, barred spiral. Milky Way: ~100,000 ly diameter, ~100 billion stars.

Hubble’s law: v = H₀d (H₀ ≈ 70 km/s/Mpc). Redshift: z = Δλ/λ = v/c. Cosmological principle: universe is homogeneous and isotropic.

Big Bang evidence: CMBR, expansion (Hubble), light element abundance, age of oldest stars. Dark matter (27%) and dark energy (68%) drive accelerated expansion.

📌 Additional notes: Exoplanets are detected by transit (dimming of star) or radial velocity (Doppler shift). The habitable zone is the region around a star where liquid water can exist. The cosmic microwave background radiation is the afterglow of the Big Bang.
🌌 v = H₀d 🌌 L = 4πR²σT⁴ 🌌 z = Δλ/λ = v/c

📐 Appendix A – Fundamental Constants

Gravitational constant G	6.67 × 10⁻¹¹ N m² kg⁻²
Acceleration due to gravity g	9.81 m s⁻²
Speed of light c	3.00 × 10⁸ m s⁻¹
Planck’s constant h	6.63 × 10⁻³⁴ J s
Boltzmann constant k	1.38 × 10⁻²³ J K⁻¹

Universal gas constant R	8.31 J mol⁻¹ K⁻¹
Avogadro’s number Nₐ	6.02 × 10²³ mol⁻¹
Elementary charge e	1.60 × 10⁻¹⁹ C
Astronomical unit AU	1.496 × 10¹¹ m
Parsec	3.26 light-years

📐 Appendix B – Essential Equations (quick reference)

Mechanics
v = u + at · s = ut + ½at² · v² = u² + 2as · F = ma · KE = ½mv² · GPE = mgh · p = mv

Waves & SHM
v = fλ · T = 1/f · a = –ω²x · ω = √(k/m) · T = 2π√(m/k) · T = 2π√(l/g)

Electricity & Magnetism
V = IR · P = IV = I²R = V²/R · R = ρL/A · ε = –NΔΦ/Δt · F = BIL sinθ

Thermodynamics & Nuclear
PV = nRT · ΔU = Q – W · N = N₀e⁻λt · t₁/₂ = ln2/λ · E = mc²

Astrophysics
F = Gm₁m₂/r² · v = √(GM/r) · T² ∝ r³ · v = H₀d

📌 Key reminders: SHM: a = –ω²x · Circular: a = v²/r · Always check sign conventions and units.

⚡ UK GCE A/L Physics – comprehensive notes covering all 12 modules · exam-ready