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# Physics Equations and Formulas ## Mechanics ### Motion * Average Velocity: $\bar{v} = \frac{\Delta x}{\Delta t}$ * Average Acceleration: $\bar{a} = \frac{\Delta v}{\Delta t}$ * Instantaneous Velocity: $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$ * Instantaneous Acceleration: $a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t}$ * Uniformly Accelerated Motion: * $v = v_0 + at$ * $x = x_0 + v_0t + \frac{1}{2}at^2$ * $v^2 = v_0^2 + 2a(x - x_0)$ * $x - x_0 = \frac{1}{2}(v + v_0)t$ ### Force and Newton's Laws * Newton's First Law: An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by a force. * Newton's Second Law: $F = ma$ * Newton's Third Law: Forces occur in pairs: $F_{AB} = -F_{BA}$ * Weight: $W = mg$ * Friction: * Static: $f_s \le \mu_s N$ * Kinetic: $f_k = \mu_k N$ ### Energy and Work * Work: $W = F \cdot d = Fd\cos\theta$ * Kinetic Energy: $K = \frac{1}{2}mv^2$ * Potential Energy: * Gravitational: $U = mgh$ * Elastic: $U = \frac{1}{2}kx^2$ * Work-Energy Theorem: $W_{net} = \Delta K$ * Power: $P = \frac{dW}{dt} = F \cdot v$ * Conservative Forces: $\Delta U = -W_c$ ### Momentum and Impulse * Momentum: $p = mv$ * Impulse: $J = \int F dt = \Delta p$ * Conservation of Momentum: $m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$ * Elastic Collision: Kinetic energy is conserved. * Inelastic Collision: Kinetic energy is not conserved. * Perfectly Inelastic Collision: Objects stick together after collision. ### Circular Motion * Angular Velocity: $\omega = \frac{d\theta}{dt}$ * Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ * Tangential Velocity: $v = r\omega$ * Tangential Acceleration: $a_t = r\alpha$ * Centripetal Acceleration: $a_c = \frac{v^2}{r} = r\omega^2$ * Centripetal Force: $F_c = ma_c = \frac{mv^2}{r}$ ### Rotational Motion * Torque: $\tau = rF\sin\theta = r_\perp F$ * Moment of Inertia: $I = \sum m_i r_i^2$ * Newton's Second Law for Rotation: $\tau_{net} = I\alpha$ * Rotational Kinetic Energy: $K = \frac{1}{2}I\omega^2$ * Angular Momentum: $L = I\omega = r \times p$ * Conservation of Angular Momentum: $I_i\omega_i = I_f\omega_f$ ### Simple Harmonic Motion * Hooke's Law: $F = -kx$ * Period of a Spring-Mass System: $T = 2\pi\sqrt{\frac{m}{k}}$ * Period of a Simple Pendulum: $T = 2\pi\sqrt{\frac{L}{g}}$ * Displacement: $x(t) = A\cos(\omega t + \phi)$ * Velocity: $v(t) = -A\omega\sin(\omega t + \phi)$ * Acceleration: $a(t) = -A\omega^2\cos(\omega t + \phi)$ ### Gravity * Newton's Law of Universal Gravitation: $F = G\frac{m_1m_2}{r^2}$ * Gravitational Potential Energy: $U = -\frac{Gm_1m_2}{r}$ * Orbital Speed: $v = \sqrt{\frac{GM}{r}}$ * Escape Speed: $v_e= \sqrt{\frac{2GM}{R}}$ ## Thermodynamics ### Temperature and Heat * Celsius to Fahrenheit: $T_F = \frac{9}{5}T_C + 32$ * Celsius to Kelvin: $T_K = T_C + 273.15$ * Heat Transfer: $Q = mc\Delta T$ * Latent Heat: $Q = mL$ ### Ideal Gas Law * Ideal Gas Law: $PV = nRT$ * Boltzmann Constant: $k_B = \frac{R}{N_A}$ * Average Kinetic Energy: $K_{avg} = \frac{3}{2}k_BT$ * Root Mean Square Speed: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3k_BT}{m}}$ ### Thermodynamics Laws * First Law: $\Delta U = Q - W$ * Work Done by a Gas: $W = \int PdV$ * Adiabatic Process: $PV^\gamma = \text{constant}$ * Carnot Efficiency: $\eta = 1 - \frac{T_c}{T_h}$ ## Waves and Optics ### Wave Motion * Wave Speed: $v = \lambda f$ * Superposition: $y = y_1 + y_2$ * Interference: Constructive/Destructive ### Sound * Speed of Sound: $v = \sqrt{\frac{B}{\rho}}$ * Intensity: $I = \frac{P}{A}$ * Sound Level (dB): $\beta = 10\log_{10}(\frac{I}{I_0})$ * Doppler Effect: * $f' = f(\frac{v \pm v_o}{v \mp v_s})$ ### Light * Index of Refraction: $n = \frac{c}{v}$ * Snell's Law: $n_1\sin\theta_1 = n_2\sin\theta_2$ * Brewster's Angle: $\theta_b = \arctan{\frac{n_2}{n_1}}$ * Thin Lens Equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$ ## Electricity and Magnetism ### Electrostatics * Coulomb's Law: $F = k\frac{|q_1q_2|}{r^2}$ * Electric Field: $E = \frac{F}{q}$ * Electric Potential: $V = \frac{U}{q}$ * Potential Energy: $U = qV$ * Capacitance: $C = \frac{Q}{V}$ * Energy of a Capacitor: $U = \frac{1}{2}CV^2$ ### Electric Circuits * Ohm's Law: $V = IR$ * Resistance: $R = \rho\frac{L}{A}$ * Power: $P = IV = I^2R = \frac{V^2}{R}$ * Series Resistors: $R_{eq} = R_1 + R_2 + \dots$ * Parallel Resistors: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots$ * Series Capacitors: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$ * Parallel Capacitors: $C_{eq} = C_1 + C_2 + \dots$ ### Magnetism * Magnetic Force on a Moving Charge: $F = qvB\sin\theta$ * Magnetic Force on a Current-Carrying Wire: $F = ILB\sin\theta$ * Magnetic Field of a Long Straight Wire: $B = \frac{\mu_0I}{2\pi r}$ * Faraday's Law of Induction: $\varepsilon = -N\frac{d\Phi}{dt}$ ## Modern Physics ### Quantum Mechanics * Photon Energy: $E = hf$ * De Broglie Wavelength: $\lambda = \frac{h}{p}$ * Heisenberg Uncertainty Principle: $\Delta x \Delta p \ge \frac{\hbar}{2}$ ### Relativity * Time Dilation: $\Delta t = \gamma \Delta t_0$ * Length Contraction: $L = \frac{L_0}{\gamma}$ * Relativistic Momentum: $p = \gamma mv$ * Relativistic Energy: $E = \gamma mc^2$ * Mass-Energy Equivalence: $E = mc^2$ ### Nuclear Physics * Radioactive Decay Law: $N(t) = N_0e^{-\lambda t}$ * Half-Life: $T_{1/2} = \frac{\ln 2}{\lambda}$ ## Constants * Gravitational Constant: $G = 6.674 \times 10^{-11} \, \text{N}\cdot\text{m}^2/\text{kg}^2$ * Speed of Light: $c = 3.00 \times 10^8 \, \text{m/s}$ * Elementary Charge: $e = 1.602 \times 10^{-19} \, \text{C}$ * Planck's Constant: $h = 6.626 \times 10^{-34} \, \text{J}\cdot\text{s}$ * Boltzmann Constant: $k_B = 1.38 \times 10^{-23} \, \text{J/K}$ * Avogadro's Number: $N_A = 6.022 \times 10^{23} \, \text{mol}^{-1}$ * Permittivity of Free Space: $\epsilon_0 = 8.854 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2$ * Permeability of Free Space: $\mu_0 = 4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A}$ * Acceleration due to Gravity: $g = 9.8 \, \text{m/s}^2$

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