Physics

Lesson 1: Kinematics — Describing Motion

Kinematics describes motion without asking why it happens. The core quantities: displacement Δx (change in position — a vector), velocity v = Δx/Δt (rate of position change), and acceleration a = Δv/Δt (rate of velocity change). Speed is the magnitude of velocity; distance is the path length, while displacement is the straight-line net change. A runner completing one 400-m lap has distance 400 m but displacement zero.

With constant acceleration, four kinematic equations rule: v = v_0 + at; Δx = v_0 t + ½at²; v² = v_0² + 2aΔx; Δx = (v_0 + v)/2 · t. Choose the equation containing your unknown but not your missing given. Dropped or thrown objects near Earth accelerate at g = 9.8 m/s² downward (ignoring air resistance) — Galileo's insight that mass doesn't matter, confirmed when Apollo astronauts dropped a hammer and feather together on the Moon.

Projectile motion splits into independent horizontal and vertical parts: horizontal velocity stays constant (no horizontal force), while vertical motion is free fall. Launch at speed v_0 and angle θ: v_0x = v_0 cosθ, v_0y = v_0 sinθ. The path is a parabola; maximum range on flat ground comes at 45°. Graphs tell the story too: the slope of a position-time graph is velocity, and the slope of a velocity-time graph is acceleration — while the area under a velocity-time graph is displacement.

Try it yourself: A ball is dropped from a 45-m cliff. How long until it hits the ground, and with what speed? A car accelerates from 10 m/s to 30 m/s over 200 m: find its acceleration.

Lesson 2: Vectors and Projectile Motion

A vector has both magnitude and direction; a scalar (like speed or mass) has only magnitude. Velocity, force, and acceleration are vectors. Vectors add tip-to-tail: a 3 N east force plus a 4 N north force gives a 5 N resultant at 53° north of east (a 3–4–5 triangle). Any vector can be split into perpendicular components: vₒ = v cos θ, vₓ = v sin θ.

Projectile motion is two independent motions at once: horizontal velocity stays constant (no horizontal force, ignoring air resistance), while vertical velocity changes by g = 9.8 m/s² downward. A ball launched at 20 m/s at 30° has vₒ = 20 cos 30° ≈ 17.3 m/s and vₓ = 20 sin 30° = 10 m/s. It reaches its peak when vₓ = 0: t = 10/9.8 ≈ 1.02 s, at height vₓ²/(2g) ≈ 5.1 m. Total flight time is twice that, ≈ 2.04 s, and the range is vₒ × 2.04 ≈ 35.3 m.

The path is a parabola, and the launch angle matters: 45° gives the maximum range on level ground (in a vacuum), while complementary angles like 30° and 60° give the same range. Air resistance shortens real trajectories and makes the descent steeper than the ascent — the reason a golf ball's flight is not a perfect parabola.

Try it yourself: A projectile launched at 25 m/s at 37° (sin 37° ≈ 0.6, cos 37° ≈ 0.8): find vₒ, vₓ, and time to peak. Answers: vₒ ≈ 20 m/s, vₓ ≈ 15 m/s, t = 15/9.8 ≈ 1.53 s.

Lesson 3: Circular Motion and Gravitation

An object moving in a circle at constant speed is still accelerating, because its direction keeps changing. Centripetal acceleration points toward the center with magnitude a = v²/r. The force producing it is centripetal force, F = mv²/r — not a new kind of force, but the role played by tension, friction, gravity, or the normal force in a given situation. A car turning on a flat road is held in its circle by friction; a satellite is held by gravity.

Newton's Law of Universal Gravitation says every mass attracts every other: F = Gm₁m₂/r², where G = 6.67 × 10⁻¹¹ N·m²/kg². Gravity weakens with the square of the distance: double the separation, quarter the force. Near Earth's surface this gives g = GM/R² ≈ 9.8 m/s². The same law holds the Moon in orbit: the Moon is in continuous free fall toward Earth, moving sideways fast enough to keep missing.

Kepler's laws describe orbits: planets move in ellipses with the Sun at one focus; a line from the Sun sweeps equal areas in equal times (so planets move faster near the Sun); and T² is proportional to a³ (period squared vs. average distance cubed). Newton's gravitation explains all three.

Try it yourself: A 0.5 kg ball on a 1 m string spins at 4 m/s. Find the tension (centripetal force). If Earth's radius doubled but mass stayed the same, what would g become? Answers: F = 0.5 × 16/1 = 8 N; g ∝ 1/R², so g would fall to 9.8/4 ≈ 2.45 m/s².

Key vocabulary: Centripetal acceleration/force — toward-the-center acceleration (v²/r) and the force causing it; universal gravitation — F = Gm₁m₂/r²; Kepler's laws — the three laws of orbital motion.

Lesson 4: Newton's Laws of Motion

Newton's First Law (inertia): objects maintain their velocity unless a net force acts. Passengers lurch forward when a bus brakes because their bodies keep moving — inertia, not a mysterious force. Newton's Second Law quantifies force: F_net = ma. One newton (1 N) accelerates 1 kg at 1 m/s². Crucially, it's the net force — the vector sum of all forces — that determines acceleration. A 2-kg block pushed with 10 N right and 4 N friction left accelerates at (10−4)/2 = 3 m/s² right.

Newton's Third Law: every action has an equal, opposite reaction — forces come in pairs acting on different objects. A rocket pushes exhaust down; exhaust pushes the rocket up. Walking pushes Earth backward; Earth pushes you forward. The pair never cancels in F_net = ma because the two forces act on different bodies — a classic trap.

Common forces: weight W = mg (gravity's pull — mass is intrinsic, weight depends on g), the normal force (perpendicular support from a surface), friction (opposes relative motion: f_s ≤ μ_s N static, f_k = μ_k N kinetic), tension (pull along ropes), and spring force (Hooke's law F = −kx). Free-body diagrams — drawing every force as an arrow — turn chaos into solvable F_net = ma equations, one axis at a time.

Try it yourself: A 70-kg person stands in an elevator accelerating upward at 2 m/s². What does the scale read? Two blocks (3 kg and 2 kg) are pushed together by a 20-N force on frictionless ice: find the acceleration and the contact force between them.

Lesson 5: Work, Energy, and Momentum

Work W = Fd cosθ is force times displacement in the force's direction — carrying a heavy box horizontally does no work against gravity (θ = 90°, cosθ = 0). Work transfers energy, measured in joules like work. Kinetic energy KE = ½mv² is motion's energy (note the v² — doubling speed quadruples energy, which is why highway crashes are so destructive). The work-energy theorem: net work equals change in kinetic energy, W_net = ΔKE.

Potential energy is stored energy: gravitational PE = mgh near Earth, elastic PE = ½kx² in springs. Mechanical energy (KE + PE) is conserved when only conservative forces act — a roller coaster trades height for speed: mgh_top = ½mv²_bottom. Friction converts mechanical energy to heat, so real coasters never quite return to their starting height. Power P = W/t (watts) is how fast work is done — a 100-W bulb uses 100 joules per second.

Momentum p = mv is motion's persistence; impulse J = FΔt = Δp changes it. Airbags extend Δt to shrink F for the same Δp. In collisions with no external impulse, total momentum is conserved — the bedrock of crash analysis and rocket propulsion. Elastic collisions also conserve kinetic energy (billiard balls, nearly); inelastic ones don't (cars crumpling, sticking together). A 2-kg cart at 3 m/s hitting a stationary 2-kg cart and sticking gives both moving at 1.5 m/s — momentum conserved, half the KE lost to deformation.

Try it yourself: How much work to lift a 15-kg box 2 m? A 1000-kg car at 20 m/s brakes to rest: how much kinetic energy becomes heat? A 0.5-kg ball at 4 m/s collides elastically head-on with an identical stationary ball: describe the outcome.

Lesson 6: Momentum and Collisions

Momentum is mass times velocity: p = mv. It is a vector, and in a closed system the total momentum is conserved — it never changes, even in violent collisions. A 2 kg cart moving at 3 m/s (p = 6 kg·m/s) colliding with a stationary 4 kg cart and sticking to it gives a combined mass of 6 kg moving at 1 m/s (p still 6). Newton's third law guarantees it: the forces the carts exert on each other are equal and opposite, so their momentum changes cancel.

Impulse, J = FΔt, equals the change in momentum: J = Δp. This is why airbags and crumple zones save lives: they lengthen the collision time, so the same Δp happens with a smaller force. A 70 kg person stopping from 20 m/s has Δp = 1,400 kg·m/s; stopping in 0.01 s takes 140,000 N, but in 0.2 s it takes only 7,000 N.

Collisions come in two flavors. In an elastic collision, kinetic energy is also conserved (billiard balls, nearly). In an inelastic collision, kinetic energy is lost to heat and deformation (a car crash, a ball of clay). Momentum is conserved in both. The maximum energy loss is a perfectly inelastic collision, where the objects stick together.

Try it yourself: A 60 kg skater pushes a 40 kg skater; both start at rest. If the 60 kg skater moves back at 2 m/s, how fast does the other move? Answers: Total momentum stays 0: 60(−2) + 40v = 0, so v = 3 m/s forward.

Key vocabulary: Momentum — p = mv; conservation of momentum — total momentum constant in a closed system; impulse — FΔt = Δp; elastic/inelastic collision — kinetic energy conserved / not conserved.

Lesson 7: Waves and Sound

A wave carries energy without carrying matter — the water bobs up and down while the wave travels on. Transverse waves oscillate perpendicular to travel (light, string waves); longitudinal waves oscillate parallel (sound's compressions and rarefactions). The big three: wavelength λ (length of one cycle), frequency f (cycles per second, Hz), amplitude (maximum displacement). They bind through the wave equation v = fλ — and period T = 1/f.

Sound is a longitudinal pressure wave needing a medium: 343 m/s in air at 20°C, ~1500 m/s in water, ~5000 m/s in steel — stiffer, denser media transmit faster. Pitch is frequency (human hearing ~20 Hz–20 kHz); loudness is amplitude (decibels, logarithmic: +10 dB sounds twice as loud but carries ten times the intensity). The Doppler effect shifts frequency for moving sources — an ambulance's pitch drops as it passes — and underlies radar guns and redshift cosmology.

Waves superpose: overlapping waves add displacements. Constructive interference (crest on crest) amplifies; destructive (crest on trough) cancels — noise-canceling headphones engineer destruction. Standing waves form when a wave reflects and interferes with itself: guitar strings vibrate in quantized patterns with nodes (still points) and antinodes. Resonance — driving at a natural frequency — builds amplitude dramatically, from shattering wine glasses to collapsing bridges (Tacoma Narrows, 1940).

Try it yourself: A wave has λ = 2.5 m and f = 136 Hz: find its speed. How far does thunder travel in the 3 s between lightning and thunder? Explain why you hear the bass from a neighbor's music but not the treble (hint: diffraction and absorption).

Lesson 8: Light, Reflection, and Refraction

Light travels in straight lines at c = 3.00 × 10⁸ m/s in a vacuum, slowing in denser media. The law of reflection says the angle of incidence equals the angle of reflection, measured from the normal (the perpendicular to the surface). A flat mirror forms a virtual image as far behind the mirror as the object is in front — which is why your reflection seems to stand behind the glass.

Refraction is light bending as it changes speed. Snell's Law, n₁ sin θ₁ = n₂ sin θ₂, relates the angles, where n is the index of refraction (n = c/v). Light entering water (n = 1.33) from air bends toward the normal; leaving water it bends away. This is why a straw in a glass looks broken and a pool looks shallower than it is. When light tries to leave a dense medium at a steep angle, it can undergo total internal reflection — the principle behind fiber optics.

Lenses use refraction to form images. A converging lens focuses parallel rays at its focal point; the thin-lens equation 1/f = 1/do + 1/di and magnification m = −di/do locate and size the image. Cameras, eyes, and telescopes are all applications of the same two equations.

Try it yourself: An object is 30 cm from a converging lens of focal length 10 cm. Where is the image, and what is the magnification? Answers: 1/di = 1/10 − 1/30 = 1/15, di = 15 cm; m = −15/30 = −0.5 (inverted, half-size).

Key vocabulary: Normal — the perpendicular to a surface; Snell's Law — n₁ sin θ₁ = n₂ sin θ₂; index of refraction — n = c/v; total internal reflection — complete reflection inside a denser medium.

Lesson 9: Electricity Basics

Electric charge comes in positive and negative; like charges repel, opposites attract, with Coulomb's law F = kq₁q₂/r² (k ≈ 9× 10⁹ N·m²/C²) — an inverse-square law like gravity. Charge is conserved and quantized in units of e ≈ 1.6× 10^−19 C. Conductors (metals) let charge flow; insulators (rubber, glass) don't; semiconductors (silicon) sit between and run the digital world.

Current I = ΔQ/Δt (amperes) is charge flow; voltage V (volts) is energy per charge — the electrical “push”; resistance R (ohms) opposes flow. Ohm's law, V = IR, governs simple circuits: a 12-V battery across 4 Ω; drives 3 A. Power P = IV = I²R = V²/R (watts) is why thin wires overheat — power grids transmit at high voltage to keep current (and I²R losses) low.

Resistors in series add (R_eq = R₁ + R₂) and share one current; in parallel, reciprocals add (1/R_eq = 1/R₁ + 1/R₂) and each gets full voltage. Household outlets are parallel — one appliance failing doesn't kill the rest. Kirchhoff's laws generalize: currents sum to zero at a junction (charge conservation), voltages sum to zero around a loop (energy conservation). Fuses and breakers exploit I²R heating to sacrifice themselves before wires do.

Try it yourself: A 9-V battery drives 0.5 A through a bulb: find the bulb's resistance and power. Two 10-Ω; resistors in series across 20 V: find current and each resistor's voltage drop. Why do birds on a power line not get shocked?

Lesson 10: Magnetism and Electromagnetism

A magnet has two poles, north and south; like poles repel, opposites attract, and a magnetic field fills the space around it. Earth's magnetic field is what makes compasses work — and it is roughly the field of a giant bar magnet tilted relative to the planet's spin axis. A moving electric charge creates a magnetic field, and a magnetic field exerts a force on a moving charge: F = qvB sin θ. This is the principle of electric motors (current in a magnetic field feels a force) and of mass spectrometers, which sort ions by bending their paths.

Electromagnetic induction runs the other way: a changing magnetic flux through a loop induces a voltage, ε = −N ΔΦ/Δt (Faraday's Law). The minus sign is Lenz's Law: the induced current opposes the change that caused it. Induction is how generators make nearly all the world's electricity — spinning a coil in a magnetic field — and how transformers step voltage up and down for power transmission.

A transformer obeys V₁/V₂ = N₁/N₂ (voltage ratio = turns ratio). Power lines carry electricity at hundreds of thousands of volts precisely because higher voltage means lower current for the same power (P = VI), and lower current means far less energy lost as heat in the wires (P_loss = I²R).

Try it yourself: A generator coil's flux changes by 0.04 Wb in 0.01 s through 100 turns. What EMF is induced? Answers: ε = 100 × 0.04/0.01 = 400 V.

Key vocabulary: Magnetic field — the region of magnetic influence; Faraday's Law — changing flux induces voltage; Lenz's Law — the induced effect opposes the change; transformer — a device changing AC voltage via turns ratio.

Lesson 11: Introduction to Modern Physics — Photons and Relativity

Classical physics fails at very small and very fast scales. The photoelectric effect showed that light delivers energy in discrete packets, photons, with E = hf (h = Planck's constant, 6.63 × 10⁻³⁴ J·s). A metal ejects electrons only when the light's frequency exceeds a threshold, no matter how bright the light is — because a dim high-frequency beam has energetic photons, while a bright low-frequency beam has many weak ones. Einstein won his Nobel Prize for this explanation, not for relativity.

Wave-particle duality extends the idea: electrons, too, behave as waves (they diffract), and light behaves as particles. The uncertainty principle sets a fundamental limit: the more precisely you know a particle's position, the less precisely you can know its momentum. Atoms have quantized energy levels; an electron dropping between levels emits a photon of exactly the energy difference — the origin of atomic spectra and of lasers.

Special relativity rests on two postulates: the laws of physics are the same in all inertial frames, and the speed of light is constant for all observers. Consequences: moving clocks run slow (time dilation, Δt = γΔt₀), moving objects shorten (length contraction), and mass and energy are interchangeable: E = mc². GPS satellites must correct for both special and general relativistic effects, or their positioning would drift by kilometers per day.

Try it yourself: A spaceship travels at 0.8c. What is γ = 1/√(1 − v²/c²)? If 10 years pass on Earth, how much time passes aboard? Answers: γ = 1/√(1 − 0.64) = 1/0.6 ≈ 1.67; aboard, 10/1.67 = 6 years.

Key vocabulary: Photon — a quantum of light, E = hf; photoelectric effect — electron ejection by light above a threshold frequency; time dilation — moving clocks run slow; E = mc² — mass-energy equivalence.

Lesson 12: Electromagnetic Waves and the Spectrum

An electromagnetic (EM) wave is a traveling disturbance of electric and magnetic fields that needs no medium — it crosses empty space at c = 3.00 × 10⁸ m/s. The fields oscillate perpendicular to each other and to the direction of travel, and they regenerate each other: a changing electric field creates a magnetic field and vice versa. That is why sunlight reaches us through the vacuum of space, while sound cannot. For all EM waves, c = fλ, so higher frequency means shorter wavelength.

The electromagnetic spectrum orders EM waves by frequency: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Our eyes detect only a narrow slice — visible light from about 380 nm (violet) to 750 nm (red). Each band has characteristic uses and hazards: radio carries broadcasts, microwaves heat food and link satellites, infrared is felt as heat and used in night-vision cameras, ultraviolet tans and sunburns skin, X-rays image bones, and gamma rays treat cancers — and all higher-energy bands can damage living tissue in large doses.

Try it yourself: A microwave oven uses 2.45 GHz waves. Find the wavelength. A green light photon has wavelength 550 nm: find its frequency and energy in eV. Answers: λ = 3.00 × 10⁸/2.45 × 10⁷ ≈ 0.122 m (12.2 cm); f = c/λ ≈ 5.45 × 10⁺⁴ Hz; E = hf ≈ 3.6 × 10⁻¹⁷ J ≈ 2.25 eV.

Real-world application: Weather radar, 5G networks, airport scanners, and the James Webb Space Telescope (which sees in infrared) all exploit different bands of the same spectrum.

Key vocabulary: Electromagnetic wave — self-propagating electric and magnetic fields, speed c in vacuum; electromagnetic spectrum — the full range of EM waves ordered by frequency; photon energy — E = hf.

Lesson 13: Simple Harmonic Motion — Pendulums and Springs

Simple harmonic motion (SHM) is periodic motion driven by a restoring force proportional to displacement: F = −kx. The minus sign means the force always pulls back toward equilibrium — stretch a spring and it pulls back; compress it and it pushes out. The result is sinusoidal motion described by x(t) = A cos(ωt + φ), where A is the amplitude, ω = 2π/T is the angular frequency, and the period T and frequency f = 1/T describe how fast the cycle repeats.

Two classic systems show SHM. A mass on a spring has period T = 2π√(m/k): heavier masses and softer springs oscillate more slowly. A simple pendulum (small swings) has T = 2π√(L/g): the period depends on string length and gravity, but — remarkably — not on the mass of the bob or (for small angles) the amplitude. Galileo noticed this watching a swinging lamp in Pisa Cathedral, and pendulum clocks ran the world for three centuries. Energy in SHM sloshes between kinetic and potential: at the extremes the mass momentarily stops (all potential), at equilibrium it moves fastest (all kinetic).

Try it yourself: A pendulum of length 0.25 m on Earth: find its period. If a spring's mass doubles, what happens to its period? Answers: T = 2π√(0.25/9.8) ≈ 1.00 s; period grows by √2 ≈ 1.41 times.

Real-world application: Car suspensions, earthquake-resistant building dampers, metronomes, and quartz watches (a vibrating crystal at 32,768 Hz) all rely on SHM.

Key vocabulary: Simple harmonic motion — periodic motion with F = −kx; amplitude — maximum displacement; period / frequency — time per cycle / cycles per second.

Lesson 14: Thermodynamics — Heat, Temperature, and the Laws

Temperature measures the average kinetic energy of particles; heat is thermal energy transferred because of a temperature difference. Heat always flows spontaneously from hot to cold, never the reverse. The specific heat capacity c tells how much energy changes a substance's temperature: Q = mcΔT. Water's unusually high c (4,186 J/kg·°C) is why coastal climates are mild and why a pot of water takes so long to boil. Latent heat is the energy of phase changes — melting ice absorbs 334,000 J/kg without warming at all.

The First Law of Thermodynamics is energy conservation for heat: ΔU = Q − W (the change in a system's internal energy equals heat added minus work done by the system). The Second Law says entropy — disorder — never decreases in an isolated system: heat engines cannot be 100% efficient, and perpetual motion machines are impossible. The maximum possible efficiency of a heat engine operating between temperatures T_hot and T_cold (in kelvin) is the Carnot efficiency, 1 − T_cold/T_hot. The Zeroth Law (objects in thermal equilibrium share a temperature) is what makes thermometers work.

Try it yourself: How much heat melts 0.50 kg of ice at 0°C? A refrigerator moves heat from cold to hot — does that violate the Second Law? Answers: Q = mL = 0.50 × 334,000 = 167,000 J. No — it consumes electrical work, and total entropy (fridge + room) still increases.

Real-world application: Refrigerators, air conditioners, car engines, and power stations are all heat engines or heat pumps governed by these laws.

Key vocabulary: Specific heat capacity — energy per kg per degree; entropy — a measure of disorder; Carnot efficiency — 1 − T_cold/T_hot, the efficiency ceiling.

Lesson 15: Fluids — Pressure, Buoyancy, and Bernoulli

Pressure is force per unit area: P = F/A. In a fluid at rest, pressure grows with depth: P = P₀ + ρgh, where ρ is density. That is why your ears pop diving to the bottom of a pool, and why dams are built thicker at the base. Pascal's principle says a pressure change in an enclosed fluid is transmitted undiminished everywhere — the basis of hydraulic lifts, where a small force on a small piston becomes a large force on a large piston.

Archimedes' principle explains floating: the buoyant force equals the weight of displaced fluid, F_b = ρVg. A steel ship floats because its hull displaces a huge volume of water, while a solid steel block sinks. Whether an object floats depends on average density, not material alone. For moving fluids, Bernoulli's principle says faster flow means lower pressure: P + ½ρv² + ρgh = constant along a streamline. Airplane wings exploit this — faster air over the curved top surface means lower pressure, producing lift.

Try it yourself: A diver is 10 m below the surface. What is the water pressure (atmospheric pressure ≈ 101 kPa)? Why does a shower curtain blow inward when the shower runs? Answers: P = 101 kPa + (1000)(9.8)(10) ≈ 199 kPa. Fast-moving shower water lowers the pressure inside, so outside air pushes the curtain in (Bernoulli).

Real-world application: Hydraulic brakes, hot-air balloons, submarines (ballast tanks), carburetors, and curveballs (the Magnus effect) all run on fluid physics.

Key vocabulary: Pressure — P = F/A; buoyant force — weight of displaced fluid; Bernoulli's principle — faster flow, lower pressure.

Lesson 16: Rotational Motion and Torque

Rotation has its own versions of the linear quantities: angular displacement θ, angular velocity ω = Δθ/Δt, and angular acceleration α = Δω/Δt. A point at radius r moves with tangential speed v = rω and centripetal acceleration a = rω². The rotational analog of F = ma is τ = Iα, where torque τ = rF sinφ is the twisting effect of a force and moment of inertia I measures resistance to angular acceleration — mass spread far from the axis is harder to spin than mass near the axis.

Angular momentum L = Iω is conserved when no net external torque acts — the reason a figure skater spins faster pulling her arms in (I shrinks, so ω grows). Static equilibrium requires both F_net = 0 and τ_net = 0: bridges, seesaws, and cranes are designed around balanced torques. Levers, wrenches, and doorknobs all trade force for distance through torque: pushing far from the hinge needs less force.

Try it yourself: A seesaw: a 30-kg child sits 2.0 m left of the pivot. Where must a 20-kg child sit on the right to balance? A bicycle wheel's rim speed is 6.0 m/s with radius 0.35 m: find ω. Answers: (30 × 2.0)/20 = 3.0 m right. ω = v/r = 6.0/0.35 ≈ 17.1 rad/s.

Real-world application: Flywheels store energy in buses, gyroscopes stabilize spacecraft and phones, and torque wrenches keep engine bolts at exact tightness.

Key vocabulary: Torque — τ = rF sinφ, the twisting effect; moment of inertia — rotational mass; angular momentum — L = Iω, conserved without external torque.

Lesson 17: DC Circuits and Kirchhoff's Laws

Real circuits mix series and parallel branches, and solving them needs Kirchhoff's laws. The junction rule (charge conservation): the sum of currents entering any junction equals the sum leaving — current is never created or destroyed. The loop rule (energy conservation): the sum of voltage changes around any closed loop is zero — every volt the battery supplies is dropped across the circuit's components. Together they let you write one equation per unknown current and solve any DC circuit.

Two practical quantities matter in every circuit. Internal resistance r inside a real battery means its terminal voltage falls under load: V = ε − Ir, so a car battery reads 12.6 V at rest but less while cranking. Electrical safety devices — fuses, breakers, and ground-fault interrupters — exploit the heating and magnetic effects of current to cut power before wires overheat or people are shocked. Household wiring is parallel (each device gets full voltage), with breakers typically rated 15–20 A.

Try it yourself: Three resistors (2 Ω, 3 Ω, 6 Ω) in parallel across 12 V: find the equivalent resistance and total current. Why does a flashlight dim as its batteries age? Answers: 1/R = 1/2 + 1/3 + 1/6 = 1, R = 1 Ω; I = 12/1 = 12 A. Internal resistance rises as batteries deplete, so more voltage is lost inside the battery.

Real-world application: Phone chargers, electric-vehicle battery packs (hundreds of cells wired in series-parallel), and home breaker panels are all Kirchhoff's laws in daily action.

Key vocabulary: Kirchhoff's junction rule — currents in = currents out; loop rule — voltages sum to zero around a loop; internal resistance — a real battery's own resistance.

Lesson 18: Optics — Mirrors, Lenses, and Instruments

Image formation follows two rules: light travels in straight lines and bends predictably at boundaries. Plane mirrors produce virtual, upright, same-size images as far behind the mirror as the object is in front. Concave mirrors converge light to a focal point and can make magnified real images (makeup mirrors, telescopes); convex mirrors diverge light, giving smaller, wider views (store security mirrors, car side mirrors — "objects are closer than they appear"). The mirror equation, 1/f = 1/d_o + 1/d_i, with magnification m = −d_i/d_o, locates every image.

Lenses obey the same equation. Converging lenses focus light (camera lenses, eyeglasses for farsightedness, magnifying glasses); diverging lenses spread it (eyeglasses for nearsightedness). The eye itself is a converging-lens camera: the cornea and lens focus an inverted real image on the retina, and the brain flips it. Optical instruments chain lenses together — a microscope pairs a short-focus objective with an eyepiece for magnifications of 1000× or more; a telescope gathers faint light with a large aperture. Lens power is measured in diopters, P = 1/f (f in metres).

Try it yourself: An object is 12 cm from a converging lens of f = 8 cm. Find d_i and m. Why are telescope mirrors so large? Answers: 1/d_i = 1/8 − 1/12 = 1/24, d_i = 24 cm; m = −2. Large apertures gather more light (brightness ∝ area) and sharpen resolution.

Real-world application: LASIK reshapes the cornea to change the eye's focal length; fiber-optic endoscopes carry images through the body by total internal reflection.

Key vocabulary: Real vs. virtual image — light actually converges / only appears to; focal length — distance to the focus; diopter — lens power, 1/f in metres.

Lesson 19: Nuclear Physics and Radioactivity

The nucleus holds protons and neutrons bound by the strong nuclear force, which beats the protons' electrical repulsion at tiny distances. Some nuclei are unstable and decay to reach stability. Alpha decay ejects a helium nucleus (2 protons, 2 neutrons), lowering Z by 2. Beta decay turns a neutron into a proton, ejecting an electron (Z rises by 1). Gamma decay releases excess energy as a high-energy photon without changing Z. Each decay is random for a single nucleus but statistically predictable for trillions of them.

Radioactive decay follows an exponential law with a characteristic half-life: the time for half a sample to decay. Carbon-14 (half-life 5,730 years) dates ancient wood and bone; uranium-238 (4.5 billion years) dates rocks and the Earth itself. Fission splits heavy nuclei (uranium-235), releasing energy and neutrons that can sustain a chain reaction — the basis of nuclear power plants and, tragically, atomic bombs. Fusion joins light nuclei (hydrogen into helium), powering the Sun and stars; controlled fusion on Earth remains a grand engineering challenge. Einstein's E = mc² sets the energy scale: a tiny mass defect becomes enormous energy.

Try it yourself: A sample drops from 64 g to 4 g in 24 days. What is its half-life? Why does fusion fuel (hydrogen) produce no long-lived radioactive waste? Answers: 64 → 32 → 16 → 8 → 4 is 4 half-lives in 24 days, so T = 6 days. Fusion's product is stable helium, not radioactive fission fragments.

Real-world application: PET scans use radioactive tracers, smoke detectors use americium-241, and radiotherapy targets cancer cells with focused radiation.

Key vocabulary: Half-life — time for half a sample to decay; fission — splitting a heavy nucleus; fusion — joining light nuclei.

Lesson 20: Astrophysics — Stars, Galaxies, and the Expanding Universe

Stars are born in collapsing gas clouds, shine by fusing hydrogen into helium, and die according to their mass. Sun-like stars swell into red giants, shed their outer layers, and fade as white dwarfs. Massive stars (>8 solar masses) explode as supernovae, leaving neutron stars or black holes — objects so dense that not even light escapes from within their event horizon. The elements heavier than iron in your body — the calcium in your bones, the iron in your blood — were forged only in supernova explosions: we are, literally, stardust.

Light from distant galaxies is redshifted — stretched to longer wavelengths — by the expansion of space itself. Hubble's law, v = H₀d, says a galaxy's recession speed grows with its distance, so the universe is expanding and was once hot and dense: the Big Bang model. Its afterglow, the cosmic microwave background (2.7 K in every direction), is direct evidence. The universe's fate depends on its contents: ordinary matter is only about 5%; dark matter (~27%) holds galaxies together gravitationally, and dark energy (~68%) is accelerating the expansion.

Try it yourself: A galaxy recedes at 21,000 km/s with H₀ = 70 km/s/Mpc. How far away is it? If the universe's expansion is accelerating, what does that imply about dark energy? Answers: d = v/H₀ = 300 Mpc. Dark energy's repulsive effect dominates and grows relative to matter as the universe expands.

Real-world application: GPS, satellite communications, and interplanetary navigation all require relativistic corrections from the physics of this course — from time dilation to orbital mechanics.

Key vocabulary: Supernova — explosive death of a massive star; redshift — wavelength stretched by expansion; Hubble's law — v = H₀d.