physics cheat sheet

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\section*{Physics Formula Sheet}
This formula sheet covers key equations for rotational kinematics, springs and harmonic motion, kinematics, forces, torque, momentum, collisions, energy, and work.
]

\formulaitem{Rotational Kinematics: Arc Length:}{S = r\Delta\theta}
\formulaitem{Angular Acceleration:}{\alpha = \frac{\Delta\omega}{\Delta t}}
\formulaitem{Angular Velocity:}{\omega = \frac{\Delta\theta}{\Delta t}}
\formulaitem{Velocity:}{V = r\omega = \frac{s}{\Delta t} = \frac{r\Delta\theta}{\Delta t}}
\formulaitem{Tangential Acceleration:}{a_{\text{tan}} = r\alpha = \frac{r\Delta v}{\Delta t} = \frac{r\Delta\omega}{\Delta t}}
\formulaitem{Centripetal Acceleration:}{a_c = r\omega^2 = \frac{v^2}{r}}
\formulaitem{Rotational Kinetic Energy:}{K_{\text{rot}} = \frac{1}{2}I\omega^2}
\formulaitem{Angular Momentum:}{L = I\omega}
\formulaitem{Change in Angular Momentum:}{\Delta L = \tau\Delta t}
\formulaitem{Relationship between Linear and Angular Quantities:}{\omega = \frac{v}{r}, \Delta\theta = \Delta x}

\formulaitem{Springs and Harmonic Motion: Displacement in SHM:}{x = A \cos(2\pi ft)}
\formulaitem{Period of a Spring:}{T_s = 2\pi \sqrt{\frac{m}{k}}}
\formulaitem{Potential Energy in a Spring:}{U_s = \frac{kx^2}{2}}
\formulaitem{Force by a Spring:}{F_s = kx}
\formulaitem{Period of a Pendulum:}{T_p = 2\pi \sqrt{\frac{l}{g}}}

\formulaitem{Kinematics: Projectile Motion:}{V_{iy} = V \sin \theta, V_{ix} = V \cos \theta}
\formulaitem{Final Velocity:}{v_f = v_i + at}
\formulaitem{Acceleration (from velocities):}{a = \frac{v_f - v_i}{t}}
\formulaitem{Initial Velocity (from final velocity):}{v_i = v_f - at}
\formulaitem{Final Velocity (from displacement):}{v_f^2 = v_i^2 + 2a\Delta x}
\formulaitem{Acceleration (from velocities and displacement):}{a = \frac{v_f^2 - v_i^2}{2\Delta x}}
\formulaitem{Displacement (from velocity and time):}{\Delta x = v_i t + \frac{1}{2}at^2}
\formulaitem{Acceleration (from displacement and time):}{a = \frac{2(\Delta x - v_i t)}{t^2}}
\formulaitem{Average Velocity (from displacement and time):}{\Delta x = \frac{(v_i + v_f)t}{2}}
\formulaitem{Final Velocity (from displacement and time):}{v_f = \frac{2\Delta x}{t} - v_i}

\formulaitem{Forces: Newton's Second Law:}{F = ma}
\formulaitem{Friction Force:}{F_f = \mu F_N}

\formulaitem{Torque and Center of Mass: Torque:}{\tau = rF \cos \Theta, \tau = rF \sin \Theta}

\formulaitem{Momentum and Collisions: Momentum:}{p = mv}
\formulaitem{Impulse:}{J = Ft = \Delta p}
\formulaitem{Elastic Collision:}{m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}}
\formulaitem{Inelastic Collision:}{m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2)v}
\formulaitem{Explosions:}{(m_1 + m_2)v = m_1 v_{1f} + m_2 v_{2f}}

\formulaitem{Energy and Work: Potential Energy:}{U = mgh}
\formulaitem{Spring Potential Energy:}{U_{\text{spring}} = \frac{1}{2}kx^2}
\formulaitem{Kinetic Energy:}{K = \frac{1}{2}mv^2}
\formulaitem{Power:}{P = \frac{W}{t} = \frac{F d}{t}}

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