---
title: "Ultimate AP Physics 1 Equation Sheets"
description: "Every single AP Physics 1 equation you need to know + 50 derived formulas to help you get a 5 on the AP Physics 1 Exam."
url: "https://nerd-notes.com/ap-physics-1-formula-sheets/"
date_modified: "2026-09-26T12:20:45+00:00"
---

# Ultimate AP Physics 1 Equation Sheets

We upgraded the official AP Physics 1 equation sheet with derived formulas that show how the equations connect, so there is nothing extra to memorize.

- [View Equations](https://nerd-notes.com/wp-content/uploads/2025/05/2025-AP-Physics-1-Base-Formulas-Nerd-Notes-nerd-notes.com_.pdf) (PDF, opens in a new tab)
- [View Derived Equations](https://nerd-notes.com/wp-content/uploads/2025/05/2025-AP-Physics-1-Derived-Equations-nerd-notes.com_.pdf) (PDF, opens in a new tab)
- [View Digital Equation Sheet](https://nerd-notes.com/ap-physics-1-formula-sheets/#eq-official) (jumps to the digital sheet at the bottom of this page)

Both PDFs are free to open, no account needed. They are Nerd Notes' 2025 editions, 4 pages each: equations beside a notes column, grouped by topic (Rotational Motion, Kinematics, Forces, Energy, Momentum, Simple Harmonic Motion, Fluids).

## [Click here to watch: base and derived formulas explained](https://www.youtube.com/watch?v=9f2UNm7taYU&list=PLXzMIT8Ipop9C6Bw95EUNE9VHFRRodVbJ)

We’re working on an updated video reflecting new equations in the new AP Physics 1 Exam. Please stay updated via [the Nerd Notes YouTube channel.](https://www.youtube.com/@nerd-notes)

## And even more free resources to help you get a 5

- For AP-style questions, [check out UBQ.](https://nerd-notes.com/ubq/)
- For AP-style Practice Exams, [check out UBQ Quiz Labs.](https://nerd-notes.com/quiz/)
- For a SPEED review of all 8 AP Physics 1 units, [click here.](https://nerd-notes.com/ap-physics-speed-review/)
- Use Phy to solve and explain hard questions. [Try here.](https://nerd-notes.com/ai/)

## Official 2026 Digital Equation Sheet

Also built into every question on [UBQ](https://nerd-notes.com/ubq/), or [download the official PDF](https://apcentral.collegeboard.org/media/pdf/ap-physics-1-equations-sheet.pdf).

On the page this is an interactive sheet: a sticky tab bar jumps between its sections, and hovering (or focusing) an equation shows its name and the AP Physics 1 unit it belongs to. Its full content follows: College Board's "AP Physics 1 2026 Exam Reference Information", transcribed, with each equation's name and CED unit as the page shows them on hover. Sections marked "Nerd Notes addition" carry a Nerd Notes badge on the page and are not part of the College Board sheet.

AP Physics 1 units: 1 Kinematics · 2 Force and Translational Dynamics · 3 Work, Energy, and Power · 4 Linear Momentum · 5 Torque and Rotational Dynamics · 6 Energy and Momentum of Rotating Systems · 7 Oscillations · 8 Fluids.

### Mechanics and Fluids Equations

- Velocity (constant acceleration) (Unit 1: Kinematics): $v_x = v_{x0} + a_x t$
- Position (constant acceleration) (Unit 1: Kinematics): $x = x_0 + v_{x0}t + \frac{1}{2}a_x t^2$
- Velocity and displacement (constant acceleration) (Unit 1: Kinematics): $v_x^2 = v_{x0}^2 + 2a_x\left(x - x_0\right)$
- Center of mass position (Unit 2: Force and Translational Dynamics): $\vec{x}_{\text{cm}} = \frac{\sum m_i \vec{x}_i}{\sum m_i}$
- Newton's second law (system) (Unit 2: Force and Translational Dynamics): $\vec{a}_{\text{sys}} = \frac{\sum \vec{F}}{m_{\text{sys}}} = \frac{\vec{F}_{\text{net}}}{m_{\text{sys}}}$
- Newton's law of universal gravitation (Unit 2: Force and Translational Dynamics): $\left|\vec{F}_g\right| = G\frac{m_1 m_2}{r^2}$
- Friction force (Unit 2: Force and Translational Dynamics): $\left|\vec{F}_f\right| \le \left|\mu \vec{F}_N\right|$
- Hooke's law (spring force) (Unit 2: Force and Translational Dynamics): $\vec{F}_s = -k\Delta\vec{x}$
- Centripetal acceleration (Unit 2: Force and Translational Dynamics): $a_c = \frac{v^2}{r}$
- Translational kinetic energy (Unit 3: Work, Energy, and Power): $K = \frac{1}{2}mv^2$
- Work done by a constant force (Unit 3: Work, Energy, and Power): $W = F_{\parallel}d = Fd\cos\theta$
- Work-energy theorem (Unit 3: Work, Energy, and Power): $\Delta K = \sum W_i = \sum F_{\parallel,i}\,d_i$
- Spring potential energy (Unit 3: Work, Energy, and Power): $U_s = \frac{1}{2}k(\Delta x)^2$
- Gravitational potential energy (general) (Unit 3: Work, Energy, and Power): $U_G = -\frac{Gm_1 m_2}{r}$
- Gravitational potential energy near Earth (Unit 3: Work, Energy, and Power): $\Delta U_g = mg\Delta y$
- Average power (Unit 3: Work, Energy, and Power): $P_{\text{avg}} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$
- Instantaneous power (Unit 3: Work, Energy, and Power): $P_{\text{inst}} = F_{\parallel}v = Fv\cos\theta$
- Momentum (Unit 4: Linear Momentum): $\vec{p} = m\vec{v}$
- Newton's second law (momentum form) (Unit 4: Linear Momentum): $\vec{F}_{\text{net}} = \frac{\Delta\vec{p}}{\Delta t} = m\frac{\Delta\vec{v}}{\Delta t} = m\vec{a}$
- Impulse-momentum theorem (Unit 4: Linear Momentum): $\vec{J} = \vec{F}_{\text{avg}}\Delta t = \Delta\vec{p}$
- Center of mass velocity (Unit 4: Linear Momentum): $\vec{v}_{\text{cm}} = \frac{\sum \vec{p}_i}{\sum m_i} = \frac{\sum m_i \vec{v}_i}{\sum m_i}$

Variables: $a$ acceleration; $d$ distance; $E$ energy; $F$ force; $J$ impulse; $k$ spring constant; $K$ kinetic energy; $m$ mass; $p$ momentum; $P$ power; $r$ radius or distance; $t$ time; $U$ potential energy; $v$ velocity or speed; $W$ work; $x$ position; $y$ vertical position; $\theta$ angle; $\mu$ coefficient of friction

- Angular velocity (constant angular acceleration) (Unit 5: Torque and Rotational Dynamics): $\omega = \omega_0 + \alpha t$
- Angular position (constant angular acceleration) (Unit 5: Torque and Rotational Dynamics): $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$
- Angular velocity and displacement (constant angular acceleration) (Unit 5: Torque and Rotational Dynamics): $\omega^2 = \omega_0^2 + 2\alpha\left(\theta - \theta_0\right)$
- Linear and angular speed (Unit 5: Torque and Rotational Dynamics): $v = r\omega$
- Tangential acceleration (Unit 5: Torque and Rotational Dynamics): $a_T = r\alpha$
- Torque (Unit 5: Torque and Rotational Dynamics): $\tau = r_{\perp}F = rF\sin\theta$
- Rotational inertia of point masses (Unit 5: Torque and Rotational Dynamics): $I = \sum m_i r_i^2$
- Parallel axis theorem (Unit 5: Torque and Rotational Dynamics): $I' = I_{\text{cm}} + Md^2$
- Newton's second law for rotation (Unit 5: Torque and Rotational Dynamics): $\alpha_{\text{sys}} = \frac{\Sigma\tau}{I_{\text{sys}}} = \frac{\tau_{\text{net}}}{I_{\text{sys}}}$
- Rotational kinetic energy (Unit 6: Energy and Momentum of Rotating Systems): $K = \frac{1}{2}I\omega^2$
- Work done by a torque (Unit 6: Energy and Momentum of Rotating Systems): $W = \tau\Delta\theta$
- Angular momentum of a rigid body (Unit 6: Energy and Momentum of Rotating Systems): $L = I\omega$
- Angular momentum of a point mass (Unit 6: Energy and Momentum of Rotating Systems): $L = rmv\sin\theta$
- Angular impulse-momentum theorem (Unit 6: Energy and Momentum of Rotating Systems): $\Delta L = \tau\Delta t$
- Rolling without slipping (Unit 6: Energy and Momentum of Rotating Systems): $\Delta x_{\text{cm}} = r\Delta\theta$
- Period and frequency (Unit 7: Oscillations): $T = \frac{1}{f}$
- Period of a mass-spring system (Unit 7: Oscillations): $T_s = 2\pi\sqrt{\frac{m}{k}}$
- Period of a simple pendulum (Unit 7: Oscillations): $T_p = 2\pi\sqrt{\frac{\ell}{g}}$
- Simple harmonic motion (cosine form) (Unit 7: Oscillations): $x = A\cos(2\pi ft)$
- Simple harmonic motion (sine form) (Unit 7: Oscillations): $x = A\sin(2\pi ft)$
- Density (Unit 8: Fluids): $\rho = \frac{m}{V}$
- Pressure (Unit 8: Fluids): $P = \frac{F_{\perp}}{A}$
- Absolute pressure at depth (Unit 8: Fluids): $P = P_0 + \rho gh$
- Gauge pressure (Unit 8: Fluids): $P_{\text{gauge}} = \rho gh$
- Buoyant force (Archimedes' principle) (Unit 8: Fluids): $F_b = \rho Vg$
- Continuity equation (Unit 8: Fluids): $A_1 v_1 = A_2 v_2$
- Bernoulli's equation (Unit 8: Fluids): $P_1 + \rho g y_1 + \frac{1}{2}\rho v_1^2 = P_2 + \rho g y_2 + \frac{1}{2}\rho v_2^2$

Variables: $a$ acceleration; $A$ amplitude or area; $d$ distance; $f$ frequency; $F$ force; $h$ height; $I$ rotational inertia; $k$ spring constant; $K$ kinetic energy; $\ell$ length; $L$ angular momentum; $m$ mass; $M$ mass; $P$ pressure; $r$ radius or distance; $t$ time; $T$ period; $v$ velocity or speed; $V$ volume; $W$ work; $x$ position; $y$ vertical position; $\alpha$ angular acceleration; $\theta$ angle or angular position; $\rho$ density; $\tau$ torque; $\omega$ angular speed

### Constants and Conversion Factors

- Universal gravitational constant: $\begin{aligned} G &= 6.67\times10^{-11}\ \mathrm{m^3/(kg\cdot s^2)} \\ &= 6.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2} \end{aligned}$
- 1 atmosphere of pressure: $1\ \mathrm{atm} = 1.0\times10^{5}\ \mathrm{N/m^2} = 1.0\times10^{5}\ \mathrm{Pa}$
- Magnitude of the acceleration due to gravity at Earth's surface: $g = 9.8\ \mathrm{m/s^2}$
- Magnitude of the gravitational field strength at Earth's surface: $g = 9.8\ \mathrm{N/kg}$

### Prefixes

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

### Unit Symbols

| Unit | Symbol |
|---|---|
| hertz | $\mathrm{Hz}$ |
| joule | $\mathrm{J}$ |
| kilogram | $\mathrm{kg}$ |
| meter | $\mathrm{m}$ |
| newton | $\mathrm{N}$ |
| pascal | $\mathrm{Pa}$ |
| second | $\mathrm{s}$ |
| watt | $\mathrm{W}$ |

### Values of Trigonometric Functions for Common Angles

| $\theta$ | $0^\circ$ | $30^\circ$ | $37^\circ$ | $45^\circ$ | $53^\circ$ | $60^\circ$ | $90^\circ$ |
|---|---|---|---|---|---|---|---|
| $\sin\theta$ | $0$ | $1/2$ | $3/5$ | $\sqrt{2}/2$ | $4/5$ | $\sqrt{3}/2$ | $1$ |
| $\cos\theta$ | $1$ | $\sqrt{3}/2$ | $4/5$ | $\sqrt{2}/2$ | $3/5$ | $1/2$ | $0$ |
| $\tan\theta$ | $0$ | $\sqrt{3}/3$ | $3/4$ | $1$ | $4/3$ | $\sqrt{3}$ | $\infty$ |

### Exam Conventions

- The frame of reference of any problem is assumed to be inertial unless otherwise stated.
- Air resistance is assumed to be negligible unless otherwise stated.
- Springs and strings are assumed to be ideal unless otherwise stated.
- Fluids are assumed to be ideal, and pipes are assumed to be completely filled by fluid, unless otherwise stated.

### Geometry and Trigonometry

**Rectangle**

- Area of a rectangle: $A = bh$

**Triangle**

- Area of a triangle: $A = \frac{1}{2}bh$

**Circle**

- Area of a circle: $A = \pi r^2$
- Circumference of a circle: $C = 2\pi r$
- Arc length: $s = r\theta$

**Rectangular Solid**

- Volume of a rectangular solid: $V = \ell wh$

**Cylinder**

- Volume of a cylinder: $V = \pi r^2\ell$
- Surface area of a cylinder: $S = 2\pi r\ell + 2\pi r^2$

**Sphere**

- Volume of a sphere: $V = \frac{4}{3}\pi r^3$
- Surface area of a sphere: $S = 4\pi r^2$

**Right Triangle**

- Pythagorean theorem: $a^2 + b^2 = c^2$
- Sine: $\sin\theta = \frac{a}{c}$
- Cosine: $\cos\theta = \frac{b}{c}$
- Tangent: $\tan\theta = \frac{a}{b}$

Variables: $A$ area; $b$ base; $C$ circumference; $h$ height; $\ell$ length; $r$ radius; $s$ arc length; $S$ surface area; $V$ volume; $w$ width; $\theta$ angle

### SI Units (Nerd Notes addition)

| Quantity | SI unit |
|---|---|
| Displacement, $x$ | meter, $\mathrm{m}$ |
| Velocity, $v$ | meters per second, $\mathrm{m/s}$ |
| Acceleration, $a$ | meters per second squared, $\mathrm{m/s^2}$ |
| Time, $t$ | second, $\mathrm{s}$ |
| Mass, $m$ | kilogram, $\mathrm{kg}$ |
| Force, $F$ | newton, $\mathrm{N} = \mathrm{kg\cdot m/s^2}$ |
| Energy and work, $E$, $K$, $U$, $W$ | joule, $\mathrm{J} = \mathrm{N\cdot m}$ |
| Power, $P$ | watt, $\mathrm{W} = \mathrm{J/s}$ |
| Momentum, $p$ | $\mathrm{kg\cdot m/s}$ |
| Angular speed, $\omega$ | radians per second, $\mathrm{rad/s}$ |
| Torque, $\tau$ | $\mathrm{N\cdot m}$ |
| Rotational inertia, $I$ | $\mathrm{kg\cdot m^2}$ |
| Frequency, $f$ | hertz, $\mathrm{Hz} = \mathrm{s^{-1}}$ |
| Pressure, $P$ | pascal, $\mathrm{Pa} = \mathrm{N/m^2}$ |

### Unit Conversion Example (Nerd Notes addition)

Convert 5 kilometers to millimeters. Multiply by conversion factors that equal 1; each unit that appears on the top and the bottom cancels:

$$5\ \cancel{\mathrm{km}}\times\frac{10^{3}\ \cancel{\mathrm{m}}}{1\ \cancel{\mathrm{km}}}\times\frac{10^{3}\ \mathrm{mm}}{1\ \cancel{\mathrm{m}}} = 5\times10^{6}\ \mathrm{mm}$$

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The equation-sheet PDFs, the video and the resources above are plain links; the digital sheet is an interactive JavaScript component whose data loads from the Nerd Notes API — its full content is reproduced above for automated readers.