📌 Kinematics

Velocity:

\( v = \frac{\Delta x}{\Delta t} \)

Acceleration:

\( a = \frac{\Delta v}{\Delta t} \)

Position (Constant Acceleration):

\( x = x_0 + v_0 t + \frac{1}{2} a t^2 \)

Velocity (Constant Acceleration):

\( v = v_0 + a t \)

Velocity Squared Equation:

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

📌 Dynamics

Newton’s Second Law:

\( F = m a \)

Momentum:

\( p = m v \)

Impulse:

\( J = F \Delta t = \Delta p \)

Force of Friction:

\( F_f = \mu N \)

Weight:

\( W = m g \)

📌 Work and Energy

Work:

\( W = F d \cos \theta \)

Kinetic Energy:

\( KE = \frac{1}{2} m v^2 \)

Potential Energy (Gravitational):

\( PE = m g h \)

Work-Energy Theorem:

\( W = \Delta KE \)

Conservation of Mechanical Energy:

\( KE_i + PE_i = KE_f + PE_f \)

Power:

\( P = \frac{W}{t} = F v \cos \theta \)

📌 Rotational Motion

Angular Velocity:

\( \omega = \frac{\Delta \theta}{\Delta t} \)

Angular Acceleration:

\( \alpha = \frac{\Delta \omega}{\Delta t} \)

Rotational Kinematics:

\( \theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2 \)

Torque:

\( \tau = r F \sin \theta \)

Moment of Inertia (Point Mass):

\( I = m r^2 \)

Angular Momentum:

\( L = I \omega \)

Rotational Kinetic Energy:

\( KE_{rot} = \frac{1}{2} I \omega^2 \)

📌 Gravitation

Gravitational Force:

\( F = G \frac{m_1 m_2}{r^2} \)

Gravitational Potential Energy:

\( U = -G \frac{m_1 m_2}{r} \)

Orbital Velocity:

\( v = \sqrt{\frac{G M}{r}} \)

Escape Velocity:

\( v_e = \sqrt{\frac{2 G M}{r}} \)

📌 Thermodynamics

Ideal Gas Law:

\( PV = nRT \)

Kinetic Theory of Gases (Average KE):

\( KE_{avg} = \frac{3}{2} k T \)

First Law of Thermodynamics:

\( \Delta U = Q - W \)

Heat Transfer (Conduction):

\( Q = k A \frac{\Delta T}{L} t \)

Efficiency of Heat Engine:

\( \eta = 1 - \frac{T_C}{T_H} \)

📌 Waves and Oscillations

Wave Speed:

\( v = f \lambda \)

Frequency of Oscillation:

\( f = \frac{1}{T} \)

Simple Harmonic Motion (Position):

\( x = A \cos(\omega t + \phi) \)

Angular Frequency (Spring):

\( \omega = \sqrt{\frac{k}{m}} \)

Angular Frequency (Pendulum):

\( \omega = \sqrt{\frac{g}{L}} \)

Wave Intensity:

\( I = \frac{P}{4\pi r^2} \)

📌 Electromagnetism

Coulomb’s Law:

\( F = k_e \frac{q_1 q_2}{r^2} \)

Electric Field:

\( E = \frac{F}{q} = k_e \frac{q}{r^2} \)

Electric Potential:

\( V = k_e \frac{q}{r} \)

Capacitance:

\( C = \frac{Q}{V} \)

Ohm’s Law:

\( V = I R \)

Magnetic Force on a Moving Charge:

\( F = q v B \sin \theta \)

Magnetic Field (Current-Carrying Wire):

\( B = \frac{\mu_0 I}{2 \pi r} \)

Faraday’s Law of Induction:

\( \mathcal{E} = - \frac{d\Phi_B}{dt} \)

📌 Optics

Lens Formula:

\( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)

Magnification:

\( M = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \)

Snell’s Law:

\( n_1 \sin \theta_1 = n_2 \sin \theta_2 \)

Critical Angle:

\( \theta_c = \sin^{-1} \left( \frac{n_2}{n_1} \right) \)

Diffraction (Single Slit):

\( \sin \theta = \frac{\lambda}{a} \)

📌 Quantum Mechanics

de Broglie Wavelength:

\( \lambda = \frac{h}{p} \)

Energy of a Photon:

\( E = h f = \frac{h c}{\lambda} \)

Uncertainty Principle:

\( \Delta x \Delta p \geq \frac{\hbar}{2} \)

Schrödinger Equation (Time-Independent):

\( -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V \psi = E \psi \)

Energy Levels (Particle in a Box):

\( E_n = \frac{n^2 h^2}{8 m L^2} \)

📌 Relativity

Time Dilation:

\( t = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \)

Length Contraction:

\( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \)

Relativistic Momentum:

\( p = \gamma m v \), where \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \)

Mass-Energy Equivalence:

\( E = m c^2 \)

Relativistic Energy:

\( E = \gamma m c^2 \)