JEE Class 11 Formulas

Cheatsheet Content

### Kinematics - **Equations of Motion:** - $v = u + at$ - $s = ut + \frac{1}{2}at^2$ - $v^2 = u^2 + 2as$ - $s_n = u + \frac{a}{2}(2n - 1)$ - **Relative Velocity:** $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$ - **Projectile Motion:** - Range $R = \frac{u^2 \sin(2\theta)}{g}$ - Max Height $H = \frac{u^2 \sin^2\theta}{2g}$ - Time of Flight $T = \frac{2u \sin\theta}{g}$ ### Newton's Laws of Motion & Friction - **Newton's Second Law:** $\vec{F} = m\vec{a}$ - **Impulse:** $\vec{J} = \Delta\vec{p} = \vec{F}_{avg} \Delta t$ - **Static Friction:** $f_s \le \mu_s N$ - **Kinetic Friction:** $f_k = \mu_k N$ ### Work, Energy & Power - **Work Done:** $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$ - **Kinetic Energy:** $K = \frac{1}{2}mv^2$ - **Potential Energy (Gravity):** $U = mgh$ - **Potential Energy (Spring):** $U = \frac{1}{2}kx^2$ - **Work-Energy Theorem:** $W_{net} = \Delta K$ - **Power:** $P = \frac{W}{t} = \vec{F} \cdot \vec{v}$ ### Centre of Mass & Collision - **Centre of Mass:** $\vec{r}_{CM} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$ - **Conservation of Momentum:** $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$ (if $F_{ext}=0$) - **Coefficient of Restitution:** $e = \frac{\text{relative velocity after}}{\text{relative velocity before}}$ - Elastic collision: $e=1$ - Inelastic collision: $0 ### Rotational Motion - **Angular Displacement:** $\Delta\theta$ - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ - **Relations:** $v = r\omega$, $a_t = r\alpha$, $a_c = \frac{v^2}{r} = \omega^2 r$ - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$ - **Moment of Inertia:** $I = \sum m_i r_i^2$ - **Kinetic Energy (Rotational):** $K_R = \frac{1}{2}I\omega^2$ - **Angular Momentum:** $\vec{L} = I\vec{\omega} = \vec{r} \times \vec{p}$ - **Conservation of Angular Momentum:** $I_1\omega_1 = I_2\omega_2$ (if $\tau_{ext}=0$) ### Gravitation - **Newton's Law:** $F = \frac{Gm_1m_2}{r^2}$ - **Gravitational Potential Energy:** $U = -\frac{Gm_1m_2}{r}$ - **Gravitational Potential:** $V = -\frac{GM}{r}$ - **Escape Velocity:** $v_e = \sqrt{\frac{2GM}{R}}$ - **Orbital Velocity:** $v_o = \sqrt{\frac{GM}{r}}$ - **Kepler's Third Law:** $T^2 \propto r^3$ ### Properties of Matter - **Stress:** $\sigma = \frac{F}{A}$ - **Strain:** $\epsilon = \frac{\Delta L}{L}$ - **Young's Modulus:** $Y = \frac{\text{Stress}}{\text{Strain}}$ - **Bulk Modulus:** $B = \frac{\text{Volume Stress}}{\text{Volume Strain}} = \frac{-\Delta P}{\Delta V/V}$ - **Surface Tension:** $T = \frac{F}{L}$ - **Capillary Rise:** $h = \frac{2T \cos\theta}{r\rho g}$ - **Bernoulli's Equation:** $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$ - **Poiseuille's Formula:** $Q = \frac{\pi Pr^4}{8\eta L}$ ### Thermodynamics - **Zeroth Law:** Defines temperature - **First Law:** $\Delta U = Q - W$ - **Work Done (Isothermal):** $W = nRT \ln\left(\frac{V_2}{V_1}\right)$ - **Work Done (Adiabatic):** $W = \frac{nR(T_1 - T_2)}{\gamma - 1}$ - **Efficiency of Heat Engine:** $\eta = 1 - \frac{Q_2}{Q_1} = 1 - \frac{T_2}{T_1}$ - **Specific Heat Capacity:** $Q = mc\Delta T$ - **Molar Specific Heat:** $C_P - C_V = R$ - **Relation for Adiabatic Process:** $PV^\gamma = \text{constant}$ ### Oscillations & Waves - **Simple Harmonic Motion (SHM):** - $y(t) = A \sin(\omega t + \phi)$ - Velocity: $v = A\omega \cos(\omega t + \phi) = \pm \omega \sqrt{A^2 - y^2}$ - Acceleration: $a = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 y$ - Time Period (Spring): $T = 2\pi\sqrt{\frac{m}{k}}$ - Time Period (Simple Pendulum): $T = 2\pi\sqrt{\frac{L}{g}}$ - **Wave Speed:** $v = f\lambda$ - **Speed of Sound in Gas:** $v = \sqrt{\frac{\gamma P}{\rho}}$ - **Standing Waves (String fixed at both ends):** $L = \frac{n\lambda}{2}$, $f_n = \frac{nv}{2L}$ - **Standing Waves (Organ pipe, open):** $L = \frac{n\lambda}{2}$, $f_n = \frac{nv}{2L}$ - **Standing Waves (Organ pipe, closed):** $L = \frac{(2n-1)\lambda}{4}$, $f_n = \frac{(2n-1)v}{4L}$ - **Beat Frequency:** $f_{beat} = |f_1 - f_2|$ - **Doppler Effect:** $f' = f \left(\frac{v \pm v_o}{v \mp v_s}\right)$ ### Basic Mathematics - **Quadratic Formula:** $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ - **Binomial Theorem:** $(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$ - **Approximations:** - $(1+x)^n \approx 1+nx$ for $|x| \ll 1$ - $\sin\theta \approx \theta$, $\tan\theta \approx \theta$, $\cos\theta \approx 1 - \frac{\theta^2}{2}$ for small $\theta$ ### Trigonometry - **Identities:** - $\sin^2\theta + \cos^2\theta = 1$ - $1 + \tan^2\theta = \sec^2\theta$ - $1 + \cot^2\theta = \csc^2\theta$ - **Sum/Difference Formulas:** - $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ - $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ - $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ - **Double Angle:** - $\sin(2A) = 2\sin A \cos A$ - $\cos(2A) = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$ ### Differentiation - **Power Rule:** $\frac{d}{dx}(x^n) = nx^{n-1}$ - **Product Rule:** $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$ - **Quotient Rule:** $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$ - **Chain Rule:** $\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$ - **Trigonometric Derivatives:** - $\frac{d}{dx}(\sin x) = \cos x$ - $\frac{d}{dx}(\cos x) = -\sin x$ - $\frac{d}{dx}(\tan x) = \sec^2 x$ ### Integration - **Power Rule:** $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \ne -1$) - **Logarithm:** $\int \frac{1}{x} dx = \ln|x| + C$ - **Trigonometric Integrals:** - $\int \sin x dx = -\cos x + C$ - $\int \cos x dx = \sin x + C$ - $\int \sec^2 x dx = \tan x + C$ - **Definite Integral:** $\int_a^b f(x)dx = F(b) - F(a)$ ### Vectors (Math) - **Magnitude:** $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$ - **Dot Product:** $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta = A_x B_x + A_y B_y + A_z B_z$ - **Cross Product:** $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ - Magnitude: $|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|\sin\theta$