Formula Sheet

Official-style AP reference. Hover a formula to copy its LaTeX.

Mechanics and Fluids — Translational Motion

v_x=v_{x0}+a_xt

Velocity under constant acceleration

x=x_0+v_{x0}t+\frac12a_xt^2

Position under constant acceleration

v_x^2=v_{x0}^2+2a_x(x-x_0)

Constant-acceleration relation without time

\vec x_{cm}=\frac{\sum_i m_i\vec x_i}{\sum_i m_i}

Center of mass

\vec a_{sys}=\frac{\sum_i\vec F_i}{m_{sys}}=\frac{\vec F_{net}}{m_{sys}}

Acceleration of a system

|\vec F_G|=G\frac{m_1m_2}{r^2}

Universal gravitation

|\vec F_f|\le\mu|\vec F_N|

Friction-force limit

\vec F_s=-k\Delta\vec x

Hooke's law

a_c=\frac{v^2}{r}

Centripetal acceleration

Mechanics and Fluids — Energy and Momentum

K=\frac12mv^2

Translational kinetic energy

W=F_{\parallel}d=Fd\cos\theta

Work by a constant force

\Delta K=\sum_iW_i=\sum_iF_{\parallel,i}d_i

Work–kinetic energy theorem

U_s=\frac12k(\Delta x)^2

Spring potential energy

U_G=-G\frac{Mm}{r}

Gravitational potential energy

\Delta U_g=mg\Delta y

Near-Earth gravitational potential-energy change

P_{avg}=\frac{W}{\Delta t}=\frac{\Delta E}{\Delta t}

Average power

P_{inst}=F_{\parallel}v=Fv\cos\theta

Instantaneous power

\vec p=m\vec v

Linear momentum

\vec F_{net}=\frac{\Delta\vec p}{\Delta t}=m\frac{\Delta\vec v}{\Delta t}=m\vec a

Newton's second law in momentum form

\vec J=\vec F_{avg}\Delta t=\Delta\vec p

Impulse–momentum theorem

\vec v_{cm}=\frac{\sum_i\vec p_i}{\sum_i m_i}=\frac{\sum_i m_i\vec v_i}{\sum_i m_i}

Center-of-mass velocity

Mechanics and Fluids — Rotation and Oscillations

\omega=\omega_0+\alpha t

Angular velocity under constant angular acceleration

\theta=\theta_0+\omega_0t+\frac12\alpha t^2

Angular position under constant angular acceleration

\omega^2=\omega_0^2+2\alpha(\theta-\theta_0)

Constant-angular-acceleration relation without time

v=r\omega

Tangential speed

a_T=r\alpha

Tangential acceleration

\tau=r_\perp F=rF\sin\theta

Torque magnitude

I=\sum_i m_i r_i^2

Rotational inertia for discrete masses

I'=I_{cm}+Md^2

Parallel-axis theorem

\alpha_{sys}=\frac{\sum\tau}{I_{sys}}=\frac{\tau_{net}}{I_{sys}}

Angular acceleration of a system

K=\frac12I\omega^2

Rotational kinetic energy

W=\tau\Delta\theta

Work by a constant torque

L=I\omega

Angular momentum of a rigid body

L=rmv\sin\theta

Angular momentum magnitude of a particle

\Delta L=\tau\Delta t

Angular impulse

\Delta x_{cm}=r\Delta\theta

Rolling displacement

T=\frac1f

Period and frequency

T_s=2\pi\sqrt{\frac{m}{k}}

Period of a mass–spring oscillator

T_p=2\pi\sqrt{\frac{\ell}{g}}

Period of a simple pendulum

x=A\cos(2\pi ft)

Cosine form of simple harmonic motion

x=A\sin(2\pi ft)

Sine form of simple harmonic motion

Mechanics and Fluids — Fluids

\rho=\frac{m}{V}

Mass density

P=\frac{F_\perp}{A}

Pressure

P=P_0+\rho gh

Absolute pressure at depth

P_{gauge}=\rho gh

Gauge pressure

F_b=\rho Vg

Buoyant force

A_1v_1=A_2v_2

Continuity equation

P_1+\rho gy_1+\frac12\rho v_1^2=P_2+\rho gy_2+\frac12\rho v_2^2

Bernoulli's equation