Physics Formula Sheet - Mechanics

1. Kinematics

Linear Motion

• Average velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$
• Average acceleration: $a_{avg} = \frac{\Delta v}{\Delta t}$
• Equations of motion (constant acceleration):
  • $v = u + at$
  • $s = ut + \frac{1}{2}at^2$
  • $v^2 = u^2 + 2as$
  • $s = \frac{(u+v)}{2} \cdot t$

Projectile Motion

• Horizontal range: $R = \frac{u^2 \sin 2\theta}{g}$
• Maximum height: $H = \frac{u^2 \sin^2 \theta}{2g}$
• Time of flight: $T = \frac{2u \sin \theta}{g}$

2. Dynamics

Newton’s Laws

• Newton’s Second Law: $F = ma$
• Weight: $W = mg$
• Friction:
  • Static friction: $f_s \leq \mu_s N$
  • Kinetic friction: $f_k = \mu_k N$

Work, Energy, and Power

• Work: $W = \vec{F} \cdot \vec{s} = Fs \cos \theta$
• Kinetic Energy: $KE = \frac{1}{2}mv^2$
• Potential Energy: $PE = mgh$
• Power: $P = \frac{W}{t} = Fv \cos \theta$

3. Circular Motion

• Angular velocity: $\omega = \frac{v}{r}$
• Centripetal acceleration: $a_c = \frac{v^2}{r} = \omega^2 r$
• Centripetal force: $F_c = \frac{mv^2}{r} = m\omega^2 r$

4. Gravitation

• Gravitational force: $F = G\frac{m_1 m_2}{r^2}$
• Gravitational potential energy: $U = -G\frac{m_1 m_2}{r}$
• Escape velocity: $v_e = \sqrt{\frac{2GM}{R}}$

5. Oscillations

Simple Harmonic Motion

• Equation of motion: $x = A \sin(\omega t + \phi)$
• Velocity: $v = A\omega \cos(\omega t + \phi)$
• Acceleration: $a = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x$
• Time period: $T = \frac{2\pi}{\omega}$
• Frequency: $f = \frac{1}{T} = \frac{\omega}{2\pi}$

Spring-Mass System

• Time period: $T = 2\pi\sqrt{\frac{m}{k}}$
• Energy: $E = \frac{1}{2}kA^2$ (total energy)

Simple Pendulum

• Time period: $T = 2\pi\sqrt{\frac{L}{g}}$ (small oscillations)

6. Properties of Matter

Elasticity

• Young’s Modulus: $Y = \frac{F/A}{\Delta L/L} = \frac{FL}{A\Delta L}$
• Bulk Modulus: $B = \frac{\Delta P}{\Delta V/V}$
• Modulus of Rigidity: $\eta = \frac{F/A}{\tan \theta}$

Fluid Mechanics

• Pressure: $P = \frac{F}{A}$
• Pressure at depth: $P = P_0 + \rho gh$
• Archimedes’ principle: $F_b = \rho_{fluid} V_{displaced} g$
• Bernoulli’s equation: $P + \frac{1}{2}\rho v^2 + \rho gh = constant$

7. Thermodynamics

Heat and Temperature

• Linear expansion: $\Delta L = L_0 \alpha \Delta T$
• Volume expansion: $\Delta V = V_0 \gamma \Delta T$
• Heat capacity: $Q = mc\Delta T$
• Latent heat: $Q = mL$

Ideal Gas Law

• Equation of state: $PV = nRT$
• Kinetic theory: $PV = \frac{1}{3}Nm\overline{v^2}$

Laws of Thermodynamics

• First law: $\Delta U = Q - W$
• Work done by gas: $W = \int_{V_1}^{V_2} P , dV$
• Efficiency of Carnot engine: $\eta = 1 - \frac{T_c}{T_h}$

This formula sheet covers essential mechanics formulas for JEE and NEET preparation. Use this for quick revision during your study sessions.