AP Physics C: Mechanics · 2026 exam
Equations and reference
Every formula, vector identity, calculus rule, and constant you'll have on test day. Same content as the official sheet, dark mode, printable.
Section 1
Mechanics
Translational Mechanics
\(v_x = v_{x0} + a_x t\)
\(x = x_0 + v_{x0}t + \frac{1}{2}a_xt^2\)
\(v_x^2 = v_{x0}^2 + 2a_x(x - x_0)\)
\(\Delta x = \int v_x(t)\,dt\)
\(\Delta v_x = \int a_x(t)\,dt\)
\(\vec{x}_{\mathrm{cm}} = \frac{\sum m_i\vec{x}_i}{\sum m_i}\)
\(\vec{r}_{\mathrm{cm}} = \frac{\int \vec{r}\,dm}{\int dm}\)
\(\lambda = \frac{d}{d\ell}m(\ell)\)
\(\vec{a}_{\mathrm{sys}} = \frac{\sum \vec{F}}{m_{\mathrm{sys}}} = \frac{\vec{F}_{\mathrm{net}}}{m_{\mathrm{sys}}}\)
\(|\vec{F}_g| = G\frac{m_1m_2}{r^2}\)
\(|\vec{F}_f| \le \mu |\vec{F}_N|\)
\(\vec{F}_s = -k\Delta\vec{x}\)
\(a_c = \frac{v^2}{r} = r\omega^2\)
\(T = \frac{1}{f}\)
\(K = \frac{1}{2}mv^2\)
\(W = \int_a^b \vec{F}\cdot d\vec{r}\)
\(\Delta K = \sum W_i = \sum F_{\parallel,i}\,d_i\)
\(\Delta U = -\int_a^b \vec{F}_{\mathrm{cf}}(r)\cdot d\vec{r}\)
\(F_x = -\frac{dU(x)}{dx}\)
\(U_s = \frac{1}{2}k(\Delta x)^2\)
\(U_G = -\frac{Gm_1m_2}{r}\)
\(\Delta U_g = mg\Delta y\)
\(P_{\mathrm{avg}} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}\)
\(P_{\mathrm{inst}} = \frac{dW}{dt}\)
\(\vec{p} = m\vec{v}\)
\(\vec{F}_{\mathrm{net}} = \frac{d\vec{p}}{dt}\)
\(\vec{J} = \int_{t_1}^{t_2} \vec{F}_{\mathrm{net}}(t)\,dt = \Delta\vec{p}\)
\(\vec{v}_{\mathrm{cm}} = \frac{\sum \vec{p}_i}{\sum m_i} = \frac{\sum m_i\vec{v}_i}{\sum m_i}\)
Symbols
\(a\) = acceleration
\(d\) = distance
\(E\) = energy
\(f\) = frequency
\(F\) = force
\(J\) = impulse
\(k\) = spring constant
\(K\) = kinetic energy
\(\ell\) = length
\(m\) = mass
\(M\) = mass
\(p\) = momentum
\(P\) = power
\(r\) = radius, distance, or position
\(t\) = time
\(T\) = period
\(U\) = potential energy
\(v\) = velocity or speed
\(W\) = work
\(x\) = position or distance
\(y\) = vertical position
\(\lambda\) = linear mass density
\(\mu\) = coefficient of friction
Rotational and Oscillation
\(\omega = \frac{d\theta}{dt}\)
\(\alpha = \frac{d\omega}{dt}\)
\(\omega = \omega_0 + \alpha t\)
\(\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2\)
\(\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)\)
\(v = r\omega\)
\(a_T = r\alpha\)
\(\vec{\tau} = \vec{r}\times\vec{F}\)
\(I_{\mathrm{tot}} = \sum I_i = \sum m_ir_i^2\)
\(I = \int r^2\,dm\)
\(I' = I_{\mathrm{cm}} + Md^2\)
\(\alpha_{\mathrm{sys}} = \frac{\sum \tau}{I_{\mathrm{sys}}} = \frac{\tau_{\mathrm{net}}}{I_{\mathrm{sys}}}\)
\(K_{\mathrm{rot}} = \frac{1}{2}I\omega^2\)
\(W = \int \tau\,d\theta\)
\(\vec{L} = \vec{r}\times\vec{p} = I\vec{\omega}\)
\(\Delta L = \int \tau\,dt\)
\(\Delta x_{\mathrm{cm}} = r\Delta\theta\)
\(T = \frac{2\pi}{\omega} = \frac{1}{f}\)
\(T_s = 2\pi\sqrt{\frac{m}{k}}\)
\(T_p = 2\pi\sqrt{\frac{\ell}{g}}\)
\(T_{\mathrm{phys}} = 2\pi\sqrt{\frac{I}{mgd}}\)
\(x = x_{\max}\cos(\omega t + \phi)\)
Symbols
\(a\) = acceleration
\(d\) = distance
\(f\) = frequency
\(F\) = force
\(I\) = rotational inertia
\(k\) = spring constant
\(K\) = kinetic energy
\(\ell\) = length
\(L\) = angular momentum
\(m\) = mass
\(M\) = mass
\(p\) = momentum
\(r\) = radius, distance, or position
\(t\) = time
\(T\) = period
\(v\) = velocity or speed
\(W\) = work
\(x\) = position or distance
\(\alpha\) = angular acceleration
\(\theta\) = angular position
\(\tau\) = torque
\(\phi\) = phase angle
\(\omega\) = angular frequency or angular speed
Section 2
Geometry and Trigonometry
Plane Figures
Rectangle: \(A = bh\)
Triangle: \(A = \frac{1}{2}bh\)
Circle: \(A = \pi r^2\)
Circle: \(C = 2\pi r\)
Arc length: \(s = r\theta\)
Solids
Rect. solid: \(V = \ell wh\)
Cylinder: \(V = \pi r^2\ell\)
Cylinder: \(S = 2\pi r\ell + 2\pi r^2\)
Sphere: \(V = \frac{4}{3}\pi r^3\)
Sphere: \(S = 4\pi r^2\)
Symbols
\(A\) = area
\(b\) = base
\(C\) = circumference
\(h\) = height
\(\ell\) = length
\(r\) = radius
\(s\) = arc length
\(S\) = surface area
\(V\) = volume
\(w\) = width
\(\theta\) = angle
Right Triangle
\(a^2 + b^2 = c^2\)
\(\sin\theta = \frac{a}{c}\)
\(\cos\theta = \frac{b}{c}\)
\(\tan\theta = \frac{a}{b}\)
Section 3
Vectors and Calculus
Vectors
\(\vec{A}\cdot\vec{B} = AB\cos\theta\)
\(|\vec{A}\times\vec{B}| = AB\sin\theta\)
\(\vec{r} = (A\hat{\imath} + B\hat{\jmath} + C\hat{k})\)
\(\vec{C} = \vec{A} + \vec{B}\)
\(\vec{C} = (A_x + B_x)\hat{\imath} + (A_y + B_y)\hat{\jmath}\)
Calculus
\(\frac{df}{dx} = \frac{df}{du}\frac{du}{dx}\)
\(\frac{d}{dx}(x^n) = nx^{n-1}\)
\(\frac{d}{dx}(e^{ax}) = ae^{ax}\)
\(\frac{d}{dx}(\ln ax) = \frac{1}{x}\)
\(\frac{d}{dx}[\sin(ax)] = a\cos(ax)\)
\(\frac{d}{dx}[\cos(ax)] = -a\sin(ax)\)
\(\int x^n\,dx = \frac{1}{n+1}x^{n+1},\ n \neq -1\)
\(\int e^{ax}\,dx = \frac{1}{a}e^{ax}\)
\(\int \frac{dx}{x+a} = \ln|x+a|\)
\(\int \cos(ax)\,dx = \frac{1}{a}\sin(ax)\)
\(\int \sin(ax)\,dx = -\frac{1}{a}\cos(ax)\)
Identities
\(\log(a\cdot b^x) = \log a + x\log b\)
\(\sin^2\theta + \cos^2\theta = 1\)
\(\sin(2\theta) = 2\sin\theta\cos\theta\)
\(\frac{\sin\theta}{\cos\theta} = \tan\theta\)
Section 4
Constants and Conversion Factors
Universal gravitational constant
\(G = 6.67 \times 10^{-11}\ \mathrm{m^3/(kg\cdot s^2)}\)
\(\phantom{G\ } = 6.67 \times 10^{-11}\ \mathrm{N\cdot m^2/kg^2}\)
Acceleration due to gravity at Earth's surface
\(g = 9.8\ \mathrm{m/s^2}\)
Gravitational field strength at Earth's surface
\(g = 9.8\ \mathrm{N/kg}\)
Section 5
Prefixes and Unit Symbols
| 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 | \(\mu\) |
| \(10^{-9}\) | nano | n |
| \(10^{-12}\) | pico | p |
| Quantity | Symbol | Quantity | Symbol |
|---|---|---|---|
| hertz | Hz | newton | N |
| joule | J | second | s |
| kilogram | kg | watt | W |
| meter | m |
Section 6
Trig Values 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
- Use an inertial frame of reference unless another frame is stated.
- Ignore air resistance unless it is stated.
- Treat springs and strings as ideal unless stated.
Based on the AP Physics C: Mechanics 2026 Exam Reference Information.
Reformatted as a study sheet, not an official College Board publication.