AP Physics 1 Equation Sheet

The same content as the sheet you get on the exam, in our own layout. Every value in it is worked out by our math engine. The practice exams show this sheet under their Reference button.

AP Physics 1 equation sheet

The constants, conversions and equations you may look up on the AP Physics 1 exam, gathered in one place. The exam gives you a table with the same content; knowing where each equation sits, and what each symbol means, saves time when you need it.

Constants and conversion factors

Quantity	Value
Universal gravitational constant	$G$ = $6.67 \times 10^{-11}\,\mathrm{m^{3}/(kg{\cdot}s^{2})}$ = $6.67 \times 10^{-11}\,\mathrm{N{\cdot}m^{2}/kg^{2}}$
Acceleration due to gravity at the Earth's surface (its magnitude)	$g$ = $9.8\,\mathrm{m/s^{2}}$
Gravitational field strength at the Earth's surface (its magnitude)	$g$ = $9.8\,\mathrm{N/kg}$
1 atmosphere of pressure	1 atm = $1.0 \times 10^{5}\,\mathrm{N/m^{2}}$ = $1.0 \times 10^{5}\,\mathrm{Pa}$

Prefixes

Factor	Prefix	Symbol
$10^{12}$	tera	T
$10^{9}$	giga	G
$10^{6}$	mega	M
$10^{3}$	kilo	k
$10^{-2}$	centi	c
$10^{-3}$	milli	m
$10^{-6}$	micro	μ
$10^{-9}$	nano	n
$10^{-12}$	pico	p

Unit symbols

Unit	Symbol	Unit	Symbol
hertz	Hz	newton	N
joule	J	pascal	Pa
kilogram	kg	second	s
meter	m	watt	W

Values of the trigonometric functions for common angles

The values at 37° and 53° are the rounded ones of the 3-4-5 right triangle, which the exam uses.

$\theta$	0°	30°	37°	45°	53°	60°	90°
$\sin\theta$	0	$\frac{1}{2}$	$\frac{3}{5}$	$\frac{\sqrt{2}}{2}$	$\frac{4}{5}$	$\frac{\sqrt{3}}{2}$	1
$\cos\theta$	1	$\frac{\sqrt{3}}{2}$	$\frac{4}{5}$	$\frac{\sqrt{2}}{2}$	$\frac{3}{5}$	$\frac{1}{2}$	0
$\tan\theta$	0	$\frac{\sqrt{3}}{3}$	$\frac{3}{4}$	1	$\frac{4}{3}$	$\sqrt{3}$	$\infty$

Conventions of the exam

The frame of reference of any problem is inertial unless the problem says otherwise.
Air resistance is negligible unless the problem says otherwise.
Springs and strings are ideal unless the problem says otherwise.
Fluids are ideal, and pipes are completely filled by fluid, unless the problem says otherwise.

Geometry and trigonometry

Shape	Equations
Rectangle	$A = bh$
Triangle	$A = \frac{1}{2}bh$
Circle	$A = \pi r^2$ , $C = 2\pi r$ , $s = r\theta$
Rectangular solid	$V = \ell w h$
Cylinder	$V = \pi r^2 \ell$ , $S = 2\pi r\ell + 2\pi r^2$
Sphere	$V = \frac{4}{3}\pi r^3$ , $S = 4\pi r^2$
Right triangle (legs $a$ and $b$ , hypotenuse $c$ , $\theta$ opposite $a$ )	$a^{2} + b^{2} = c^{2}$ , $\sin\theta = \frac{a}{c}$ , $\cos\theta = \frac{b}{c}$ , $\tan\theta = \frac{a}{b}$

$A$ is area, $b$ base, $C$ circumference, $h$ height, $\ell$ length, $r$ radius, $s$ arc length, $S$ surface area, $V$ volume, $w$ width and $\theta$ angle.

Mechanics and fluids: motion and force

Equation	What it relates
$v_x = v_{x0} + a_x t$	velocity after a time, at constant acceleration
$x = x_0 + v_{x0}t + \frac{1}{2}a_x t^2$	position after a time, at constant acceleration
$v_x^2 = v_{x0}^2 + 2a_x(x - x_0)$	velocity and position, with no time
$\vec{x}_{cm} = \dfrac{\sum m_i \vec{x}_i}{\sum m_i}$	the center of mass
$\vec{a}_{sys} = \dfrac{\sum \vec{F}}{m_{sys}} = \dfrac{\vec{F}_{net}}{m_{sys}}$	Newton's second law for a system
$\lvert\vec{F}_g\rvert = G\dfrac{m_1 m_2}{r^2}$	the gravitational force between two masses
$\lvert\vec{F}_f\rvert \le \lvert\mu \vec{F}_N\rvert$	friction (equal for kinetic friction)
$\vec{F}_s = -k\Delta\vec{x}$	the spring force
$a_c = \dfrac{v^2}{r}$	centripetal acceleration

Energy, power and momentum

Equation	What it relates
$K = \frac{1}{2}mv^2$	translational kinetic energy
$W = F_{\parallel}d = Fd\cos\theta$	the work of a constant force
$\Delta K = \sum W_i = \sum F_{\parallel,i}d_i$	the work-energy theorem
$U_s = \frac{1}{2}k(\Delta x)^2$	the potential energy of a spring
$U_G = -\dfrac{Gm_1m_2}{r}$	gravitational potential energy of two masses
$\Delta U_g = mg\Delta y$	gravitational potential energy near the Earth
$P_{avg} = \dfrac{W}{\Delta t} = \dfrac{\Delta E}{\Delta t}$	average power
$P_{inst} = F_{\parallel}v = Fv\cos\theta$	instantaneous power
$\vec{p} = m\vec{v}$	momentum
$\vec{F}_{net} = \dfrac{\Delta\vec{p}}{\Delta t} = m\dfrac{\Delta\vec{v}}{\Delta t} = m\vec{a}$	net force and the change in momentum
$\vec{J} = \vec{F}_{avg}\Delta t = \Delta\vec{p}$	impulse
$\vec{v}_{cm} = \dfrac{\sum \vec{p}_i}{\sum m_i} = \dfrac{\sum m_i\vec{v}_i}{\sum m_i}$	the velocity of the center of mass

Rotation

Equation	What it relates
$\omega = \omega_0 + \alpha t$	angular velocity after a time, at constant angular acceleration
$\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$	angular position after a time
$\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$	angular velocity and angle, with no time
$v = r\omega$	linear and angular speed
$a_T = r\alpha$	tangential and angular acceleration
$\tau = r_{\perp}F = rF\sin\theta$	torque
$I = \sum m_i r_i^2$	rotational inertia
$I' = I_{cm} + Md^2$	the parallel-axis theorem
$\alpha_{sys} = \dfrac{\sum\tau}{I_{sys}} = \dfrac{\tau_{net}}{I_{sys}}$	Newton's second law in rotational form
$K = \frac{1}{2}I\omega^2$	rotational kinetic energy
$W = \tau\Delta\theta$	the work of a torque
$L = I\omega$	angular momentum of a rotating object
$L = rmv\sin\theta$	angular momentum of a moving point mass
$\Delta L = \tau\Delta t$	angular impulse
$\Delta x_{cm} = r\Delta\theta$	rolling without slipping

Oscillations

Equation	What it relates
$T = \dfrac{1}{f}$	period and frequency
$T_s = 2\pi\sqrt{\dfrac{m}{k}}$	the period of a mass on a spring
$T_p = 2\pi\sqrt{\dfrac{\ell}{g}}$	the period of a simple pendulum
$x = A\cos(2\pi ft)$	position in simple harmonic motion, starting at $+A$
$x = A\sin(2\pi ft)$	position in simple harmonic motion, starting at 0

Fluids

Equation	What it relates
$\rho = \dfrac{m}{V}$	density
$P = \dfrac{F_{\perp}}{A}$	pressure
$P = P_0 + \rho gh$	absolute pressure at a depth
$P_{gauge} = \rho gh$	gauge pressure
$F_b = \rho Vg$	the buoyant force
$A_1v_1 = A_2v_2$	continuity
$P_1 + \rho gy_1 + \frac{1}{2}\rho v_1^2 = P_2 + \rho gy_2 + \frac{1}{2}\rho v_2^2$	Bernoulli's equation

The symbols: $a$ acceleration, $A$ amplitude or area, $d$ distance, $E$ energy, $f$ frequency, $F$ force, $h$ height, $I$ rotational inertia, $J$ impulse, $k$ spring constant, $K$ kinetic energy, $\ell$ length, $L$ angular momentum, $m$ and $M$ mass, $p$ momentum, $P$ power or pressure, $r$ radius or distance, $t$ time, $T$ period, $U$ potential energy, $v$ velocity or speed, $V$ volume, $W$ work, $x$ position, $y$ vertical position, $\alpha$ angular acceleration, $\theta$ angle or angular position, $\mu$ coefficient of friction, $\rho$ density, $\tau$ torque and $\omega$ angular speed.