AP Physics C Formula Sheet: Calculus Equations for Both Exams

An AP Physics C formula sheet differs from the algebra based sheets in one way above all: its key relations are derivatives and integrals, starting with $v = \frac{dx}{dt}$ . AP Physics C is two separate courses with two separate exams, Mechanics and Electricity and Magnetism, and both expect you to set up and evaluate simple calculus as part of the physics.

This page is a study overview written for this site. For the exam itself, the board publishes its own equation tables for each exam. Here you will find the calculus backbone shared by both courses, a worked example of each style of question, and links to the detailed physics C mechanics formula sheet and physics C E&M formula sheet. The full list of course pages is on the AP physics formula sheet hub.

Quick facts

What this is
A study overview, not an official sheet
Two exams
Mechanics, and Electricity and Magnetism
Math level
Derivatives and integrals, usually of simple functions
Core idea
Rates use derivatives; totals use integrals
Mechanics key
$\vec{F}_{net} = \frac{d\vec{p}}{dt}$
E&M key
$\oint \vec{E}\cdot d\vec{A} = \frac{q_{enc}}{\varepsilon_0}$

The calculus pattern behind an AP Physics C formula sheet

Almost every relation follows one of two patterns. A rate is a derivative: velocity is the rate of change of position, current is the rate of flow of charge, induced emf is the rate of change of flux. A total is an integral: work adds up force over distance, charge adds up current over time, and a field adds up contributions from tiny pieces of charge.

The algebra versions you met earlier, such as $v = \frac{\Delta x}{\Delta t}$ , are special cases where the rate is constant. Once a quantity varies, the calculus form is the one that is always true.

Shared relations across mechanics and E&M

Derivatives and integrals that appear again and again
Calculus relation	What it describes	Unit	Linked page
$v = \frac{dx}{dt},\ a = \frac{dv}{dt}$	motion from a position function	m/s, m/s²	Instantaneous velocity
$\vec{F}_{net} = \frac{d\vec{p}}{dt}$	Newton's second law in its general form	N	Net force
$W = \int \vec{F}\cdot d\vec{r}$	work by a varying force	J	Work
$P = \frac{dW}{dt}$	power as the rate of work	W	Work and energy
$\tau_{net} = \frac{dL}{dt}$	torque changes angular momentum	N m	Angular momentum
$I = \int r^2\,dm$	rotational inertia of a continuous body	kg m²	Moment of inertia
$I = \frac{dq}{dt}$	electric current	A	Current
$\Delta V = -\int \vec{E}\cdot d\vec{r}$	potential difference from a field	V	Voltage
$\varepsilon = -\frac{d\Phi_B}{dt}$	Faraday's law of induction	V	Magnetic flux

The pairs mirror each other. Force is minus the slope of potential energy, $F_x = -\frac{dU}{dx}$ , just as the electric field is minus the slope of potential, $E_x = -\frac{dV}{dx}$ .

Worked examples in AP Physics C style

Example 1: motion from a position function

A particle moves along x with position $x(t) = 2t^3 - 4t$ (x in metres, t in seconds). What are its velocity and acceleration when t = 2.0 s?

Differentiate: $v = \frac{dx}{dt} = 6t^2 - 4$ and $a = \frac{dv}{dt} = 12t$ .

At t = 2.0 s: $v = 6(4) - 4 = 20\ \text{m/s}$ and $a = 12(2.0) = 24\ \text{m/s}^2$ .

Answer: 20 m/s and 24 m/s². The constant acceleration equations would be wrong here, since a changes with time.

Example 2: charge from a changing current

The current in a wire is $I(t) = 3.0t^2$ amperes. How much charge passes in the first 2.0 s?

$q = \int_0^{2.0} 3.0t^2\,dt = \left[t^3\right]_0^{2.0} = 8.0\ \text{C}$ .

Answer: 8.0 C. Compare: the average current over the interval is 8.0/2.0 = 4.0 A, not the final 12 A.

How the C courses compare with AP Physics 1 and 2

Course comparison in general terms
Course	Main content	Math used
AP Physics 1	mechanics, rotation, oscillations, fluids	algebra, trigonometry
AP Physics 2	thermodynamics, E&M, optics, waves, modern physics	algebra, trigonometry
AP Physics C: Mechanics	mechanics including rotation and oscillations	calculus
AP Physics C: E&M	electrostatics, circuits, magnetism, induction	calculus

Many students take C Mechanics after or alongside a first physics course, then E&M. The C courses cover fewer topics than AP Physics 2 but in more depth. The algebra versions are on the pages for AP Physics 1 and for AP Physics 2.

Common slips on calculus based questions

Using constant acceleration equations when a depends on t. Integrate instead.
Losing the constant of integration. Use the initial condition to fix it, such as x = 0 at t = 0.
Integrating the wrong variable. Write dm, dq or dx in terms of one variable before integrating.
Ignoring symmetry. In Gauss's and Ampère's law questions, the right surface or loop does most of the work.

A good check on any calculus answer: differentiate it again and see whether you get back what the question gave.

Frequently asked questions

Is there one AP Physics C formula sheet or two?

AP Physics C is two separate exams, Mechanics and Electricity and Magnetism, and the exam board provides equation tables for each. Check the current versions on the exam board's materials for your exam.

Is this the official physics C formula sheet?

No. Think of it as a study overview that explains the calculus relations the courses use. Use the official tables when you practise past questions.

How much calculus do you need for AP Physics C?

You should be comfortable differentiating and integrating polynomials, simple exponentials and trigonometric functions, and setting up integrals over a length, area or volume.

Should I take AP Physics C mechanics before E&M?

Most students do, because E&M reuses ideas such as work, energy and force in a field. Your school's order may differ.