AP Physics 1 Formula Sheet 2026 (with Explanations)
The complete AP Physics 1 2026 equation sheet — every formula explained in plain English across all 8 College Board units, with constants & conversion factors, SI prefixes, common-angle trigonometric values, and geometry formulas all in one place.
This page is a readable companion to the official College Board AP Physics 1 2026 equation sheet. Every formula on the printed sheet appears below in clean LaTeX, alongside a short explanation of what it means and a link to the matching Wikipedia article so you can dive deeper.
Quick facts about the 2026 sheet
Provided on the exam?
Yes. Both the multiple-choice and free-response sections give you the same equations, constants, and tables shown here.
Calculator allowed?
Yes — a scientific or graphing calculator is allowed on every section. The equation sheet does not replace your calculator; you still evaluate every numeric expression yourself.
What still needs memorizing?
Free-body diagrams, the conditions for conservation of energy and momentum, and moments of inertia for common shapes (ring, disk, sphere, rod). The sheet only lists the algebra.
Equations, unit by unit
Unit 1 — Kinematics
The three constant-acceleration kinematic equations describe one-dimensional motion when net force (and therefore acceleration) does not change with time.
Velocity from acceleration
Explanation
When acceleration a_x is constant, the final velocity v_x equals the initial velocity v_x0 plus a_x · t. The graph of v_x versus t is a straight line with slope a_x.
When to use
Use when you know acceleration and time but do not need displacement directly.
Position from acceleration
Explanation
Position grows linearly with the initial velocity term and quadratically with the acceleration term — that is the shape of a parabola in an x-vs-t graph for free fall and other constant-acceleration motion.
When to use
Use when you know acceleration and time and want the position (or displacement) but do not need the final velocity.
Time-independent kinematic equation
Explanation
Algebraic eliminator of t. Multiply, expand, and the time variable cancels, leaving a relationship between velocities and displacement only.
When to use
Use when the problem doesn't mention time and you have to relate v, v0, a, and Δx directly (e.g., minimum stopping distance).
Center of mass position
Explanation
The x-coordinate of a system's center of mass is the mass-weighted average of the particles' x-positions.
When to use
Use for multi-object systems when the motion of the whole system can be represented by its center of mass.
Unit 2 — Forces and Translational Dynamics
Newton's second law and the specific forces it most commonly involves: weight, friction, the spring restoring force, universal gravitation, and the centripetal force needed to keep an object on a circular path.
Newton's second law
Explanation
The net external force on a system determines the acceleration of that system's center of mass. Solve in components — sum the x-forces for a_x and the y-forces for a_y.
When to use
Every dynamics problem starts here, after you have drawn a free-body diagram and chosen a coordinate system.
Weight (gravitational force near Earth)
Explanation
Near a planet's surface, gravity pulls down with a force equal to the object's mass times the local gravitational field strength g. On Earth, g ≈ 9.8 N/kg points toward the center of the Earth.
Friction force
Explanation
Friction is proportional to the magnitude of the normal force. Static friction is an inequality — it adjusts up to its maximum value μs|FN| to keep the surfaces from sliding. Kinetic friction is an equality — once sliding begins, it equals μk|FN| regardless of speed.
When to use
If the surfaces aren't sliding (yet), use the static inequality with the equality reserved for the verge of sliding. If they are sliding, use the kinetic equation.
Hooke's law (spring force)
Explanation
An ideal spring exerts a restoring force proportional to its displacement x from equilibrium and opposite to it. The spring constant k (in N/m) measures stiffness.
When to use
Whenever a spring is part of the system, including the spring-mass oscillator and any "effective spring" approximation for small oscillations.
Newton's law of universal gravitation
Explanation
Two point masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centres. The same equation gives the gravitational field strength g at distance r from a planet of mass M.
When to use
Use for orbits, satellites, and any problem where the object is not near a planet's surface (so g ≠ 9.8 N/kg).
Centripetal acceleration and centripetal force
Explanation
An object moving in a circle of radius r at speed v accelerates toward the centre at v²/r, even when its speed is constant. Multiplying by mass gives the net inward force required to sustain the circular motion.
When to use
Identify what real force (tension, normal, gravity, friction…) is supplying the centripetal force, then set its magnitude equal to mv²/r.
Unit 3 — Work, Energy, and Power
Forms of mechanical energy (kinetic, gravitational, elastic), the work done by a constant force, the work-energy theorem, and the rate at which energy is transferred.
Kinetic energy
Explanation
The energy associated with motion. Kinetic energy is a scalar — direction does not matter. Doubling speed quadruples kinetic energy.
Gravitational potential energy (near Earth)
Explanation
Near a planet's surface (constant g), only the vertical change in height matters. For two masses separated by distance r, the universal gravitational potential energy is negative and approaches zero at infinite separation.
When to use
Use ΔUg = mgΔy close to a planet's surface. Use UG = −Gm1m2/r for orbital or large-distance gravitational-energy problems.
Elastic (spring) potential energy
Explanation
Energy stored in an ideal spring deformed by x from equilibrium. Stretch or compress the same amount and you store the same energy.
Work done by a force
Explanation
Work is the dot product of force and displacement: only the component of force along the direction of motion does work. Forces perpendicular to the motion (such as the normal force on a flat surface, or gravity for purely horizontal motion) do zero work.
When to use
When energy methods are easier than Newton's second law — typically when you know start and end states but not the details in between.
Work–kinetic energy theorem
Explanation
The change in kinetic energy equals the sum of the work done by all forces. Each work term uses only the component of force parallel to the displacement.
When to use
Apply when you can identify the forces doing work and want to connect them directly to a change in speed.
Power
Explanation
Average power is work or energy transfer divided by elapsed time. Instantaneous power from a force is the parallel component of the force times speed. Units: 1 W = 1 J/s.
Unit 4 — Linear Momentum
Linear momentum, the impulse–momentum theorem, and conservation of momentum in isolated systems (the workhorse for collision and explosion problems).
Linear momentum
Explanation
A vector quantity equal to mass times velocity. Has the same direction as the velocity vector.
Impulse–momentum theorem
Explanation
Impulse is the average force times the time interval over which it acts, and equals the change in momentum. Dividing the momentum change by time gives the net force during that interval.
When to use
Best for collisions, kicks, throws — anything where a force acts briefly and you care about the change in motion that results.
Conservation of linear momentum
Explanation
If no net external force acts on a system, its total momentum stays constant. In a collision between two objects A and B, the total momentum of A + B before equals the total momentum of A + B after.
When to use
Always conserve in 1D and 2D collisions and explosions. Conserve x-momentum and y-momentum separately when the collision is two-dimensional.
Center of mass velocity
Explanation
The velocity of a system's center of mass is the mass-weighted average of the particles' velocities.
When to use
Use when combining the motion of several objects into one system-level velocity, especially in collision and explosion problems.
Unit 5 — Torque and Rotational Dynamics
The rotational analogues of Newton's laws: arc-length / tangential-velocity / tangential-acceleration links between linear and angular motion, the rotational kinematic equations, torque, and the rotational form of Newton's second law.
Linking linear and rotational quantities
Explanation
Arc length s = rθ relates how far a point on a rotating object travels to the angle (in radians) the object has turned through. Differentiating gives v = rω and a_t = rα; for rolling without slipping, the center-of-mass displacement satisfies Δxcm = rΔθ.
When to use
Use any time you need to translate between angular and linear quantities — most commonly in rolling-without-slipping problems where v_cm = rω.
Rotational kinematic equations
Explanation
Direct rotational analogues of the three linear kinematic equations: replace x → θ, v → ω, a → α. They describe rotation under constant angular acceleration.
Torque
Explanation
Torque is the rotational analogue of force. r⊥ is the lever arm — the perpendicular distance from the rotation axis to the line of action of F. A force directed straight at the pivot produces zero torque.
When to use
Use whenever you need to determine if (or how fast) something will rotate; sum torques about the same axis for static equilibrium.
Newton's second law for rotation
Explanation
Net external torque on a system determines the system's angular acceleration. Direct analogue of F = ma, with rotational inertia I playing the role of mass.
Rotational inertia
Explanation
Rotational inertia measures how hard it is to change an object's rotational motion. For point masses, add miri²; the parallel-axis theorem shifts a known center-of-mass inertia to a parallel axis a distance d away.
When to use
Use when mass distribution matters, especially rolling objects, compound systems, and rotation about an axis that is not through the center of mass.
Unit 6 — Energy and Momentum of Rotating Systems
Rotational kinetic energy, angular momentum, the rotational analogue of impulse, and conservation of angular momentum for systems with no external torque.
Rotational kinetic energy
Explanation
Energy of a rotating object — analogous to ½mv² but with I and ω instead of m and v.
When to use
For rolling objects you need both translational and rotational KE: K_total = ½ m v² + ½ I ω².
Work done by a torque
Explanation
A constant torque does work equal to torque times angular displacement, the rotational analogue of W = Fd.
When to use
Use when a force causes rotation through an angle and you want to track energy transfer.
Angular momentum
Explanation
Angular momentum L is the rotational analogue of linear momentum p. For a rigid body rotating about a fixed axis, L = Iω; for a particle moving past an axis, L = rmv sinθ.
Angular impulse
Explanation
The change in angular momentum equals the (net) torque applied multiplied by the time it acts. The rotational analogue of J = FΔt = Δp.
When to use
When no external torque acts, angular momentum is conserved (L_initial = L_final). Classic example: a figure skater spinning faster after pulling their arms in (smaller I → larger ω).
Unit 7 — Oscillations
Simple harmonic motion: the periods of the spring–mass and simple-pendulum systems, the link between frequency and period, and the sine/cosine position equations for a SHM oscillator.
Period of a spring–mass oscillator
Explanation
The period (time for one full oscillation) of a mass m attached to an ideal spring of stiffness k. Period grows with mass and shrinks with stiffness — and crucially does not depend on amplitude.
When to use
Whenever a problem is about a horizontally or vertically hung mass on a spring undergoing small oscillations.
Period of a simple pendulum
Explanation
For a small-angle simple pendulum of length ℓ, the period depends only on ℓ and the local gravitational field strength g. Mass and amplitude do not matter (small-angle approximation).
When to use
Valid for small swing angles (less than about 15°). For larger angles the motion is no longer simple harmonic and the period depends on amplitude.
Frequency from period
Explanation
Frequency f and period T are reciprocals. f is in Hertz (Hz), the number of full oscillations per second.
Position in simple harmonic motion
Explanation
The position of a simple harmonic oscillator can be written with either sine or cosine. The choice depends on where the oscillator is at t = 0.
Unit 8 — Fluids (new for 2026)
Fluids are new to AP Physics 1 since the 2024–25 redesign: density, pressure, hydrostatic pressure with depth, the buoyant force from Archimedes' principle, plus the continuity and Bernoulli equations for ideal flowing fluids.
Density
Explanation
Mass per unit volume. SI unit kg/m³. Water has ρ ≈ 1000 kg/m³; air at sea level has ρ ≈ 1.2 kg/m³.
Pressure
Explanation
Force per unit area. SI unit pascal (1 Pa = 1 N/m²). Pressure is a scalar — at any point in a fluid it pushes equally in all directions.
Hydrostatic pressure (pressure with depth)
Explanation
In an incompressible fluid at rest, pressure increases linearly with depth h below the surface. P0 is the pressure at the top, and gauge pressure is the extra pressure due only to the fluid column.
When to use
Use P = P0 + ρgh for absolute pressure and Pgauge = ρgh for pressure measured relative to the surface pressure.
Buoyant force (Archimedes' principle)
Explanation
A submerged or floating object feels an upward force equal to the weight of the fluid it displaces. Determines whether an object floats (ρ_object < ρ_fluid), sinks (ρ_object > ρ_fluid), or is neutrally buoyant.
When to use
For any floating, sinking, or fully submerged object in a fluid — pair with Newton's second law to find apparent weight or net acceleration.
Continuity equation
Explanation
For an incompressible fluid in steady flow through a pipe, the volume flow rate A·v is the same everywhere along the pipe. Squeeze the pipe and the fluid speeds up; widen it and the fluid slows down.
When to use
Whenever fluid flows through a pipe whose cross-sectional area changes.
Bernoulli's equation
Explanation
Conservation of energy per unit volume for an ideal flowing fluid. The three terms are pressure energy, kinetic energy per volume, and gravitational potential energy per volume. Their sum is the same at any two points along a streamline.
When to use
When you know two points in a flow (e.g., top and bottom of a tank, narrow and wide sections of a pipe) and want to relate pressure, speed, or height between them.
Reference tables
Key reference boxes from the official PDF — constants & conversion factors, SI prefixes, common-angle trig values, and geometry & trigonometry — recreated here for quick lookup.
Constants & conversion factors
These values are printed on the AP Physics 1 exam sheet — you do not need to memorize them. Use them when a problem requires gravity, the gravitational constant, or atmospheric pressure.
| Quantity | Symbol | Approximate value |
|---|---|---|
| Acceleration due to gravity at Earth's surface | 9.8 m/s² = 9.8 N/kg | |
| Universal gravitational constant | 6.67 × 10⁻¹¹ N·m²/kg² | |
| Atmospheric pressure (1 atm) | 1.0 × 10⁵ Pa = 1 atm |
SI prefixes
Standard decimal prefixes for SI units. "50 km" is 50 × 10³ m = 5 × 10⁴ m; "3 µN" is 3 × 10⁻⁶ N.
| Factor | Prefix | Symbol |
|---|---|---|
| 10¹² | tera | T |
| 10⁹ | giga | G |
| 10⁶ | mega | M |
| 10³ | kilo | k |
| 10⁻² | centi | c |
| 10⁻³ | milli | m |
| 10⁻⁶ | micro | μ |
| 10⁻⁹ | nano | n |
| 10⁻¹² | pico | p |
Trigonometric values for common angles
Exact values for the seven angles that appear most often in AP Physics 1 problems. The 37° / 53° row reflects the classic 3–4–5 right triangle — useful for inclined-plane problems where sin and cos work out to clean fractions.
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | |||
| 30° | |||
| 37° | |||
| 45° | |||
| 53° | |||
| 60° | |||
| 90° | undefined |
Geometry & trigonometry
Area, surface area, and volume of common shapes, plus the relations among the sides and angles of a right triangle.
Rectangle
Triangle
Circle
Rectangular solid
Cylinder
Sphere
Right triangle