Science Reference

Physics Equations & Formulas

Classical Mechanics through Modern Physics — with LaTeX syntax and MATLAB commands

Conventions

Vectors are bold ($\mathbf{F}$) or arrow-notation ($\vec{F}$). Scalar magnitudes are italic ($F$). SI units throughout. Time derivatives use dot notation ($\dot{x} = dx/dt$) or Leibniz notation as appropriate.

1. Physical Constants

Constant	Symbol	Value	MATLAB
Speed of light	$c$	$2.998 \times 10^8$ m/s	`c = 2.998e8;`
Gravitational constant	$G$	$6.674 \times 10^{-11}$ N·m²/kg²	`G = 6.674e-11;`
Surface gravity (Earth)	$g$	$9.807$ m/s²	`g = 9.807;`
Boltzmann constant	$k_B$	$1.381 \times 10^{-23}$ J/K	`kB = 1.381e-23;`
Stefan–Boltzmann	$\sigma$	$5.670 \times 10^{-8}$ W/(m²·K⁴)	`sigma = 5.670e-8;`
Planck's constant	$h$	$6.626 \times 10^{-34}$ J·s	`h = 6.626e-34;`
Reduced Planck	$\hbar$	$1.055 \times 10^{-34}$ J·s	`hbar = h/(2*pi);`
Elementary charge	$e$	$1.602 \times 10^{-19}$ C	`e = 1.602e-19;`
Coulomb constant	$k_e$	$8.988 \times 10^9$ N·m²/C²	`ke = 8.988e9;`
Permittivity of free space	$\varepsilon_0$	$8.854 \times 10^{-12}$ F/m	`eps0 = 8.854e-12;`
Permeability of free space	$\mu_0$	$4\pi \times 10^{-7}$ H/m	`mu0 = 4pi1e-7;`
Electron mass	$m_e$	$9.109 \times 10^{-31}$ kg	`me = 9.109e-31;`
Proton mass	$m_p$	$1.673 \times 10^{-27}$ kg	`mp = 1.673e-27;`
Avogadro's number	$N_A$	$6.022 \times 10^{23}$ mol⁻¹	`NA = 6.022e23;`
Gas constant	$R$	$8.314$ J/(mol·K)	`R = 8.314;`
Earth mass	$M_E$	$5.972 \times 10^{24}$ kg	`ME = 5.972e24;`
Earth radius	$R_E$	$6.371 \times 10^6$ m	`RE = 6.371e6;`

2. Kinematics

1D Constant Acceleration

The Big Four Kinematic Equations

$$v = v_0 + at$$

$$x = x_0 + v_0 t + \tfrac{1}{2}at^2$$

$$v^2 = v_0^2 + 2a(x - x_0)$$

$$x = x_0 + \tfrac{1}{2}(v_0 + v)t$$

LaTeX: v = v_0 + at \qquad x = x_0 + v_0 t + \tfrac{1}{2}at^2

v0 = 0; a = 9.81; t = 5;          % free fall from rest
v = v0 + a*t;                      % 49.05 m/s
x = v0*t + 0.5*a*t^2;             % 122.6 m

Projectile Motion

Projectile Equations (no air resistance)

$$x(t) = v_0 \cos\theta \cdot t \qquad y(t) = v_0 \sin\theta \cdot t - \tfrac{1}{2}gt^2$$

$$R = \frac{v_0^2 \sin 2\theta}{g} \qquad h_{\max} = \frac{v_0^2 \sin^2\theta}{2g} \qquad t_{\text{flight}} = \frac{2v_0\sin\theta}{g}$$

v0 = 30; theta = deg2rad(45); g = 9.81;
R = v0^2*sin(2*theta)/g;           % 91.74 m range
h_max = v0^2*sin(theta)^2/(2*g);   % 22.94 m max height
t_flight = 2*v0*sin(theta)/g;      % 4.33 s total flight

Circular Motion

Uniform Circular Motion

$$a_c = \frac{v^2}{r} = \omega^2 r \qquad v = \omega r \qquad T = \frac{2\pi}{\omega} = \frac{2\pi r}{v}$$

r = 5; v = 10;
ac = v^2/r;                        % 20 m/s² centripetal
omega = v/r;                       % 2 rad/s
T = 2*pi/omega;                    % 3.14 s period

Relative Motion

Galilean Velocity Addition

$$\mathbf{v}_{AC} = \mathbf{v}_{AB} + \mathbf{v}_{BC}$$

Velocity of A relative to C = velocity of A relative to B + velocity of B relative to C.

3. Newton's Laws & Forces

Newton's Three Laws

$$\text{1st: } \sum \mathbf{F} = 0 \;\Rightarrow\; \mathbf{a} = 0 \qquad \text{(inertia)}$$

$$\text{2nd: } \mathbf{F}_{\text{net}} = m\mathbf{a} \qquad \text{(fundamental)}$$

$$\text{3rd: } \mathbf{F}_{AB} = -\mathbf{F}_{BA} \qquad \text{(action–reaction)}$$

m = 10; a = 3;
F = m*a;                           % 30 N

Weight

$$W = mg$$

m = 75; g = 9.81;
W = m*g;                           % 735.75 N

Friction

$$f_s \leq \mu_s N \qquad f_k = \mu_k N$$

$f_s$ = static friction (maximum), $f_k$ = kinetic friction, $N$ = normal force, $\mu$ = coefficient.

mu_s = 0.6; mu_k = 0.4; N = 100;
f_s_max = mu_s * N;                % 60 N (max before sliding)
f_k = mu_k * N;                    % 40 N (while sliding)

Spring Force (Hooke's Law)

$$F = -kx$$

$k$ = spring constant (N/m), $x$ = displacement from equilibrium. Negative = restoring.

k = 200; x = 0.05;
F = -k*x;                          % -10 N (restoring)

Drag Force

$$F_D = \frac{1}{2}C_D \rho A v^2 \qquad v_{\text{terminal}} = \sqrt{\frac{2mg}{C_D \rho A}}$$

Cd = 0.47; rho = 1.225; A = 0.01; v = 30;   % sphere in air
Fd = 0.5*Cd*rho*A*v^2;             % 2.59 N

Inclined Plane

$$F_\parallel = mg\sin\theta \qquad F_\perp = N = mg\cos\theta \qquad a = g(\sin\theta - \mu_k\cos\theta)$$

m = 5; g = 9.81; theta = deg2rad(30); mu_k = 0.2;
a = g*(sin(theta) - mu_k*cos(theta));   % 3.21 m/s²

4. Work, Energy & Power

Work

$$W = \mathbf{F} \cdot \mathbf{d} = Fd\cos\theta \qquad W = \int_{x_1}^{x_2} F(x)\,dx$$

F = 50; d = 10; theta = deg2rad(30);
W = F*d*cos(theta);                 % 433 J

Kinetic & Potential Energy

$$KE = \frac{1}{2}mv^2 \qquad PE_{\text{grav}} = mgh \qquad PE_{\text{spring}} = \frac{1}{2}kx^2$$

m = 2; v = 10; h = 5; g = 9.81; k = 500; x = 0.1;
KE = 0.5*m*v^2;                    % 100 J
PE_grav = m*g*h;                    % 98.1 J
PE_spring = 0.5*k*x^2;             % 2.5 J

Work–Energy Theorem

$$W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$

Conservation of Mechanical Energy

$$KE_i + PE_i = KE_f + PE_f \qquad \text{(no non-conservative forces)}$$

$$\frac{1}{2}mv_i^2 + mgh_i = \frac{1}{2}mv_f^2 + mgh_f$$

% Ball dropped from 20 m — find speed at ground
g = 9.81; h = 20; vi = 0;
vf = sqrt(vi^2 + 2*g*h);           % 19.8 m/s

Power

$$P = \frac{W}{t} = \frac{dW}{dt} = \mathbf{F} \cdot \mathbf{v} = Fv\cos\theta$$

W = 5000; t = 10;
P = W/t;                           % 500 W
% or: force times velocity
F = 200; v = 2.5;
P2 = F*v;                          % 500 W

5. Momentum & Collisions

Linear Momentum & Impulse

$$\mathbf{p} = m\mathbf{v} \qquad \mathbf{J} = \mathbf{F}\Delta t = \Delta \mathbf{p} \qquad \mathbf{F} = \frac{d\mathbf{p}}{dt}$$

m = 0.145; v = 40;                  % baseball
p = m*v;                            % 5.8 kg·m/s
F = 5000; dt = p/F;                 % 1.16 ms contact time

Conservation of Momentum

$$m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$$

Collision Types

Type	Momentum	Kinetic Energy	Formula
Elastic	Conserved	Conserved	Both $p$ and $KE$ equations
Inelastic	Conserved	Not conserved	$p$ equation only
Perfectly inelastic	Conserved	Max loss	$v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}$

% Perfectly inelastic: 2 kg at 5 m/s hits 3 kg at rest
m1 = 2; v1 = 5; m2 = 3; v2 = 0;
vf = (m1*v1 + m2*v2)/(m1+m2);      % 2 m/s
KE_lost = 0.5*m1*v1^2 - 0.5*(m1+m2)*vf^2;  % 15 J lost

Elastic Collision — 1D Closed-Form

$$v_{1f} = \frac{m_1 - m_2}{m_1 + m_2}v_{1i} + \frac{2m_2}{m_1 + m_2}v_{2i}$$

$$v_{2f} = \frac{2m_1}{m_1 + m_2}v_{1i} + \frac{m_2 - m_1}{m_1 + m_2}v_{2i}$$

m1 = 2; m2 = 3; v1i = 6; v2i = 0;
v1f = ((m1-m2)*v1i + 2*m2*v2i)/(m1+m2);   % -1.2 m/s (bounces back)
v2f = (2*m1*v1i + (m2-m1)*v2i)/(m1+m2);   % 4.8 m/s

Center of Mass

$$x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i} = \frac{m_1 x_1 + m_2 x_2 + \cdots}{m_1 + m_2 + \cdots}$$

m = [2 5 3]; x = [1 4 7];
x_cm = sum(m.*x)/sum(m);           % 4.3 m

6. Rotational Mechanics

Rotational Kinematics

$$\omega = \omega_0 + \alpha t \qquad \theta = \omega_0 t + \tfrac{1}{2}\alpha t^2 \qquad \omega^2 = \omega_0^2 + 2\alpha\theta$$

Direct analogs of translational kinematics with $\theta \leftrightarrow x$, $\omega \leftrightarrow v$, $\alpha \leftrightarrow a$.

Torque

$$\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} \qquad \tau = rF\sin\theta \qquad \tau_{\text{net}} = I\alpha$$

r = 0.5; F = 100; theta = deg2rad(90);
tau = r*F*sin(theta);               % 50 N·m

Moment of Inertia — Common Shapes

Shape	Axis	$I$
Point mass	At distance $r$	$mr^2$
Solid cylinder/disk	Central axis	$\frac{1}{2}mR^2$
Hollow cylinder	Central axis	$mR^2$
Solid sphere	Through center	$\frac{2}{5}mR^2$
Hollow sphere	Through center	$\frac{2}{3}mR^2$
Thin rod (center)	Perpendicular, center	$\frac{1}{12}mL^2$
Thin rod (end)	Perpendicular, end	$\frac{1}{3}mL^2$

Parallel Axis Theorem

$$I = I_{\text{cm}} + md^2$$

Rotational Kinetic Energy & Angular Momentum

$$KE_{\text{rot}} = \frac{1}{2}I\omega^2 \qquad L = I\omega \qquad \tau = \frac{dL}{dt}$$

Rolling without slipping: $KE_{\text{total}} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$, where $v = R\omega$.

m = 5; R = 0.2; v = 4;             % solid sphere rolling
I = (2/5)*m*R^2;
omega = v/R;
KE_trans = 0.5*m*v^2;              % 40 J
KE_rot = 0.5*I*omega^2;            % 16 J
KE_total = KE_trans + KE_rot;      % 56 J

7. Gravitation

Newton's Law of Universal Gravitation

$$F = G\frac{m_1 m_2}{r^2} \qquad g = \frac{GM}{R^2}$$

G = 6.674e-11; ME = 5.972e24; RE = 6.371e6;
g = G*ME/RE^2;                      % 9.82 m/s² (surface gravity)

Gravitational Potential Energy

$$U = -\frac{Gm_1 m_2}{r} \qquad \text{(near surface: } U \approx mgh\text{)}$$

Orbital Mechanics

$$v_{\text{orbit}} = \sqrt{\frac{GM}{r}} \qquad T = 2\pi\sqrt{\frac{r^3}{GM}} \qquad v_{\text{escape}} = \sqrt{\frac{2GM}{r}}$$

G = 6.674e-11; ME = 5.972e24; r = 6.371e6 + 400e3;  % ISS ~400 km
v_orbit = sqrt(G*ME/r);             % 7,670 m/s
T = 2*pi*sqrt(r^3/(G*ME));          % 5,558 s = 92.6 min
v_esc = sqrt(2*G*ME/r);             % 10,848 m/s

Kepler's Third Law

$$T^2 = \frac{4\pi^2}{GM}r^3 \qquad \frac{T_1^2}{r_1^3} = \frac{T_2^2}{r_2^3}$$

8. Oscillations & Waves

Simple Harmonic Motion (SHM)

$$x(t) = A\cos(\omega t + \phi) \qquad v(t) = -A\omega\sin(\omega t + \phi) \qquad a(t) = -\omega^2 x$$

$$\omega = 2\pi f = \frac{2\pi}{T} \qquad v_{\max} = A\omega \qquad a_{\max} = A\omega^2$$

Mass–Spring System & Simple Pendulum

$$T_{\text{spring}} = 2\pi\sqrt{\frac{m}{k}} \qquad T_{\text{pendulum}} = 2\pi\sqrt{\frac{L}{g}}$$

m = 0.5; k = 200;
T_spring = 2*pi*sqrt(m/k);          % 0.314 s
L = 1; g = 9.81;
T_pend = 2*pi*sqrt(L/g);            % 2.006 s

Damped Oscillation

$$x(t) = A_0 e^{-\gamma t}\cos(\omega_d t + \phi) \qquad \omega_d = \sqrt{\omega_0^2 - \gamma^2} \qquad \gamma = \frac{b}{2m}$$

Wave Equation & Properties

$$v = f\lambda \qquad y(x,t) = A\sin(kx - \omega t) \qquad k = \frac{2\pi}{\lambda}$$

f = 440; v = 343;                   % A4 note in air
lambda = v/f;                        % 0.78 m wavelength

Wave on a String

$$v = \sqrt{\frac{T}{\mu}} \qquad \mu = \frac{m}{L}$$

$T$ = tension (N), $\mu$ = linear mass density (kg/m).

Standing Waves

$$f_n = \frac{nv}{2L} \quad \text{(both ends fixed)} \qquad f_n = \frac{nv}{4L} \quad \text{(one end open, } n = 1,3,5\ldots\text{)}$$

v = 343; L = 0.65;                   % guitar string
f = (1:6)*v/(2*L);                   % first 6 harmonics

Sound Intensity & Decibels

$$\beta = 10\log_{10}\!\left(\frac{I}{I_0}\right) \qquad I_0 = 10^{-12} \text{ W/m}^2$$

I = 1e-5;                            % W/m²
beta = 10*log10(I/1e-12);           % 70 dB

Doppler Effect

$$f' = f\frac{v \pm v_{\text{observer}}}{v \mp v_{\text{source}}}$$

Upper signs: approaching. Lower signs: receding. $v$ = speed of sound.

f = 500; v = 343; vs = 30;          % source approaching
f_obs = f*(v)/(v - vs);              % 548 Hz (higher pitch)

Beats

$$f_{\text{beat}} = |f_1 - f_2|$$

9. Fluid Mechanics

Pressure & Hydrostatic Pressure

$$P = \frac{F}{A} \qquad P = P_0 + \rho g h$$

rho = 1000; g = 9.81; h = 10;
P = 101325 + rho*g*h;               % 199,425 Pa at 10m depth

Pascal's Principle

$$\frac{F_1}{A_1} = \frac{F_2}{A_2} \qquad \text{(hydraulic press)}$$

Archimedes' Principle (Buoyancy)

$$F_B = \rho_{\text{fluid}} V_{\text{displaced}} g$$

rho_water = 1000; V = 0.01; g = 9.81;
Fb = rho_water*V*g;                  % 98.1 N

Continuity Equation & Bernoulli's Equation

$$A_1 v_1 = A_2 v_2 \qquad \text{(continuity)}$$

$$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant} \qquad \text{(Bernoulli)}$$

% Bernoulli: find velocity from pressure drop
P1 = 200000; P2 = 150000; rho = 1000;   % Pa
v2 = sqrt(2*(P1-P2)/rho);                % 10.0 m/s

Viscous Flow (Poiseuille's Law)

$$Q = \frac{\pi r^4 \Delta P}{8\eta L}$$

$Q$ = volume flow rate, $r$ = pipe radius, $\eta$ = dynamic viscosity, $L$ = pipe length.

10. Thermodynamics

Heat Transfer

$$Q = mc\Delta T \qquad Q = mL \;\text{(phase change)}$$

$c$ = specific heat, $L$ = latent heat (fusion or vaporization).

m = 0.5; c = 4184; dT = 80;         % heat 500g water by 80°C
Q = m*c*dT;                          % 167,360 J
% Phase change: melt ice
L_fus = 334000;                      % J/kg for water
Q_melt = m*L_fus;                    % 167,000 J

Ideal Gas Law & Kinetic Theory

$$PV = nRT = Nk_BT \qquad KE_{\text{avg}} = \frac{3}{2}k_BT \qquad v_{\text{rms}} = \sqrt{\frac{3k_BT}{m}}$$

Laws of Thermodynamics

$$\text{1st: } \Delta U = Q - W \qquad \text{(energy conservation)}$$

$$\text{2nd: } \Delta S_{\text{universe}} \geq 0 \qquad S = k_B \ln \Omega$$

$$\text{3rd: } S \to 0 \text{ as } T \to 0 \text{ K}$$

Thermodynamic Processes (Ideal Gas)

Process	Constant	Work $W$	Heat $Q$
Isothermal	$T$	$nRT\ln(V_f/V_i)$	$W$
Isobaric	$P$	$P\Delta V$	$nC_P\Delta T$
Isochoric	$V$	$0$	$nC_V\Delta T$
Adiabatic	$Q = 0$	$-\Delta U$	$0$

Adiabatic Process

$$PV^\gamma = \text{const} \qquad TV^{\gamma-1} = \text{const} \qquad \gamma = \frac{C_P}{C_V}$$

Monatomic: $\gamma = 5/3$. Diatomic: $\gamma = 7/5$.

Heat Engine Efficiency & Carnot

$$\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H} \qquad \eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$$

TH = 600; TC = 300;                  % Kelvin
eta_carnot = 1 - TC/TH;             % 0.50 = 50% max possible

Heat Transfer Mechanisms

$$\text{Conduction: } \frac{Q}{t} = \frac{kA\Delta T}{L} \qquad \text{Radiation: } P = \varepsilon\sigma AT^4$$

% Radiation from a blackbody
sigma = 5.670e-8; T = 5778;          % Sun surface temp
P_per_m2 = sigma * T^4;              % 6.32e7 W/m² (solar luminosity/area)

11. Electrostatics & Magnetism

Coulomb's Law & Electric Field

$$F = k_e\frac{q_1 q_2}{r^2} \qquad \mathbf{E} = k_e\frac{q}{r^2}\hat{r} \qquad \mathbf{F} = q\mathbf{E}$$

Electric Potential & Potential Energy

$$V = k_e\frac{q}{r} \qquad U = k_e\frac{q_1 q_2}{r} \qquad V = -\int \mathbf{E}\cdot d\mathbf{l} \qquad E = -\frac{dV}{dx}$$

ke = 8.988e9; q = 1e-6; r = 0.5;
V = ke*q/r;                          % 17,976 V

Capacitance & Energy

$$C = \frac{Q}{V} = \frac{\varepsilon_0 A}{d} \qquad U = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$$

Ohm's Law & Resistivity

$$V = IR \qquad R = \frac{\rho L}{A} \qquad P = IV = I^2R = \frac{V^2}{R}$$

rho = 1.68e-8; L = 100; A = 1e-6;   % copper wire
R = rho*L/A;                         % 1.68 Ω

Magnetic Force

$$\mathbf{F} = q\mathbf{v} \times \mathbf{B} \qquad F = qvB\sin\theta$$

$$\text{On a wire: } F = I L B\sin\theta$$

Magnetic Field Sources

$$\text{Long wire: } B = \frac{\mu_0 I}{2\pi r} \qquad \text{Solenoid: } B = \mu_0 n I \qquad \text{Loop center: } B = \frac{\mu_0 I}{2R}$$

mu0 = 4*pi*1e-7; I = 10; r = 0.05;
B_wire = mu0*I/(2*pi*r);             % 4.0e-5 T

Faraday's & Lenz's Law

$$\mathcal{E} = -\frac{d\Phi_B}{dt} = -N\frac{d(BA\cos\theta)}{dt}$$

Electromagnetic Wave

$$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = f\lambda \qquad E = cB \qquad I = \frac{1}{2}c\varepsilon_0 E_0^2$$

12. Optics

Snell's Law (Refraction)

$$n_1\sin\theta_1 = n_2\sin\theta_2 \qquad n = \frac{c}{v}$$

n1 = 1.0; theta1 = deg2rad(45); n2 = 1.5;  % air → glass
theta2 = asin(n1*sin(theta1)/n2);            % 28.1°

Total Internal Reflection

$$\theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right) \qquad \text{(only when } n_1 > n_2\text{)}$$

n1 = 1.5; n2 = 1.0;
theta_c = rad2deg(asin(n2/n1));      % 41.8° critical angle

Thin Lens & Mirror Equation

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \qquad m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}$$

Sign conventions: real image $d_i > 0$, virtual $d_i < 0$. Converging lens $f > 0$, diverging $f < 0$.

f = 0.10; do = 0.25;                 % 10 cm lens, object at 25 cm
di = 1/(1/f - 1/do);                 % 0.167 m = 16.7 cm
m = -di/do;                           % -0.667 (inverted, smaller)

Lensmaker's Equation

$$\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

Double-Slit Interference (Young's)

$$d\sin\theta = m\lambda \;\text{(maxima)} \qquad d\sin\theta = \left(m + \tfrac{1}{2}\right)\lambda \;\text{(minima)}$$

$$y_m = \frac{m\lambda L}{d} \qquad \text{(fringe position on screen)}$$

lambda = 550e-9; d = 0.1e-3; L = 2;  % green light, 0.1mm slits, 2m screen
y1 = 1*lambda*L/d;                     % 0.011 m = 1.1 cm spacing

Single-Slit Diffraction

$$a\sin\theta = m\lambda \quad \text{(minima, } m = \pm 1, \pm 2, \ldots\text{)}$$

Diffraction Grating

$$d\sin\theta = m\lambda \qquad R = mN \;\text{(resolving power)}$$

Brewster's Angle (Polarization)

$$\tan\theta_B = \frac{n_2}{n_1}$$

13. Modern Physics & Relativity

Special Relativity

Lorentz Factor

$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}} \qquad \beta = \frac{v}{c}$$

c = 2.998e8; v = 0.9*c;
gamma = 1/sqrt(1 - (v/c)^2);        % 2.294

Time Dilation & Length Contraction

$$\Delta t = \gamma \Delta t_0 \qquad L = \frac{L_0}{\gamma}$$

$\Delta t_0$ = proper time (in rest frame). $L_0$ = proper length (in rest frame). Moving clocks run slow; moving objects are shortened.

dt0 = 1; gamma = 2.294;
dt = gamma*dt0;                       % 2.294 s observed
L0 = 100;
L = L0/gamma;                         % 43.6 m observed length

Relativistic Energy & Momentum

$$E = \gamma mc^2 \qquad E_0 = mc^2 \qquad KE = (\gamma - 1)mc^2$$

$$p = \gamma mv \qquad E^2 = (pc)^2 + (mc^2)^2$$

m = 9.109e-31; v = 0.9*2.998e8; c = 2.998e8;
gamma = 1/sqrt(1-(v/c)^2);
E = gamma*m*c^2;                      % 1.88e-13 J
E0 = m*c^2;                           % 8.19e-14 J (rest energy)
KE = (gamma-1)*m*c^2;                 % 1.06e-13 J

Relativistic Velocity Addition

$$u' = \frac{u + v}{1 + uv/c^2}$$

Replaces Galilean addition at high speeds. Ensures $u' \leq c$ always.

Quantum Mechanics

Photon Energy & Momentum

$$E = h\nu = \frac{hc}{\lambda} \qquad p = \frac{h}{\lambda} = \frac{E}{c}$$

Photoelectric Effect

$$KE_{\max} = h\nu - \phi = eV_0$$

$\phi$ = work function (minimum energy to eject electron). $V_0$ = stopping voltage.

h = 6.626e-34; c = 2.998e8; e = 1.602e-19;
lambda = 250e-9;                       % UV light
phi = 4.3*e;                           % sodium work function (4.3 eV)
KE_max = h*c/lambda - phi;             % 1.07e-19 J = 0.67 eV

de Broglie Wavelength

$$\lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2mKE}}$$

Heisenberg Uncertainty Principle

$$\Delta x \cdot \Delta p \geq \frac{\hbar}{2} \qquad \Delta E \cdot \Delta t \geq \frac{\hbar}{2}$$

Schrödinger Equation

$$i\hbar\frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi \qquad \text{(time-dependent)}$$

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\psi = E\psi \qquad \text{(time-independent, 1D)}$$

LaTeX: i\hbar\frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi

Particle in a Box (Infinite Square Well)

$$E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} = \frac{n^2 h^2}{8mL^2} \qquad n = 1, 2, 3, \ldots$$

$$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right)$$

h = 6.626e-34; me = 9.109e-31; L = 1e-9;  % 1 nm box
n = 1:5;
E = (n.^2 * h^2) / (8*me*L^2);            % energy levels (J)
E_eV = E / 1.602e-19;                       % [0.38, 1.50, 3.38, 6.01, 9.40] eV

Nuclear & Particle Physics

Mass–Energy Equivalence

$$E = \Delta m \cdot c^2 \qquad 1 \text{ amu} = 931.5 \text{ MeV/c}^2$$

Radioactive Decay

$$N(t) = N_0 e^{-\lambda t} \qquad \lambda = \frac{\ln 2}{t_{1/2}} \qquad A = \lambda N$$

Compton Scattering

$$\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta) = \lambda_C(1 - \cos\theta)$$

Compton wavelength: $\lambda_C = h/(m_e c) = 2.426 \times 10^{-12}$ m.

h = 6.626e-34; me = 9.109e-31; c = 2.998e8;
lambda_C = h/(me*c);                   % 2.426e-12 m
theta = deg2rad(90);
d_lambda = lambda_C*(1 - cos(theta));  % 2.426e-12 m shift

Wien's Displacement Law & Planck's Law

$$\lambda_{\max} T = 2.898 \times 10^{-3} \text{ m·K}$$

$$B(\lambda, T) = \frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/(\lambda k_B T)} - 1}$$

T_sun = 5778;
lambda_max = 2.898e-3 / T_sun;        % 501 nm (green-yellow peak)

Connection to your study path

This reference spans the physics you'll encounter from introductory mechanics through the first courses in modern physics. The equations connect directly to your learning path diagram:

Chapters 2–9 = Physics 1 & 2 (Classical Mechanics, Waves, Fluids, Thermodynamics)

Chapter 11 = Physics 2 + E&M Theory (Electrostatics, Magnetism, Maxwell's equations)

Chapter 12 = Physics 2 (Optics)

Chapter 13 = Modern Physics (Relativity, Quantum Mechanics — prerequisites for your Kalman filter work via state-space models and for the unified physics path via QFT and General Relativity)