Physics Deep Dives

Every topic rebuilt 3Blue1Brown-style: foundations from zero, the derivation in pictures, worked examples for every scenario, graded exercises, and concept-trap question banks.

536 topics · 2677 pages — more land as generation runs

1.1

Measurement, Vectors & Kinematics

22 deep-dive topics

1.1.1Physical quantities — fundamental and derived

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.2SI units — seven base units and all derived units

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.3Dimensional analysis — checking equations, deriving relations

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.4Significant figures — rules for operations

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.5Errors — absolute, relative, percentage; systematic vs random

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.6Scalars vs vectors — definition, examples

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.7Vector representation — magnitude, direction, components

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.8Vector addition — triangle law, parallelogram law

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.9Resolution of vectors — into components (any axes)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.10Unit vectors — î, ĵ, k̂; constructing unit vector

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.11Dot product — formula, geometric meaning, work calculation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.12Cross product — formula, direction (right-hand rule), torque - area calculation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.13Position vector, displacement, distance

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.14Average velocity vs instantaneous velocity

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.15Average acceleration vs instantaneous acceleration

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.16Equations of motion (SUVAT) — derivations from calculus

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.17Free fall — g = 9.8 m - s², sign conventions

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.18Graphs — x-t, v-t, a-t; areas and slopes meaning

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.19Projectile motion — horizontal - vertical independence, full derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.20Range, max height, time of flight — all derived

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.21Relative motion — 1D and 2D; river-boat problems

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.1.22Reference frames — Galilean transformations

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2

Newton's Laws & Dynamics

25 deep-dive topics

1.2.1Newton's first law — inertia, operational definition of force

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.2Newton's second law — F = ma (net force), impulse-momentum form

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.3Newton's third law — action-reaction, common misconceptions

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.4Free body diagrams — systematic drawing technique

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.5Normal force — reaction force, not always = mg

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.6Friction — static (maximum), kinetic, rolling

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.7Coefficients of friction — measurement, material dependence

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.8Angle of friction, angle of repose — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.9Tension in inextensible strings

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.10Atwood machine — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.11Inclined planes — with and without friction

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.12Pulley systems — mechanical advantage

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.13Non-inertial reference frames — pseudo forces

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.14Rotating frames — centrifugal force, Coriolis force

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.15Circular motion — centripetal acceleration derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.16Centripetal force — what provides it in various situations

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.17Banking of roads — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.18Vertical circular motion — minimum speed conditions

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.19Newton's law of gravitation — universal, action at distance

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.20Gravitational field intensity g = GM - r²

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.21Variation of g — with altitude, latitude, depth

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.22Gravitational potential energy — U = −GMm - r (not mgh)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.23Escape velocity — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.24Orbital velocity for circular orbit — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.2.25Weightlessness — true (free fall) vs apparent

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3

Work, Energy & Power

13 deep-dive topics

1.3.1Work — definition, dot product F·d, sign convention

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3.2Work done by variable force — integration

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3.3Work-energy theorem — derivation from Newton's second law

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3.4Kinetic energy — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3.5Potential energy — definition, gravitational (mgh and −GMm - r), elastic (½kx²)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3.6Conservative forces — path-independent work, potential energy defined

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3.7Non-conservative forces — friction, air drag

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3.8Conservation of mechanical energy — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3.9Power — average and instantaneous, units

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3.10Efficiency

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3.11Hooke's law — spring force F = −kx

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3.12Spring potential energy — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.3.13Spring-mass systems — collision problems

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.4

Momentum & Collisions

12 deep-dive topics

1.4.1Linear momentum p = mv

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.4.2Impulse-momentum theorem — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.4.3Conservation of linear momentum — derivation from Newton's third law

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.4.4System with external forces — conditions for conservation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.4.5Elastic collisions — 1D - solve for final velocities

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.4.6Elastic collisions — 2D - angle relationship

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.4.7Perfectly inelastic collisions — maximum KE loss

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.4.8Coefficient of restitution e = (v₂ − v₁) - (u₁ − u₂)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.4.9Centre of mass — definition for system of particles

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.4.10Centre of mass — derivation for common shapes (rod, triangle, semicircle, hemisphere)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.4.11Motion of centre of mass — external force determines a_CM

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.4.12Systems with variable mass — rocket equation derivation preview

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5

Rotational Mechanics

18 deep-dive topics

1.5.1Rigid body — definition, degrees of freedom

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.2Angular displacement θ, angular velocity ω, angular acceleration α

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.3Relation to linear quantities - v = rω, a_t = rα, a_c = rω²

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.4Torque τ = r × F — definition, physical meaning

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.5Moment of inertia I = Σmᵢrᵢ² — concept

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.6Parallel axis theorem — I = I_CM + Md² — proof

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.7Perpendicular axis theorem — I_z = I_x + I_y — proof, restrictions

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.8Moment of inertia of - rod (about end, centre), disk, ring, sphere (solid, hollow), cylinder

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.9Rotational kinetic energy = ½Iω²

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.10Angular momentum L = Iω (fixed axis), L = r × p (general)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.11Torque = dL - dt

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.12Conservation of angular momentum — conditions

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.13Rolling without slipping — v = Rω condition

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.14Rolling KE = ½mv² + ½Iω²

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.15Acceleration of rolling objects on inclines — comparison

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.16Gyroscopic effect — precession of spinning top

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.17Gyroscope in spacecraft attitude control — preview

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.5.18Equilibrium of rigid bodies — translational + rotational

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6

Oscillations & Waves

23 deep-dive topics

1.6.1Simple harmonic motion — definition, restoring force F = −kx

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.2SHM differential equation — solution - x = A cos(ωt + φ)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.3ω, T, f relationships

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.4Velocity and acceleration in SHM — v = ω√(A² − x²)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.5Energy in SHM — KE + PE = ½kA² (constant)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.6Simple pendulum — small angle approximation, T = 2π√(L - g) derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.7Physical pendulum — compound pendulum

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.8Spring-mass system — horizontal, vertical

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.9Damped oscillations — underdamped, critically damped, overdamped

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.10Q factor — quality of oscillator

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.11Forced oscillations — driving frequency

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.12Resonance — physical consequences, design implications

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.13Mechanical waves — transverse and longitudinal

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.14Wave parameters — amplitude, wavelength, frequency, period, wave speed

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.15Wave equation — derivation for string

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.16Superposition principle

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.17Interference — constructive, destructive conditions

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.18Standing waves — formation, nodes, antinodes

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.19Harmonics and overtones — on strings and in pipes

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.20Beats — derivation, applications

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.21Doppler effect — all cases - source moving, observer moving, both

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.22Shock waves — Mach number, Mach cone — - CRITICAL for rockets -

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.6.23Sound intensity — decibels (logarithmic scale)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7

Thermodynamics

26 deep-dive topics

1.7.1Temperature — thermal equilibrium, thermometers, scales

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.2Zeroth law — transitivity of thermal equilibrium

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.3Heat and internal energy — microscopic vs macroscopic

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.4Specific heat capacity — calorimetry

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.5Latent heat — phase transitions

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.6Heat transfer — conduction (Fourier's law k), convection, radiation (Stefan-Boltzmann σT⁴)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.7Thermal expansion — linear, area, volumetric

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.8Ideal gas law PV = nRT — derivation from kinetic theory

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.9Kinetic theory — pressure derivation, temperature as mean KE

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.10Internal energy of ideal gas U = (f - 2)nRT

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.11Mean free path, mean speed, RMS speed — derivations

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.12Maxwell-Boltzmann speed distribution — derivation (key for propulsion)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.13First law of thermodynamics — dU = dQ − dW, sign conventions

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.14Thermodynamic processes — isothermal (T const), isochoric (V const), isobaric (P const), adiabatic (Q = 0)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.15Work done in each process — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.16Adiabatic relations — PV^γ = const, TV^(γ−1) = const (derivation)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.17γ = Cp - Cv — for monatomic, diatomic, polyatomic

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.18Second law — Kelvin-Planck statement, Clausius statement

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.19Heat engines — efficiency η = 1 − Q_C - Q_H

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.20Refrigerators and heat pumps — COP

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.21Carnot cycle — full derivation, efficiency = 1 − T_C - T_H

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.22Entropy — Clausius definition dS = dQ_rev - T

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.23Entropy change in irreversible processes — always - 0

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.24Entropy and disorder — Boltzmann S = k·ln(W)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.25Third law of thermodynamics — S → 0 as T → 0

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.7.26Thermodynamic potentials — U, H, F, G (preview)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8

Electromagnetism

36 deep-dive topics

1.8.1Electric charge — properties, quantization, conservation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.2Coulomb's law — force, comparison with gravity

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.3Superposition principle for forces

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.4Electric field — definition, field lines, superposition

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.5Electric field of point charge, dipole, ring, disk, line charge (Gauss's law)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.6Gauss's law — integral form, choosing Gaussian surfaces

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.7Applications — sphere, cylinder, infinite plane

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.8Electric potential — definition V = −∫E·dl

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.9Potential of point charge, potential from field and vice versa

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.10Equipotential surfaces — perpendicular to field

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.11Capacitance — parallel plate derivation, cylindrical, spherical

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.12Series and parallel capacitors — derivations

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.13Energy stored in capacitor U = ½CV²

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.14Dielectrics — polarization, dielectric constant, effect on capacitance

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.15Drift velocity, mobility, conductivity

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.16Ohm's law — microscopic origin, resistivity

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.17Series and parallel resistance

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.18Kirchhoff's current law (KCL), Kirchhoff's voltage law (KVL)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.19RC circuits — charging, discharging, time constant τ = RC

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.20Magnetic force on charge — F = qv × B

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.21Magnetic force on current-carrying conductor

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.22Biot-Savart law — magnetic field from current element

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.23Ampere's circuital law — magnetostatic form

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.24Magnetic field of straight wire, circular loop, solenoid, toroid

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.25Magnetic flux Φ = ∫B·dA

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.26Faraday's law — EMF = −dΦ - dt

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.27Lenz's law — opposing induced current

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.28Self-inductance L, mutual inductance M

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.29RL circuit — growth and decay of current

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.30LC circuit — oscillations (electrical analog of SHM)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.31Maxwell's equations — integral form, all four

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.32Displacement current — Maxwell's addition to Ampere's law

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.33Electromagnetic waves — derivation from Maxwell's equations

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.34Speed of light c = 1 - √(ε₀ μ₀)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.35EM spectrum — all bands and applications

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

1.8.36Poynting vector — energy flux in EM waves

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1

Analytical Mechanics

25 deep-dive topics

2.1.1Constraints — holonomic vs non-holonomic, rheonomic vs scleronomic

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.2Generalized coordinates — choosing them, degrees of freedom

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.3Kinetic energy in generalized coordinates

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.4Lagrangian L = T − V

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.5Derivation of Euler-Lagrange equations from D'Alembert's principle

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.6Applying E-L equations to various systems

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.7Generalized momenta and generalized forces

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.8Cyclic coordinates — corresponding conservation law

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.9Noether's theorem — symmetry ↔ conservation law

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.10Constraints using Lagrange multipliers

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.11Hamiltonian — definition H = Σpᵢq̇ᵢ − L

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.12Hamilton's equations of motion

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.13Phase space — trajectories, phase portraits

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.14Liouville's theorem — phase space volume conservation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.15Poisson brackets — definition, properties, connection to commutators

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.16Canonical transformations — generating functions

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.17Hamilton-Jacobi equation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.18Action-angle variables — integrable systems

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.19Principle of least action — Hamilton's principle derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.20Normal modes — coupled oscillators, normal coordinates

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.21Rigid body dynamics — Euler angles, Euler's equations of motion

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.22Inertia tensor — principal axes, principal moments

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.23Torque-free rotation — Euler's equations, asymmetric top

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.24Gyroscope — steady precession derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.1.25Chaotic systems — sensitivity to initial conditions, Lyapunov exponents

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2

Fluid Mechanics

30 deep-dive topics

2.2.1Fluid definition — shear stress, no fixed shape

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.2Density, specific gravity

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.3Viscosity — dynamic μ, kinematic ν = μ - ρ; Newtonian vs non-Newtonian

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.4Surface tension — origin, Young-Laplace equation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.5Hydrostatics — pressure = ρgh, derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.6Pascal's law — pressure transmits equally

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.7Buoyancy — Archimedes' principle, derivation from pressure difference

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.8Manometers, barometers

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.9Fluid kinematics — Eulerian vs Lagrangian description

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.10Streamlines, pathlines, streaklines

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.11Stream function, velocity potential

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.12Continuity equation — derivation (conservation of mass), ρAv = const

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.13Reynolds transport theorem

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.14Bernoulli's equation — derivation from F = ma along streamline

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.15Assumptions in Bernoulli — steady, inviscid, incompressible, along streamline

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.16Applications — Pitot tube, Venturi meter, orifice flow

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.17Viscous flow — Poiseuille flow, velocity profile in pipe

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.18Navier-Stokes equations — derivation from Newton's second law for fluid

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.19Reynolds number Re = ρvL - μ — laminar vs turbulent criterion

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.20Boundary layer — Prandtl's concept, growth along flat plate

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.21Boundary layer thickness, displacement thickness, momentum thickness

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.22Blasius solution — exact laminar boundary layer solution

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.23Boundary layer separation — adverse pressure gradient

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.24Drag — pressure (form) drag, skin friction drag

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.25Lift — Kutta-Joukowski theorem L = ρV∞Γ

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.26Dimensional analysis — Buckingham π theorem

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.27Similarity — geometric, kinematic, dynamic; Reynolds similarity

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.28Potential flow — irrotational, inviscid; superposition of basic flows

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.29Vorticity — ω = ∇ × v, circulation Γ

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.2.30Kelvin's circulation theorem

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3

Modern Physics

33 deep-dive topics

2.3.1Blackbody radiation — Planck's quantum hypothesis

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.2Photoelectric effect — Einstein's explanation, work function

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.3Photon properties — E = hf, p = h - λ

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.4Compton scattering — wavelength shift derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.5De Broglie hypothesis — matter waves λ = h - p

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.6Davisson-Germer experiment — electron diffraction

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.7Heisenberg uncertainty principle — Δx Δp ≥ ℏ - 2, ΔE Δt ≥ ℏ - 2

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.8Wave function ψ — probability density - ψ - ²

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.9Schrödinger equation — time-dependent, time-independent

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.10Particle in a box — solving TISE, energy levels, wavefunctions

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.11Quantum tunneling — concept, transmission coefficient

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.12Hydrogen atom — solving in spherical coordinates

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.13Quantum numbers n, l, mₗ, mₛ

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.14Hydrogen energy levels Eₙ = −13.6 - n² eV

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.15Spectral series — Lyman, Balmer, Paschen

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.16Pauli exclusion principle

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.17Spin — intrinsic angular momentum

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.18Nuclear structure — protons, neutrons, nuclear forces

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.19Binding energy — mass defect, BE per nucleon curve

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.20Nuclear reactions — Q-value calculation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.21Radioactive decay — alpha, beta, gamma — mechanisms

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.22Decay law — N = N₀ e^(−λt), half-life, activity

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.23Fission — chain reaction, critical mass

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.24Fusion — solar fusion, tokamak (concept)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.25Special relativity — Michelson-Morley experiment

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.26Postulates of SR

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.27Simultaneity — relativity of simultaneity

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.28Lorentz transformation — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.29Time dilation — derivation, twin paradox

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.30Length contraction — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.31Relativistic momentum p = γmv

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.32Mass-energy equivalence E² = (pc)² + (mc²)²

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.3.33General relativity — equivalence principle, curved spacetime (overview)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4

Thermodynamics & Statistical Mechanics (Advanced)

19 deep-dive topics

2.4.1Thermodynamic potentials — U (internal), H (enthalpy), F (Helmholtz), G (Gibbs)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.2Legendre transforms connecting them

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.3Maxwell relations — derivation from each potential

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.4Gibbs-Helmholtz equation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.5Chemical potential μ = (∂G - ∂N)_{T,P}

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.6Phase equilibrium — Clausius-Clapeyron equation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.7Phase rule — Gibbs phase rule

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.8Statistical mechanics — microstate, macrostate

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.9Boltzmann's entropy S = k_B ln(Ω)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.10Canonical ensemble — partition function Z

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.11Average energy from partition function

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.12Free energy from partition function

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.13Maxwell-Boltzmann distribution — full derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.14Equipartition theorem — ½k_BT per quadratic degree of freedom

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.15Quantum statistics — distinguishable vs indistinguishable particles

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.16Bose-Einstein statistics — bosons

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.17Fermi-Dirac statistics — fermions, Fermi energy

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.18Bose-Einstein condensation — concept

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.4.19Blackbody radiation from statistical mechanics — Planck distribution

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5

Optics

18 deep-dive topics

2.5.1Geometric optics — rectilinear propagation, reflection, refraction

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.2Mirrors — plane, concave, convex; mirror equation 1 - v + 1 - u = 1 - f

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.3Sign convention for mirrors and lenses

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.4Snell's law — derivation from Fermat's principle

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.5Total internal reflection — critical angle derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.6Thin lenses — lens equation, lens maker's equation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.7Power of a lens, combination of lenses

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.8Optical instruments — human eye, simple microscope, compound microscope, telescope

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.9Aberrations — chromatic, spherical (concepts)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.10Huygens' principle — wavefront propagation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.11Young's double slit — fringe width derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.12Thin film interference — reflected and transmitted

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.13Newton's rings — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.14Diffraction — single slit intensity pattern derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.15Diffraction grating — condition for maxima

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.16Resolving power — Rayleigh criterion

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.17Polarization — Malus's law, Brewster's angle derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

2.5.18Birefringence — ordinary and extraordinary rays

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1

Compressible Flow & Aerodynamics

30 deep-dive topics

3.1.1Review of thermodynamics applied to flow — first law for open systems

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.2Stagnation (total) quantities — T₀, P₀, ρ₀ — derivations

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.3Speed of sound — a = √(γRT) — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.4Mach number M = V - a — subsonic ( - 1), transonic (~1), supersonic ( - 1), hypersonic ( - 5)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.5Area-velocity relation — dA - A = (M² − 1)(dV - V) — derivation (explains de Laval nozzle)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.6Area-Mach number relation A - A - = f(M) — isentropic flow

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.7Isentropic flow tables — P - P₀, T - T₀, ρ - ρ₀ as functions of M

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.8Choked flow — condition M = 1 at throat, maximum mass flow

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.9Converging nozzle — subsonic flow, Mach 1 at exit

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.10Converging-diverging (de Laval) nozzle — subsonic, supersonic flow

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.11Normal shock waves — Rankine-Hugoniot relations (all 5) — derivations

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.12Normal shock properties — M₂, P₂ - P₁, T₂ - T₁, ρ₂ - ρ₁, P₀₂ - P₀₁

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.13Oblique shock waves — θ-β-M relation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.14Shock wave angle, deflection angle

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.15Detached bow shock

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.16Prandtl-Meyer expansion waves — isentropic, supersonic turning

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.17Prandtl-Meyer function ν(M)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.18Over - under expanded nozzle flows

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.19Airfoil aerodynamics — camber, chord, thickness

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.20Angle of attack, lift coefficient, drag coefficient

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.21Thin airfoil theory — lift per unit span = πρV²(α + 2β - π c)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.22Finite wing theory — induced drag, Prandtl's lifting line

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.23Aspect ratio — effect on induced drag

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.24Critical Mach number — onset of local supersonic flow

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.25Wave drag — transonic and supersonic

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.26Area rule — Whitcomb's rule for transonic drag reduction

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.27Hypersonic flow — Mach 5+, high temperature effects

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.28Aerodynamic heating — recovery temperature, heat flux

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.29Aerodynamic coefficients — CN, CA, CL, CD, Cm as functions of angle of attack, Mach

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.1.30Computational aerodynamics — panel method (intro), CFD overview

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2

Orbital Mechanics & Astrodynamics

40 deep-dive topics

3.2.1Two-body problem — equations of motion, reduction to one-body

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.2Conservation of energy and angular momentum in gravitational field

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.3Orbit equation r = p - (1 + e·cos θ) — derivation from equations of motion

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.4Orbit shape from eccentricity — circle (e=0), ellipse (0 - e - 1), parabola (e=1), hyperbola (e - 1)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.5Kepler's first law — orbits are conic sections

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.6Kepler's second law — equal areas in equal times, from angular momentum conservation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.7Kepler's third law — T² ∝ a³ — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.8Orbital elements (Keplerian) — semi-major axis a, eccentricity e, inclination i, RAAN Ω, argument of perigee ω, true ano

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.9Physical meaning of each orbital element

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.10Vis-viva equation v² = GM(2 - r − 1 - a) — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.11Specific orbital energy ε = −GM - 2a

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.12Specific angular momentum h = √(GMp)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.13Circular orbit — velocity, period, energy

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.14Kepler's equation M = E − e·sin E — derivation, eccentric anomaly

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.15Solving Kepler's equation — Newton-Raphson iteration

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.16True anomaly from eccentric anomaly

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.17Converting between orbital elements and state vectors (r, v)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.18Orbit determination — Gauss's method, Gibbs method

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.19Hohmann transfer — derivation, minimum energy transfer

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.20Hohmann Δv calculation — both maneuvers

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.21Bi-elliptic transfer — when it wins over Hohmann

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.22Plane change maneuvers — Δv = 2v·sin(Δi - 2)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.23Combined maneuvers — optimal split between plane change and velocity change

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.24Gravity assist (slingshot) — patched conic, v-infinity vectors

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.25Sphere of influence — radius derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.26Patched conic method — interplanetary trajectory design

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.27Pork chop plots — Δv vs launch - arrival date

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.28Lambert's problem — connecting two positions in given time

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.29Gauss's method for Lambert's problem

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.30Lagrange points L1–L5 — derivation, stability

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.31Halo orbits — linearized motion near Lagrange points

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.32Three-body problem — restricted (CR3BP), characteristic equation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.33Orbital perturbations — J2 effect (oblateness), derivation of nodal precession

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.34Atmospheric drag — exponential atmosphere model, orbit decay

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.35Solar radiation pressure

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.36Third-body perturbations

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.37Orbit types — LEO, MEO, GEO, HEO, SSO, Molniya

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.38Groundtrack analysis — swath, revisit

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.39Launch window — phasing with target orbit

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.2.40Rendezvous and proximity operations — Clohessy-Wiltshire equations

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3

Rocket Propulsion

50 deep-dive topics

3.3.1Tsiolkovsky rocket equation — full first-principles derivation from momentum

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.2Δv = v_e · ln(m₀ - m_f) — understanding each term

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.3Mass ratio m₀ - m_f — why it's so critical

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.4Specific impulse Isp = v_e - g₀ — definition, physical meaning, units

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.5Typical Isp values — solid (~260s), LOX - RP1 (~311s), LOX - LH2 (~450s), ion engines (~3000s)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.6Thrust equation F = ṁv_e + (P_e − P_a)A_e — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.7Mass flow rate ṁ and its relation to throat area

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.8Effective exhaust velocity c = v_e + (P_e − P_a)A_e - ṁ

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.9Thrust coefficient C_F = F - (P_c A - ) — derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.10Characteristic velocity c - = P_c A - ṁ — derivation, combustion efficiency measure

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.11Nozzle thermodynamics — isentropic expansion from chamber to exit

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.12Chamber-to-exit relation - all quantities as f(M_e, γ)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.13Optimum expansion — P_e = P_a for maximum thrust

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.14Over-expanded nozzle — oblique shocks in plume, efficiency loss

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.15Under-expanded nozzle — Prandtl-Meyer expansion, efficiency loss

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.16Altitude compensation methods — nozzle extension, aerospike

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.17De Laval nozzle geometry — conical, bell (Rao contour), 80% bell

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.18Nozzle area ratio ε = A_e - A - — choosing for optimal performance

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.19Combustion thermodynamics — stoichiometry, adiabatic flame temperature

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.20Real gas effects — dissociation, recombination

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.21Characteristic velocity c - and its relation to flame temperature, MW

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.22Staged combustion cycle — full flow, fuel-rich, oxidizer-rich preburners

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.23Gas generator cycle — performance penalty vs simplicity

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.24Expander cycle — hydrogen-cooled nozzle drives turbine

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.25Pressure-fed cycle — simplest, used in upper stages

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.26Electric pump-fed cycle — modern innovation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.27Turbopump design — centrifugal pump, axial turbine stages, NPSH

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.28Regenerative cooling — heat flux, coolant flow, pressure drop

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.29Film cooling — effectiveness, coverage fraction

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.30Ablative cooling — charring, blowing

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.31Transpiration cooling

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.32Combustion instability — low-frequency (chugging), high-frequency (screaming)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.33Acoustic modes in combustion chamber — cause of instability

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.34Injector design — impinging, coaxial, swirl injectors

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.35Solid propellants — fuel + oxidizer in polymer matrix

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.36Burn rate r = a·P^n — Vieille's law

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.37Grain geometry — BATES, star, wagon wheel; neutral - progressive - regressive burn

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.38Solid rocket Isp derivation from grain properties

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.39Hybrid engines — advantages, disadvantages

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.40Electric propulsion — thrust, power, Isp trade-off

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.41Ion engine — ionization, acceleration grid, neutralizer

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.42Hall-effect thruster — cross-field discharge, annular channel

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.43FEEP, MEMS thrusters — micro-propulsion

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.44Nuclear thermal propulsion — NTR Isp ~900 s concept

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.45Rocket staging — series staging, parallel staging

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.46Optimal staging — equal mass ratios (for same Isp)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.47Payload fraction as function of Δv and Isp

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.48Propellant properties — density, freezing point, toxicity, storability

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.49Cryogenic propellants — handling, insulation, boil-off

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.3.50Hypergolic propellants — N2O4 - UDMH, MMH

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4

Rocket Flight Mechanics

26 deep-dive topics

3.4.1Coordinate systems — Earth-Centered Inertial (ECI), Earth-Centered Earth-Fixed (ECEF), North-East-Down (NED), launch, bo

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.2Transformation between frames — direction cosine matrices

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.3Forces on a rocket in flight — thrust, aerodynamic (normal, axial), gravity

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.4Equations of motion — 3DOF point mass (trajectory analysis)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.56DOF equations — translational (Newton), rotational (Euler's equations)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.6Mass properties — CG location, inertia tensor changing with propellant depletion

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.7Aerodynamic coefficients — CA (axial force), CN (normal force), Cm (pitching moment)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.8Barrowman equations — centre of pressure calculation for finned rockets

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.9Static margin = (XCP − XCG) - d — must be positive (at least 1 caliber)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.10Static stability — weather-cocking tendency

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.11Dynamic stability — pitch - yaw damping derivatives

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.12Propulsive forces — thrust misalignment, gimbal angle

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.13Gravity turn trajectory — pitch rate from aerodynamic angle of attack = 0

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.14Pitch program — open-loop pitch-over

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.15Trajectory optimization — minimum gravity loss, minimum drag loss

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.16Max-Q — maximum dynamic pressure q = ½ρv²; structural limit

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.17Staging events — separation dynamics, thrust tail-off

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.18Fairing separation — altitude, dynamic pressure requirements

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.19Reentry mechanics — ballistic coefficient β = m - (C_D A)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.20Reentry corridor — angle of attack constraints

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.21Aerodynamic heating during reentry — stagnation point heat flux Chapman equation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.22Thermal protection systems — ablators (PICA, SLA), metallic tiles, RCC

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.23Plasma sheath — communications blackout

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.24Aerocapture — using atmosphere to decelerate into orbit

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.25Aerobraking — gradual orbit lowering using atmospheric drag

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.4.26Terminal landing — propulsive descent, suicide burn

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5

Guidance, Navigation & Control (GNC)

55 deep-dive topics

3.5.1Reference frames — body frame, inertial frame; rotation between them

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.2Euler angles — roll φ, pitch θ, yaw ψ; rotation sequence (3-2-1 convention)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.3Direction cosine matrix (DCM) — construction from Euler angles

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.4DCM kinematics — Ċ = −[ω×]C

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.5Gimbal lock — problem with Euler angles at θ = ±90°

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.6Quaternions — definition q = (q₀, q₁, q₂, q₃), unit quaternion constraint

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.7Quaternion product — Hamilton product

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.8Quaternion rotation formula — rotating vector v by quaternion q

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.9Quaternion kinematics — q̇ = ½ Ξ(q) ω

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.10Converting between DCM, quaternions, Euler angles

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.11Modified Rodrigues parameters — singularity-free, compact

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.12Attitude estimation — triad method (two vector measurements)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.13Inertial navigation — accelerometer measures non-gravitational specific force

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.14Gyroscope — angular velocity measurement, bias, noise

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.15IMU — integrated accelerometer + gyroscope

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.16Mechanization equations — integrating IMU to get position, velocity, attitude

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.17INS error propagation — error state equations

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.18GPS — pseudorange, trilateration, dilution of precision

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.19GNSS — GPS, GLONASS, Galileo, BeiDou

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.20Sensor fusion — complementary filter (simple), Kalman filter (optimal)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.21Kalman filter derivation — predict step, update step

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.22Kalman gain — minimizes trace of covariance

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.23Observability — when KF can estimate state

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.24Extended Kalman Filter (EKF) — linearization, Jacobians

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.25Unscented Kalman Filter (UKF) — sigma points, better for nonlinear

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.26Control system fundamentals — plant, actuator, sensor, controller

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.27Transfer function — Laplace domain, poles and zeros

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.28Block diagram algebra

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.29State-space representation — x' = Ax + Bu, y = Cx + Du

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.30Eigenvalues of A — system modes, stability

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.31Lyapunov stability — Lyapunov function, positive definiteness

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.32Controllability matrix — rank test

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.33Observability matrix — rank test

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.34Pole placement — Ackermann's formula

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.35Linear Quadratic Regulator (LQR) — Riccati equation, optimal gains

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.36LQG — LQR + Kalman filter, separation principle

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.37H∞ control — robust to uncertainty (intro)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.38PID control — proportional, integral, derivative terms

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.39PID tuning — Ziegler-Nichols, loop shaping

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.40Root locus — Evans' method, rules for sketching

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.41Bode plot — magnitude and phase vs frequency

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.42Gain margin, phase margin — stability margins

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.43Nyquist stability criterion — encirclements of −1

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.44Thrust vector control — single-gimbal, dual-gimbal; TVC angles

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.45TVC dynamics — gimbal servo bandwidth, time delay

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.46Reaction control system — thruster selection, plume impingement limits

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.47Attitude control modes — spin stabilization, 3-axis active

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.48Reaction wheels — momentum management, zero-crossing

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.49Control moment gyroscopes (CMG) — high torque, singularity

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.50Proportional navigation guidance — N·V_c·λ̇, derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.51Augmented proportional navigation — gravity compensation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.52Optimal guidance — ZEM - ZEV formulation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.53Powered descent guidance — G-FOLD algorithm (convex optimization)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.54Terminal descent — velocity vector alignment, touchdown constraints

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.5.55Autonomous GNC for reusable rockets — SpaceX approach overview

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6

Spacecraft Structures & Systems Engineering

35 deep-dive topics

3.6.1Structural loads — axial (thrust), bending (wind shear), dynamic (vibration, acoustics, shock)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.2Structural design process — load cases, FOS (factor of safety)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.3Stress and strain — σ = F - A, ε = ΔL - L, Young's modulus E

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.4Hooke's law in 3D — generalized stress-strain (tensor)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.5Yield stress, ultimate stress — material behavior

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.6Buckling — Euler column buckling load derivation

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.7Shell buckling — thin-walled cylinder under axial load

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.8Fatigue — S-N curves, Miner's rule

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.9Fracture mechanics — stress intensity factor K, toughness K_IC

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.10Modal analysis — natural frequencies, mode shapes

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.11Random vibration — PSD, RMS acceleration

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.12Acoustic loads — SPL, octave band analysis

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.13Shock response spectrum (SRS)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.14Thermal analysis — conduction in structures, thermal stress

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.15Composite materials — fiber-matrix, ply properties, laminate theory

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.16Classical laminate theory — ABD matrix

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.17Sandwich structures — face sheets, core

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.18Finite element method — nodes, elements, stiffness matrix

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.19FEM for structures — assembling global stiffness

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.20FEM software — NASTRAN, ABAQUS (concepts and use)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.21Spacecraft bus — structure, power, thermal, ADCS, C&DH, comms, propulsion

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.22Power systems — solar arrays (I-V curve, power tracking), batteries (DoD, cycles), RTG

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.23Thermal control — multi-layer insulation (MLI), heaters, heat pipes, radiators

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.24Mass budgets — dry mass, wet mass, margin

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.25Link budget — path loss, EIRP, G - T, Eb - N0

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.26Systems engineering — V-model, requirements traceability

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.27Requirements — SMART (Specific, Measurable, Achievable, Relevant, Testable)

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.28Verification methods — analysis, test, inspection, demonstration

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.29FMEA — failure mode, effect, severity, detection, RPN

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.30Fault tree analysis (FTA) — top-down, AND - OR gates

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.31Reliability — MTTF, MTBF, exponential failure model

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.32Redundancy — cold standby, hot standby, active redundancy

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.33Environmental testing — thermal vacuum (TVAC), vibration, acoustic, EMC - EMI

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.34Space environment — LEO radiation (SAA, Van Allen), atomic oxygen, MMOD debris

D1 · Foundations D2 · Visual walkthrough D3 · Worked examples D4 · Exercises D5 · Question bank

3.6.35Radiation effects — TID, SEE, displacement damage

D1 · Foundations D5 · Question bank