AP® Physics 1 Beta
The equations and constants sheet, explained
What each equation and constant on the exam sheet is for, and when to use it.
Three habits that save points
- Use the g the question gives. The sheet prints g = 9.8 m/s²; 10 m/s² is accepted. Say which one you use, and keep it the same through the whole answer.
- SI units before you substitute. Centimeters to meters, grams to kilograms, minutes to seconds. A newton is a kg·m/s² and a joule is a N·m.
- Degree mode for angles in degrees. sin 30° = 0.5 only in degree mode; rotation formulas such as s = rθ need the angle in radians.
Constants and conventions
| Equation | What it is | When you use it | Learn it in |
|---|---|---|---|
| g = 9.8 m/s² = 9.8 N/kg | The size of the acceleration of a freely falling object near Earth's surface, and the strength of Earth's gravitational field there (the same number in two units). | Weight (F_g = mg), free fall, projectiles, U_g = mgΔy, pendulums, pressure with depth. Use the value the question gives; 10 m/s² is accepted when it does not. Say which you use. | Units 1, 2 Gravitational ForcePractice it: Projectile lab, Free-body diagrams |
| G = 6.67 × 10⁻¹¹ N·m²/kg² | The universal gravitational constant. | Only with the full law of gravitation, F_g = Gm₁m₂/r², and U_G = −Gm₁m₂/r: planets, moons, satellites. Near Earth's surface use g instead. | Unit 2 Gravitational Force |
| 1 atm = 1.0 × 10⁵ Pa | Atmospheric pressure at sea level, in pascals (1 Pa = 1 N/m²). | The P₀ in P = P₀ + ρgh, the absolute pressure at the surface of a liquid open to the air. | Unit 8 Pressure |
| Conventions | Unless the question says otherwise: the frame of reference is inertial, air resistance is negligible, springs and strings are ideal, and fluids are ideal with pipes completely full. | Read them as permission to simplify. If a question adds friction, a massive pulley or drag, it says so; then the convention no longer holds. | Units 1, 2, 8 Newton's Third Law |
Motion
| Equation | What it is | When you use it | Learn it in |
|---|---|---|---|
| v_x = v_x0 + a_x t | Velocity after a time t, for constant acceleration. | Constant acceleration along one axis (including free fall) when the displacement is not given or not wanted. For a projectile, use it on each axis separately (a_x = 0, a_y = −g). | Unit 1 Representing MotionPractice it: Motion graphs, Projectile lab |
| x = x₀ + v_x0 t + ½a_x t² | Position after a time t, for constant acceleration. | Finding where an object is, or how long it takes to get there, without its final velocity: a dropped object, a projectile's fall time, a car braking for a given time. | Unit 1 Representing MotionPractice it: Motion graphs, Projectile lab |
| v_x² = v_x0² + 2a_x(x − x₀) | Velocity and displacement linked with no time in it. | Stopping distances, the height a ball rises, any constant-acceleration problem where the time is neither given nor asked. | Unit 1 Representing MotionPractice it: Motion graphs |
| x_cm = Σm_i x_i / Σm_i | The position of a system's center of mass: the mass-weighted average position. | Where a system balances or acts as one object; locating the center of mass of up to five objects. The center of mass moves only when there is a net external force. | Unit 2 Systems and Center of Mass |
Forces
| Equation | What it is | When you use it | Learn it in |
|---|---|---|---|
| a_sys = ΣF / m_sys = F_net / m_sys | Newton's second law for a system: the net external force sets the acceleration. | Every force problem with an acceleration (or with zero acceleration, where ΣF = 0). Draw the free-body diagram first, choose axes, and sum forces along each axis. | Unit 2 Newton's Second LawPractice it: Free-body diagrams, Scaling drills |
| |F_g| = G m₁m₂ / r² | The gravitational force between two masses, with r measured between their centers. | Planets, moons, satellites and how the force scales: twice the distance gives a quarter of the force. Near the surface it becomes mg. | Unit 2 Gravitational ForcePractice it: Scaling drills |
| |F_f| ≤ |μ F_N| | Friction: kinetic friction equals μ_k F_N; static friction is as large as needed, up to μ_s F_N. | Sliding objects (kinetic friction, the equal sign) and objects about to slip (static friction at its maximum). A box at rest with a small push has static friction equal to the push, not μ_s F_N. | Unit 2 Kinetic and Static FrictionPractice it: Free-body diagrams |
| F_s = −k Δx | Hooke's law: an ideal spring pulls or pushes back toward its relaxed length, in proportion to how far it is stretched or compressed. | Spring scales, springs holding a mass at rest (kΔx = mg), finding k from a force-stretch graph (the slope). The minus sign says the force points opposite the displacement. | Unit 2 Spring ForcesPractice it: Free-body diagrams, Oscillator lab |
| a_c = v² / r | The size of the centripetal acceleration of an object moving in a circle, pointing to the center. | Cars on curves, loops, orbits, balls on strings. Set the sum of the real forces toward the center equal to mv²/r; never add a separate "centripetal force" to a free-body diagram. | Unit 2 Circular MotionPractice it: Free-body diagrams |
Energy, work and power
| Equation | What it is | When you use it | Learn it in |
|---|---|---|---|
| K = ½mv² | Translational kinetic energy. | Speed from energy, and energy from speed. It depends on v², so twice the speed gives four times the kinetic energy; ΔK is not ½m(Δv)². | Unit 3 Translational Kinetic EnergyPractice it: Energy bar charts, Collision lab |
| W = F_∥ d = F d cos θ | The work done by a constant force: the part of the force along the displacement, times the distance. | How much energy a force transfers. Positive when the force has a part along the motion, negative against it, zero when it is perpendicular (a normal force on a level floor, the string on a ball in a circle). For a changing force, use the area under the force-position graph. | Unit 3 WorkPractice it: Energy bar charts |
| ΔK = ΣW_i = ΣF_∥,i d_i | The work-energy theorem: the net work on an object, the sum of the work each force does, equals its change in kinetic energy. | Speeds after a push, stopping distances with friction, and anything with a force-position graph. | Unit 3 WorkPractice it: Energy bar charts |
| U_s = ½k(Δx)² | The elastic potential energy stored in an ideal spring. | Spring launchers and bumpers, and oscillators. It depends on (Δx)², so twice the compression stores four times the energy. | Unit 3 Potential EnergyPractice it: Energy bar charts, Oscillator lab |
| U_G = −G m₁m₂ / r | The gravitational potential energy of two masses, zero when they are infinitely far apart. | Satellites and orbits far from a surface, and escape. It is negative and rises toward zero as the masses separate. | Unit 3 Potential EnergyPractice it: Rotation lab |
| ΔU_g = m g Δy | The change in gravitational potential energy near Earth's surface. | Anything that rises or falls a height Δy near the ground: ramps, hills, pendulums. Only changes matter, so choose the zero height that makes the problem simplest. | Unit 3 Potential EnergyPractice it: Energy bar charts |
| P_avg = W / Δt = ΔE / Δt | Average power: how fast energy is transferred. | Motors, people climbing stairs, lifting loads. The same work done in half the time takes twice the power. | Unit 3 PowerPractice it: Energy bar charts |
| P_inst = F_∥ v = F v cos θ | Instantaneous power: the part of a force along the velocity, times the speed. | The power a force delivers at one moment, such as a car's engine at a given speed. At constant power, the driving force falls as the speed rises. | Unit 3 Power |
Momentum
| Equation | What it is | When you use it | Learn it in |
|---|---|---|---|
| p = m v | Momentum, a vector in the direction of the velocity. | Every collision and explosion. Keep the sign: a ball that bounces back changes its momentum by more than one that stops. | Unit 4 Linear MomentumPractice it: Collision lab |
| F_net = Δp / Δt = m Δv / Δt | Newton's second law in terms of momentum: the net force is the rate of change of momentum. | Average forces in collisions, catches and landings; why a longer stopping time means a smaller force. | Unit 4 Change in Momentum and ImpulsePractice it: Collision lab |
| J = F_avg Δt = Δp | Impulse, and the impulse-momentum theorem. | Finding a change in momentum from a force and a time, or from the area under a force-time graph. | Unit 4 Change in Momentum and ImpulsePractice it: Collision lab |
| v_cm = Σp_i / Σm_i = Σm_i v_i / Σm_i | The velocity of a system's center of mass. | It stays constant whenever the net external force on the system is zero, even through a collision or explosion; the velocity of two objects that stick together. | Unit 4 Linear MomentumPractice it: Collision lab |
Rotation
| Equation | What it is | When you use it | Learn it in |
|---|---|---|---|
| ω = ω₀ + α t | Angular velocity after a time t, for constant angular acceleration. | Wheels and fans speeding up or slowing down. The rotational partner of v = v₀ + at. | Unit 5 Rotational KinematicsPractice it: Rotation lab |
| θ = θ₀ + ω₀ t + ½α t² | Angular position after a time t, for constant angular acceleration. | The angle (in radians) or number of turns a wheel makes while it speeds up. Divide by 2π to get revolutions. | Unit 5 Rotational Kinematics |
| ω² = ω₀² + 2α(θ − θ₀) | Angular velocity and angle linked with no time in it. | How many turns a wheel makes while it stops, when the time is not given. | Unit 5 Rotational Kinematics |
| v = r ω; a_T = r α | The link between a point's linear motion and the object's rotation, at a distance r from the axis. | Strings unwinding from pulleys, points on a wheel, rolling without slipping (v_cm = Rω). ω and α must be in rad/s and rad/s². | Unit 5 Linking Linear and Rotational MotionPractice it: Rotation lab |
| τ = r_⊥ F = r F sin θ | Torque: how strongly a force turns an object about an axis, set by the force and its lever arm. | Balanced beams and seesaws, wrenches, hinged rods. Choose the axis where an unknown force acts to remove it from the sum. | Unit 5 TorquePractice it: Torque balance |
| I = Σm_i r_i² | The rotational inertia of a set of point masses about an axis. | Comparing how hard objects are to spin up: mass farther from the axis counts more (r²). For a disk, rod, hoop or sphere the question gives I. | Unit 5 Rotational InertiaPractice it: Torque balance, Rotation lab |
| I' = I_cm + M d² | The parallel-axis theorem: the rotational inertia about an axis a distance d from a parallel axis through the center of mass. | A rod turning about its end instead of its center, a disk pivoted at its rim. | Unit 5 Rotational Inertia |
| α_sys = Στ / I_sys = τ_net / I_sys | Newton's second law in rotational form. | Pulleys with mass, wheels driven by a torque, rods falling about a pivot. Write it alongside ΣF = ma for any object that also moves. | Unit 5 Newton's Second Law for RotationPractice it: Torque balance, Rotation lab |
| K = ½ I ω² | Rotational kinetic energy. | Spinning objects, and rolling objects, whose total kinetic energy is ½Mv² + ½Iω². | Unit 6 Rotational Kinetic EnergyPractice it: Rotation lab |
| W = τ Δθ | The work done by a constant torque turning through an angle Δθ (in radians). | Motors and brakes on wheels; the rotational work-energy theorem, τ_net Δθ = ΔK (with the net torque). | Unit 6 Torque and Work |
| L = I ω | The angular momentum of a rotating object. | Spinning skaters and divers, turntables that catch a falling lump, merry-go-rounds. It stays constant when the net external torque is zero. | Unit 6 Angular Momentum and Angular ImpulsePractice it: Rotation lab |
| L = r m v sin θ | The angular momentum of a point object about an axis. | A child jumping onto a merry-go-round, a ball on a string, a satellite (constant because gravity points at the planet). | Unit 6 Angular Momentum and Angular Impulse |
| ΔL = τ Δt | Angular impulse: a torque acting for a time changes the angular momentum. | Spinning a wheel up or down with a known torque for a known time. | Unit 6 Angular Momentum and Angular Impulse |
| Δx_cm = r Δθ | For an object rolling without slipping, the distance its center moves is the radius times the angle it turns. | Rolling without slipping; with v = rω it gives v_cm = Rω. | Unit 6 RollingPractice it: Rotation lab |
Oscillations
| Equation | What it is | When you use it | Learn it in |
|---|---|---|---|
| T = 1 / f | The period (time for one cycle) and the frequency (cycles per second, Hz) are reciprocals. | Every oscillation and circular motion. Time ten cycles and divide by ten for an accurate period. | Units 2, 7 Frequency and Period of SHMPractice it: Oscillator lab |
| T_s = 2π √(m / k) | The period of a mass on an ideal spring. | Spring oscillators, horizontal or hanging. It does not depend on the amplitude or on g. Graph T² against m for a straight line with slope 4π²/k. | Unit 7 Frequency and Period of SHMPractice it: Oscillator lab, Scaling drills |
| T_p = 2π √(ℓ / g) | The period of a simple pendulum swinging through small angles. | Pendulums only (not springs). It does not depend on the bob's mass or the amplitude. Graph T² against ℓ for a line with slope 4π²/g. | Unit 7 Frequency and Period of SHMPractice it: Oscillator lab, Scaling drills, Lab design |
| x = A cos(2π f t) or x = A sin(2π f t) | Position against time in simple harmonic motion, with amplitude A and frequency f. | Cosine when the object starts at x = A (released from rest), sine when it starts at x = 0 moving in the positive direction. Calculator in radian mode. | Unit 7 Representing and Analyzing SHMPractice it: Oscillator lab |
Fluids
| Equation | What it is | When you use it | Learn it in |
|---|---|---|---|
| ρ = m / V | Density: mass per unit volume, in kg/m³. | Whether something floats, the mass of a displaced fluid, and every fluid equation. | Unit 8 Internal Structure and DensityPractice it: Fluids tank |
| P = F_⊥ / A | Pressure: the perpendicular force per unit area, in pascals. | Forces on surfaces such as a window in a tank, a piston or a diver's mask. Pressure is not a force; multiply by the area to get one. | Unit 8 PressurePractice it: Fluids tank |
| P = P₀ + ρ g h | Absolute pressure at a depth h below the surface of a fluid at rest. | Pressure at the bottom of a pool or a lake. It depends on depth, not on the container's shape. A graph of P against h has intercept P₀ and slope ρg. | Unit 8 PressurePractice it: Fluids tank |
| P_gauge = ρ g h | Gauge pressure: the pressure above atmospheric. | What a tire or diving gauge reads; zero at the surface of an open liquid. | Unit 8 PressurePractice it: Fluids tank |
| F_b = ρ V g | The buoyant force: the weight of the fluid displaced. ρ is the fluid's density and V the submerged volume. | Floating and sinking. A floating object at rest has F_b equal to its weight. Draw F_b and the gravitational force as two separate forces in opposite directions. | Unit 8 Fluids and Newton's LawsPractice it: Fluids tank |
| A₁v₁ = A₂v₂ | The continuity equation: an ideal fluid in a full pipe has the same volume flow rate everywhere. | Pipes that narrow or widen, nozzles, a falling stream that thins. Half the area means twice the speed. | Unit 8 Fluids and Conservation LawsPractice it: Fluids tank |
| P₁ + ρgy₁ + ½ρv₁² = P₂ + ρgy₂ + ½ρv₂² | Bernoulli's equation: energy conservation per unit volume along a flow of ideal fluid. | Pressure differences in pipes, water leaving a hole in a tank (it gives Torricelli's result), why faster flow means lower pressure in a level pipe. | Unit 8 Fluids and Conservation LawsPractice it: Fluids tank |
Geometry and trigonometry
| Equation | What it is | When you use it | Learn it in |
|---|---|---|---|
| sin θ = a/c; cos θ = b/c; tan θ = a/b; c² = a² + b² | Right-triangle trigonometry and the Pythagorean theorem. The sheet also lists sin, cos and tan of 0°, 30°, 37°, 45°, 53°, 60° and 90° (37° and 53° are the 3-4-5 triangle). | Components of forces and velocities, heights from ramp lengths (h = d sin θ), torques at an angle. Check the calculator is in degree mode. | Math you need Right-Triangle Trigonometry and Vector ComponentsPractice it: Projectile lab, Scaling drills |
| s = r θ; C = 2πr; A = πr² | Arc length (θ in radians), circumference and area of a circle. | Distance around a wheel, the period of circular motion (v = 2πr/T), pipe cross-sections for continuity. | Units 2, 5 Linking Linear and Rotational Motion |
| A = bh; A = ½bh; V = ℓwh; V = πr²ℓ; S = 2πrℓ + 2πr²; V = 4/3 πr³; S = 4πr² | Areas of a rectangle and triangle; volumes of a rectangular solid, cylinder and sphere; surface areas of a cylinder and a sphere. | Areas under graphs (displacement, work, impulse) and volumes for density and buoyancy. | Math you need Slopes and Areas of Graphs |
What is not on the sheet
The exam gives you these values in the question, or expects you to reason without them. None of them is printed on the sheet.
- Rotational inertia of a disk, rod, hoop or sphere. The question gives it when it is needed. Know that I = Σmr² for point masses, and that mass farther from the axis means a larger I. Rotational Inertia
- Escape speed. Derive it: set the total energy ½mv² − GMm/r to zero and solve for v. Motion of Orbiting Satellites
- Torricelli's result. Derive it from Bernoulli's equation: water leaving a hole a depth h below the surface moves at √(2gh). Fluids and Conservation Laws
- Kepler's third law. Derive it: set the gravitational force equal to m·v²/r for a circular orbit, with v = 2πr/T. Circular Motion
- The minimum speed at the top of a loop. Derive it: at the slowest speed, gravity alone supplies the centripetal force, so mg = mv²/r and v = √(gr). Circular Motion
- The greatest speed of an oscillator. Derive it from energy: all of ½kA² becomes ½mv² at equilibrium, so v_max = A√(k/m) = 2πfA. Energy of Simple Harmonic Oscillators
On exam day you get this sheet of equations and constants, on paper and in the testing app. The sheet itself is published by the College Board; this page is our own guide to using it.
The sheet also lists the conventions every question assumes unless it says otherwise (inertial frames, no air resistance, ideal springs, strings and fluids), unit symbols, the metric prefixes from tera to pico, and sine, cosine and tangent at 0°, 30°, 37°, 45°, 53°, 60° and 90°.
Practice with it
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