2027 exam · Equation sheet

AP® Physics 1: Algebra-Based equation sheet (2027), explained

The official Physics 1 sheet is two pages. The first has constants, metric prefixes, unit symbols, a table of sine, cosine and tangent for common angles (including the 37° and 53° of a 3-4-5 triangle) and the geometry rules. The second has every mechanics and fluids equation in one table. Below, we go through all of it in the order you learn it, with what each letter means, its unit, when to use it, the mistake people make most, and the topic that teaches it.

The official sheet: AP Physics 1: Algebra-Based Exam Reference Information (PDF, College Board) (opens in a new tab). Keep it open next to this page. We link to it instead of copying it, so you always see College Board's current version.

Checked against the 2027 version on October 5, 2026. The explanations are ours, not College Board's.

Using the sheet on exam day

You get the sheet for the whole exam, on paper and in the testing app. Learn where things are before May, so you don't lose minutes hunting.
The sheet gives equations, not judgment. It never says when an equation applies, like the kinematics equations needing constant acceleration. That's the part to study.
Its conventions are worth knowing: the frame of reference is inertial, air resistance is negligible, springs and strings are ideal, and fluids are ideal and fill their pipes, unless a question says otherwise.
Use the trig table for the 37° and 53° angles: sin 37° = 3/5 and cos 37° = 4/5, so a 3-4-5 triangle hides in a lot of questions.
Many questions want reasoning, not a number. Even then, the equation tells you how one quantity depends on another, like the period of a spring not depending on gravity.

Constants and conversions

Universal gravitational constant: $\displaystyle G = 6.67 \times 10^{-11}\ \text{m}^3/(\text{kg}\cdot\text{s}^2) = 6.67 \times 10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$; Use it in Newton's law of gravitation and gravitational potential energy, for planets, moons and satellites. It's tiny, which is why you don't feel the pull of the person next to you.
One atmosphere of pressure: $\displaystyle 1\ \text{atm} = 1.0 \times 10^5\ \text{N/m}^2 = 1.0 \times 10^5\ \text{Pa}$; The air pressure at sea level. It's the $P_0$ at the surface of an open container of water.
Acceleration due to gravity at Earth's surface: $\displaystyle g = 9.8\ \text{m/s}^2$; How fast a falling object's speed grows when only gravity acts on it near Earth's surface.
Gravitational field strength at Earth's surface: $\displaystyle g = 9.8\ \text{N/kg}$; The same number seen as a field: each kilogram near Earth's surface feels 9.8 N of gravity, so a 2.0 kg book weighs 19.6 N.

Motion in a straight line

The three kinematics equations. They work in one direction at a time, and only when the acceleration is constant.

Velocity with constant acceleration

\displaystyle v_x = v_{x0} + a_x t

What the symbols mean

$v_x$: velocity along x at time t; Unit: m/s
$v_{x0}$: starting velocity along x; Unit: m/s
$a_x$: acceleration along x, which must be constant; Unit: m/s²
$t$: time since the start; Unit: s

Use it when: Use it when you know or want the velocity after a certain time, and position doesn't come into it.

Watch out: Dropping signs. Pick a positive direction first; with up as positive, an object thrown upward has $a_y = -9.8\ \text{m/s}^2$ the whole time, even at the top where it stops for an instant.

Try it: A car moving at 4.0 m/s speeds up at a steady 2.0 m/s² for 3.0 s. How fast is it going?

Answer: $v_x = 4.0 + (2.0)(3.0) = 10\ \text{m/s}$ .

Learn it: 1.2 Displacement, Velocity, and Acceleration · 1.3 Representing Motion

Position with constant acceleration

\displaystyle x = x_0 + v_{x0} t + \frac{1}{2} a_x t^2

What the symbols mean

$x$: position at time t; Unit: m
$x_0$: starting position; Unit: m
$v_{x0}$: starting velocity; Unit: m/s
$a_x$: constant acceleration; Unit: m/s²
$t$: time; Unit: s

Use it when: Use it when time is involved and you want where the object is, or how long it takes to get somewhere. It's the one for projectiles: use it once for x and once for y.

Watch out: Forgetting to square the time, or the $\tfrac{1}{2}$ . For the car above, after 3.0 s it has gone $4.0(3.0) + \tfrac{1}{2}(2.0)(3.0)^2 = 21\ \text{m}$ .

Learn it: 1.2 Displacement, Velocity, and Acceleration · 1.5 Vectors and Motion in Two Dimensions

Velocity and position, no time

\displaystyle v_x^2 = v_{x0}^2 + 2a_x(x - x_0)

What the symbols mean

$v_x$: final velocity; Unit: m/s
$v_{x0}$: starting velocity; Unit: m/s
$a_x$: constant acceleration; Unit: m/s²
$x - x_0$: displacement; Unit: m

Use it when: Use it when time isn't given and isn't asked for, like a braking distance or how high a ball goes.

Watch out: Taking a square root and forgetting it could be negative, or forgetting that braking means $a_x$ and $v_{x0}$ have opposite signs.

Learn it: 1.2 Displacement, Velocity, and Acceleration

Forces

Newton's second law and the forces you'll draw on free-body diagrams.

Center of mass

\displaystyle \vec{x}_{cm} = \frac{\sum m_i \vec{x}_i}{\sum m_i}

What the symbols mean

$\vec{x}_{cm}$: position of the system's center of mass; Unit: m
$m_i$: mass of each object; Unit: kg
$\vec{x}_i$: position of each object; Unit: m

Use it when: Use it to find the balance point of a set of objects, like masses on a light rod. The center of mass moves as if all the mass and all the outside forces were there.

Watch out: Measuring positions from different origins. Pick one origin and measure every position from it.

Learn it: 2.1 Systems and Center of Mass

Newton's second law

\displaystyle \vec{a}_{sys} = \frac{\sum \vec{F}}{m_{sys}} = \frac{\vec{F}_{net}}{m_{sys}}

What the symbols mean

$\vec{a}_{sys}$: acceleration of the system's center of mass; Unit: m/s²
$\sum \vec{F}$: the vector sum of the forces from outside the system; Unit: N
$\vec{F}_{net}$: net force, the same sum; Unit: N
$m_{sys}$: total mass of the system; Unit: kg

Use it when: Use it after a free-body diagram: add up the forces along each direction and divide by mass. For two blocks pushed together, treat them as one system first.

Watch out: Putting a single force on top instead of the net force, or counting forces the objects in the system exert on each other. Internal forces cancel out.

Try it: A 15 N push moves a 2.0 kg block and a 3.0 kg block, touching, across a frictionless floor. What is their acceleration?

Answer: As one system, $a = 15 / (2.0 + 3.0) = 3.0\ \text{m/s}^2$ .

Learn it: 2.5 Newton’s Second Law · 2.2 Forces and Free-Body Diagrams

Newton's law of gravitation

\displaystyle \lvert \vec{F}_g \rvert = G\frac{m_1 m_2}{r^2}

What the symbols mean

$\vec{F}_g$: gravitational force each mass exerts on the other; Unit: N
$G$: universal gravitational constant; Unit: N·m²/kg²
$m_1, m_2$: the two masses; Unit: kg
$r$: distance between their centers; Unit: m

Use it when: Use it for planets, moons and satellites, where the distance changes enough to matter. Near Earth's surface it simplifies to the weight mg.

Watch out: Using the height above the surface for r. It's center to center: a satellite's r is Earth's radius plus its altitude.

Learn it: 2.6 Gravitational Force

Friction

\displaystyle \lvert \vec{F}_f \rvert \le \lvert \mu \vec{F}_N \rvert

What the symbols mean

$\vec{F}_f$: friction force; Unit: N
$\mu$: coefficient of friction, static or kinetic; Unit: none
$\vec{F}_N$: normal force; Unit: N

Use it when: Kinetic friction (sliding) equals $\mu_k F_N$ . Static friction is anything up to $\mu_s F_N$ : it's only as big as it needs to be to stop slipping.

Watch out: Writing static friction as $\mu_s F_N$ when nothing says the object is about to slip, or assuming $F_N = mg$ on a ramp, where it's $mg\cos\theta$ .

Learn it: 2.7 Kinetic and Static Friction

Spring force (Hooke's law)

\displaystyle \vec{F}_s = -k\Delta\vec{x}

What the symbols mean

$\vec{F}_s$: force the spring exerts; Unit: N
$k$: spring constant, how stiff the spring is; Unit: N/m
$\Delta\vec{x}$: stretch or compression from the spring's natural length; Unit: m

Use it when: Use it for the force from a stretched or squeezed spring, and to find k from the slope of a force–stretch graph.

Watch out: Using the spring's total length instead of how far it's stretched. The minus sign just means the force points back toward the natural length.

Learn it: 2.8 Spring Forces

Centripetal acceleration

\displaystyle a_c = \frac{v^2}{r}

What the symbols mean

$a_c$: acceleration toward the center of the circle; Unit: m/s²
$v$: speed; Unit: m/s
$r$: radius of the circle; Unit: m

Use it when: Use it with Newton's second law for anything moving in a circle: the net force toward the center equals $m v^2 / r$ .

Watch out: Drawing a separate centripetal force on the free-body diagram. It's not a new force; tension, gravity, friction or the normal force supplies it.

Learn it: 2.9 Circular Motion

Work, energy and power

Energy is often the quickest route when a problem asks for a speed or a height and doesn't care about time.

Kinetic energy

\displaystyle K = \frac{1}{2}mv^2

What the symbols mean

$K$: translational kinetic energy; Unit: J
$m$: mass; Unit: kg
$v$: speed; Unit: m/s

Use it when: Use it for the energy an object has because it's moving. It's never negative.

Watch out: Forgetting to square v. Doubling the speed makes the kinetic energy four times as big.

Learn it: 3.1 Translational Kinetic Energy

Work done by a constant force

\displaystyle W = F_\parallel d = Fd\cos\theta

What the symbols mean

$W$: work done by the force; Unit: J
$F_\parallel$: the part of the force along the motion; Unit: N
$d$: distance moved; Unit: m
$\theta$: angle between the force and the motion; Unit: ° or rad

Use it when: Use it for the energy a force adds to or takes from an object as it moves. Only the part of the force along the motion does work.

Watch out: Using the angle with the floor when it isn't the angle between the force and the motion. A force at 90° to the motion, like the normal force on a flat floor, does no work.

Learn it: 3.2 Work

Work-energy theorem

\displaystyle \Delta K = \sum W_i = \sum F_{\parallel,i}\, d_i

What the symbols mean

$\Delta K$: change in kinetic energy; Unit: J
$W_i$: work done by each force; Unit: J
$F_{\parallel,i}$: the part of each force along the motion; Unit: N
$d_i$: distance each force acts over; Unit: m

Use it when: Use it when you know the forces and the distance and want the change in speed, like a box slowing to a stop under friction.

Watch out: Leaving out a force. Every force does work (positive, negative or zero), and they all go in the sum.

Learn it: 3.2 Work · 3.4 Conservation of Energy

Spring potential energy

\displaystyle U_s = \frac{1}{2}k(\Delta x)^2

What the symbols mean

$U_s$: energy stored in the spring; Unit: J
$k$: spring constant; Unit: N/m
$\Delta x$: stretch or compression from natural length; Unit: m

Use it when: Use it for energy stored in a stretched or squeezed spring, like a launcher or a bouncing mass.

Watch out: Using the change in stretch instead of the stretch itself. Going from 0.1 m to 0.2 m stores ½k(0.2² − 0.1²), not ½k(0.1)².

Learn it: 3.3 Potential Energy · 7.4 Energy of Simple Harmonic Oscillators

Gravitational potential energy between two masses

\displaystyle U_G = -\frac{G m_1 m_2}{r}

What the symbols mean

$U_G$: gravitational potential energy of the pair; Unit: J
$G$: universal gravitational constant; Unit: N·m²/kg²
$m_1, m_2$: the two masses; Unit: kg
$r$: distance between their centers; Unit: m

Use it when: Use it for satellites and planets, where the distance changes a lot. Zero is set at infinite distance, so it's always negative.

Watch out: Dividing by $r^2$ . That's the force; the energy has $r$ to the first power.

Learn it: 3.3 Potential Energy · 6.6 Motion of Orbiting Satellites

Gravitational potential energy near Earth

\displaystyle \Delta U_g = mg\Delta y

What the symbols mean

$\Delta U_g$: change in gravitational potential energy; Unit: J
$m$: mass; Unit: kg
$g$: gravitational field strength, 9.8 N/kg; Unit: N/kg
$\Delta y$: change in height; Unit: m

Use it when: Use it whenever something changes height near Earth's surface, usually with conservation of energy to find a speed.

Watch out: Using the path length instead of the change in height. On a slide or a ramp, only the vertical drop counts.

Try it: A 0.50 kg ball drops 2.0 m from rest. How fast is it going just before it lands?

Answer: It loses $mg\Delta y = (0.50)(9.8)(2.0) = 9.8\ \text{J}$ of potential energy, all of it now kinetic: $v = \sqrt{2(9.8)/0.50} \approx 6.3\ \text{m/s}$ .

Learn it: 3.3 Potential Energy · 3.4 Conservation of Energy

Average power

\displaystyle P_{avg} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}

What the symbols mean

$P_{avg}$: average power; Unit: W (J/s)
$W$: work done; Unit: J
$\Delta E$: energy transferred; Unit: J
$\Delta t$: time taken; Unit: s

Use it when: Use it for how fast energy is transferred, like a motor lifting a load or a person running up stairs.

Watch out: Thinking more power means more work. Two people can do the same work; the one who does it faster has more power.

Learn it: 3.5 Power

Instantaneous power

\displaystyle P_{inst} = F_\parallel v = Fv\cos\theta

What the symbols mean

$P_{inst}$: power at one instant; Unit: W
$F_\parallel$: the part of the force along the velocity; Unit: N
$v$: speed at that instant; Unit: m/s
$\theta$: angle between the force and the velocity; Unit: ° or rad

Use it when: Use it when a force pushes something at a known speed, like a car's engine at highway speed.

Watch out: Using an average speed when the question asks about one moment.

Learn it: 3.5 Power

Momentum

Momentum is the tool for collisions and explosions, where forces are huge and brief.

Momentum

\displaystyle \vec{p} = m\vec{v}

What the symbols mean

$\vec{p}$: momentum; Unit: kg·m/s
$m$: mass; Unit: kg
$\vec{v}$: velocity; Unit: m/s

Use it when: Use it for any collision or explosion. Momentum is a vector, so direction matters.

Watch out: Treating it like a speed. A ball bouncing back with the same speed has changed its momentum by 2mv, not zero.

Learn it: 4.1 Linear Momentum

Net force and momentum

\displaystyle \vec{F}_{net} = \frac{\Delta\vec{p}}{\Delta t} = m\frac{\Delta\vec{v}}{\Delta t} = m\vec{a}

What the symbols mean

$\vec{F}_{net}$: net force; Unit: N
$\Delta\vec{p}$: change in momentum; Unit: kg·m/s
$\Delta t$: time interval; Unit: s
$\vec{a}$: acceleration; Unit: m/s²

Use it when: Use it to connect a force to how quickly momentum changes. It's Newton's second law written for momentum.

Watch out: Forgetting it's the net force: if two forces act, the momentum changes because of their sum.

Learn it: 4.2 Change in Momentum and Impulse · 2.5 Newton’s Second Law

Impulse

\displaystyle \vec{J} = \vec{F}_{avg}\Delta t = \Delta\vec{p}

What the symbols mean

$\vec{J}$: impulse; Unit: N·s (same as kg·m/s)
$\vec{F}_{avg}$: average force during the collision; Unit: N
$\Delta t$: how long the force acts; Unit: s
$\Delta\vec{p}$: change in momentum; Unit: kg·m/s

Use it when: Use it for a short, hard hit: a bat on a ball, an airbag, a landing. It's also the area under a force–time graph.

Watch out: Saying an airbag reduces the change in momentum. It doesn't; it makes Δt longer, so the average force is smaller.

Try it: A 0.15 kg ball moving at 20 m/s is caught and stopped in 0.010 s. What average force acts on it?

Answer: $\Delta p = 0.15(0 - 20) = -3.0\ \text{N·s}$ , so $F_{avg} = -3.0 / 0.010 = -300\ \text{N}$ : 300 N, opposite its motion.

Learn it: 4.2 Change in Momentum and Impulse

Velocity of the center of mass

\displaystyle \vec{v}_{cm} = \frac{\sum \vec{p}_i}{\sum m_i} = \frac{\sum m_i\vec{v}_i}{\sum m_i}

What the symbols mean

$\vec{v}_{cm}$: velocity of the system's center of mass; Unit: m/s
$\vec{p}_i$: momentum of each object; Unit: kg·m/s
$m_i$: mass of each object; Unit: kg
$\vec{v}_i$: velocity of each object; Unit: m/s

Use it when: Use it for collisions: with no outside force, the center of mass keeps the same velocity before and after. In a perfectly inelastic collision, the stuck-together objects move at exactly this velocity.

Watch out: Averaging the speeds without weighting by mass, or ignoring direction. Give velocities to the left a minus sign.

Learn it: 4.3 Conservation of Linear Momentum · 4.4 Elastic and Inelastic Collisions

Rotation

Each rotation equation matches a straight-line one: θ for x, ω for v, α for a, torque for force, rotational inertia for mass.

Angular velocity with constant angular acceleration

\displaystyle \omega = \omega_0 + \alpha t

What the symbols mean

$\omega$: angular velocity at time t; Unit: rad/s
$\omega_0$: starting angular velocity; Unit: rad/s
$\alpha$: constant angular acceleration; Unit: rad/s²
$t$: time; Unit: s

Use it when: Use it for a spinning object speeding up or slowing down at a steady rate, when angle doesn't come into it.

Watch out: Mixing rpm or degrees with radians. Convert first: one revolution is 2π rad.

Learn it: 5.1 Rotational Kinematics

Angle with constant angular acceleration

\displaystyle \theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2

What the symbols mean

$\theta$: angular position at time t; Unit: rad
$\theta_0$: starting angular position; Unit: rad
$\omega_0$: starting angular velocity; Unit: rad/s
$\alpha$: constant angular acceleration; Unit: rad/s²

Use it when: Use it for how far something has turned after a given time.

Watch out: Giving the answer in radians when the question asks for revolutions. Divide by 2π.

Learn it: 5.1 Rotational Kinematics

Angular velocity and angle, no time

\displaystyle \omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)

What the symbols mean

$\omega$: final angular velocity; Unit: rad/s
$\omega_0$: starting angular velocity; Unit: rad/s
$\alpha$: constant angular acceleration; Unit: rad/s²
$\theta - \theta_0$: angle turned; Unit: rad

Use it when: Use it when time isn't given or asked for, like how many turns a wheel makes while it stops.

Watch out: Using it when the angular acceleration isn't constant, like a wheel pushed by a changing torque.

Learn it: 5.1 Rotational Kinematics

Linear speed from angular speed

\displaystyle v = r\omega

What the symbols mean

$v$: speed of a point on the rotating object; Unit: m/s
$r$: distance of that point from the axis; Unit: m
$\omega$: angular speed; Unit: rad/s

Use it when: Use it to connect spinning to moving: the edge of a wheel, a string unwinding from a pulley, a point on a merry-go-round.

Watch out: Assuming every point on a turning object has the same speed. They share ω, but points farther out move faster.

Learn it: 5.2 Connecting Linear and Rotational Motion

Tangential acceleration

\displaystyle a_T = r\alpha

What the symbols mean

$a_T$: acceleration along the circle (speeding up or slowing down); Unit: m/s²
$r$: distance from the axis; Unit: m
$\alpha$: angular acceleration; Unit: rad/s²

Use it when: Use it to link a hanging mass's acceleration to a pulley's angular acceleration when the string doesn't slip.

Watch out: Mixing it up with centripetal acceleration $v^2/r$ , which points toward the center and changes direction, not speed.

Learn it: 5.2 Connecting Linear and Rotational Motion

Torque

\displaystyle \tau = r_\perp F = rF\sin\theta

What the symbols mean

$\tau$: torque, the turning effect of a force; Unit: N·m
$r_\perp$: lever arm: the perpendicular distance from the axis to the force's line; Unit: m
$r$: distance from the axis to where the force acts; Unit: m
$F$: the force; Unit: N
$\theta$: angle between r and F; Unit: ° or rad

Use it when: Use it for anything that turns: doors, seesaws, wrenches, beams resting on supports.

Watch out: Using the angle between the force and a line perpendicular to r, which swaps sin for cos. θ is measured between r and F, so a force pointing straight along r ( $\theta = 0$ ) makes no torque at all.

Learn it: 5.3 Torque

Rotational inertia of point masses

\displaystyle I = \sum m_i r_i^2

What the symbols mean

$I$: rotational inertia, how hard it is to change the spin; Unit: kg·m²
$m_i$: each mass; Unit: kg
$r_i$: each mass's distance from the axis; Unit: m

Use it when: Use it for small masses on a light rod or at the edge of a disk. It shows why mass far from the axis makes something harder to spin.

Watch out: Measuring r from the end of the rod instead of from the axis it actually turns around.

Learn it: 5.4 Rotational Inertia

Parallel-axis theorem

\displaystyle I' = I_{cm} + Md^2

What the symbols mean

$I'$: rotational inertia about the new axis; Unit: kg·m²
$I_{cm}$: rotational inertia about a parallel axis through the center of mass; Unit: kg·m²
$M$: total mass; Unit: kg
$d$: distance between the two axes; Unit: m

Use it when: Use it when an object turns about an axis that isn't through its center of mass, like a rod pivoting at one end.

Watch out: Using it between two axes when neither goes through the center of mass. One of them has to.

Learn it: 5.4 Rotational Inertia

Newton's second law for rotation

\displaystyle \alpha_{sys} = \frac{\sum\tau}{I_{sys}} = \frac{\tau_{net}}{I_{sys}}

What the symbols mean

$\alpha_{sys}$: angular acceleration of the system; Unit: rad/s²
$\sum\tau$: sum of the torques, counting direction; Unit: N·m
$\tau_{net}$: net torque; Unit: N·m
$I_{sys}$: rotational inertia of the system; Unit: kg·m²

Use it when: Use it for pulleys with mass and anything spun by a torque. If the net torque is zero, the object is in rotational equilibrium.

Watch out: Adding torques without signs. Choose clockwise or counterclockwise as positive, and take every torque about the same axis.

Try it: A net torque of 1.0 N·m acts on a wheel with rotational inertia 0.20 kg·m², starting from rest. How fast is it spinning after 2.0 s?

Answer: $\alpha = 1.0/0.20 = 5.0\ \text{rad/s}^2$ , so $\omega = 0 + (5.0)(2.0) = 10\ \text{rad/s}$ .

Learn it: 5.6 Newton’s Second Law in Rotational Form · 5.5 Rotational Equilibrium and Newton’s First Law in Rotational Form

Rotational kinetic energy

\displaystyle K = \frac{1}{2}I\omega^2

What the symbols mean

$K$: kinetic energy of spinning; Unit: J
$I$: rotational inertia; Unit: kg·m²
$\omega$: angular speed; Unit: rad/s

Use it when: Use it with energy conservation for anything that spins. A rolling ball has both kinds: $\tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$ .

Watch out: Forgetting the spinning energy for a rolling object, which makes your predicted speed at the bottom of a ramp too high.

Learn it: 6.1 Rotational Kinetic Energy · 6.5 Rolling

Work done by a torque

\displaystyle W = \tau\Delta\theta

What the symbols mean

$W$: work done; Unit: J
$\tau$: constant torque; Unit: N·m
$\Delta\theta$: angle turned; Unit: rad

Use it when: Use it for the energy a torque adds as it turns something, the rotational version of force times distance.

Watch out: Using degrees or revolutions for the angle. Only radians give joules.

Learn it: 6.2 Torque and Work

Angular momentum of a spinning object

\displaystyle L = I\omega

What the symbols mean

$L$: angular momentum; Unit: kg·m²/s
$I$: rotational inertia; Unit: kg·m²
$\omega$: angular velocity; Unit: rad/s

Use it when: Use it with conservation of angular momentum: a skater pulling in their arms lowers I, so ω goes up.

Watch out: Assuming kinetic energy is conserved too. When the skater pulls in, L stays the same but kinetic energy increases.

Learn it: 6.3 Angular Momentum and Angular Impulse · 6.4 Conservation of Angular Momentum

Angular momentum of a moving point mass

\displaystyle L = rmv\sin\theta

What the symbols mean

$L$: angular momentum about a chosen point; Unit: kg·m²/s
$r$: distance from that point to the object; Unit: m
$m$: mass; Unit: kg
$v$: speed; Unit: m/s
$\theta$: angle between r and v; Unit: ° or rad

Use it when: Use it when something moving in a straight line hits or grabs a rotating object, like a ball striking the end of a rod.

Watch out: Thinking an object needs to move in a circle to have angular momentum. Anything moving past a point has it about that point.

Learn it: 6.3 Angular Momentum and Angular Impulse · 6.4 Conservation of Angular Momentum

Angular impulse

\displaystyle \Delta L = \tau\Delta t

What the symbols mean

$\Delta L$: change in angular momentum; Unit: kg·m²/s
$\tau$: net torque; Unit: N·m
$\Delta t$: time the torque acts; Unit: s

Use it when: Use it when a torque acts for a known time. It's also the area under a torque–time graph.

Watch out: Forgetting that zero net torque means zero change: that's exactly when angular momentum is conserved.

Learn it: 6.3 Angular Momentum and Angular Impulse

Rolling without slipping

\displaystyle \Delta x_{cm} = r\Delta\theta

What the symbols mean

$\Delta x_{cm}$: distance the center moves; Unit: m
$r$: radius of the rolling object; Unit: m
$\Delta\theta$: angle it turns; Unit: rad

Use it when: Use it for wheels and balls that roll without slipping. It also gives $v_{cm} = r\omega$ and $a_{cm} = r\alpha$ .

Watch out: Using it when the object skids. If it slips, the distance moved and the turning aren't linked this way.

Learn it: 6.5 Rolling

Oscillations

Simple harmonic motion: a mass on a spring and a pendulum swinging through a small angle.

Period and frequency

\displaystyle T = \frac{1}{f}

What the symbols mean

$T$: period, the time for one full cycle; Unit: s
$f$: frequency, cycles per second; Unit: Hz (1/s)

Use it when: Use it to switch between how long one cycle takes and how many cycles happen each second.

Watch out: Counting a half swing as a full cycle. One period is there and back again.

Learn it: 7.2 Frequency and Period of SHM

Period of a mass on a spring

\displaystyle T_s = 2\pi\sqrt{\frac{m}{k}}

What the symbols mean

$T_s$: period of the spring-mass system; Unit: s
$m$: mass on the spring; Unit: kg
$k$: spring constant; Unit: N/m

Use it when: Use it for any mass bouncing on an ideal spring, horizontal or vertical.

Watch out: Thinking amplitude or gravity changes it. They don't: the same spring and mass have the same period on the Moon.

Try it: A 0.50 kg mass bounces on a spring with k = 200 N/m. What is its period?

Answer: $T_s = 2\pi\sqrt{0.50/200} = 2\pi(0.050) \approx 0.31\ \text{s}$ .

Learn it: 7.2 Frequency and Period of SHM

Period of a simple pendulum

\displaystyle T_p = 2\pi\sqrt{\frac{\ell}{g}}

What the symbols mean

$T_p$: period of the pendulum; Unit: s
$\ell$: length of the string, to the center of the bob; Unit: m
$g$: gravitational field strength; Unit: m/s²

Use it when: Use it for a small bob on a light string swinging through a small angle. A 1.0 m pendulum on Earth has a period of about 2.0 s.

Watch out: Thinking a heavier bob changes the period. Mass isn't in the equation; only length and g matter.

Learn it: 7.2 Frequency and Period of SHM

Position in simple harmonic motion

\displaystyle x = A\cos(2\pi ft) \qquad x = A\sin(2\pi ft)

What the symbols mean

$x$: position from equilibrium at time t; Unit: m
$A$: amplitude, the biggest distance from equilibrium; Unit: m
$f$: frequency; Unit: Hz
$t$: time; Unit: s

Use it when: Use cosine when the object starts at its farthest point (x = A at t = 0), and sine when it starts at equilibrium moving in the positive direction.

Watch out: Leaving your calculator in degrees. The 2πft inside is in radians.

Learn it: 7.3 Representing and Analyzing SHM

Fluids

Density, pressure, buoyancy and flowing fluids, the newest unit in Physics 1.

Density

\displaystyle \rho = \frac{m}{V}

What the symbols mean

$\rho$: density; Unit: kg/m³
$m$: mass; Unit: kg
$V$: volume; Unit: m³

Use it when: Use it to get a mass from a volume or the other way round. Water's density is about 1000 kg/m³.

Watch out: Using g/cm³ in a formula that needs kg/m³. 1 g/cm³ is 1000 kg/m³.

Learn it: 8.1 Internal Structure and Density

Pressure

\displaystyle P = \frac{F_\perp}{A}

What the symbols mean

$P$: pressure; Unit: Pa (N/m²)
$F_\perp$: force at a right angle to the surface; Unit: N
$A$: area the force spreads over; Unit: m²

Use it when: Use it for the force a fluid puts on a surface, like water on a dam or air on a window.

Watch out: Using the whole force when it's at an angle. Only the part perpendicular to the surface counts.

Learn it: 8.2 Pressure

Pressure at a depth

\displaystyle P = P_0 + \rho gh

What the symbols mean

$P$: absolute pressure at depth h; Unit: Pa
$P_0$: pressure at the surface (often 1 atm); Unit: Pa
$\rho$: density of the fluid; Unit: kg/m³
$g$: gravitational field strength; Unit: m/s²
$h$: depth below the surface; Unit: m

Use it when: Use it for the total pressure in a liquid, like on a diver.

Watch out: Leaving out the air pressure on top when the question asks for absolute pressure.

Try it: What is the absolute pressure 10 m under the surface of a lake (water's density 1000 kg/m³)?

Answer: $P = 1.0 \times 10^5 + (1000)(9.8)(10) = 1.98 \times 10^5\ \text{Pa}$ , about 2.0 × 10⁵ Pa: roughly double the pressure at the surface.

Learn it: 8.2 Pressure

Gauge pressure

\displaystyle P_{gauge} = \rho gh

What the symbols mean

$P_{gauge}$: pressure above the surrounding air pressure; Unit: Pa
$\rho$: fluid density; Unit: kg/m³
$g$: gravitational field strength; Unit: m/s²
$h$: depth; Unit: m

Use it when: Use it when a question asks how much extra pressure the liquid itself adds, on top of the air pressure at its surface.

Watch out: Mixing up gauge and absolute pressure. Absolute is gauge plus the air pressure above.

Learn it: 8.2 Pressure

Buoyant force

\displaystyle F_b = \rho Vg

What the symbols mean

$F_b$: upward buoyant force; Unit: N
$\rho$: density of the fluid, not the object; Unit: kg/m³
$V$: volume of fluid pushed aside (the submerged volume); Unit: m³
$g$: gravitational field strength; Unit: m/s²

Use it when: Use it for anything floating or submerged. A floating object's buoyant force equals its weight.

Watch out: Using the object's density, or its whole volume when it's only partly underwater.

Learn it: 8.3 Fluids and Newton’s Laws

Continuity equation

\displaystyle A_1 v_1 = A_2 v_2

What the symbols mean

$A_1, A_2$: cross-sectional areas of the pipe at two points; Unit: m²
$v_1, v_2$: fluid speeds at those points; Unit: m/s

Use it when: Use it when a pipe narrows or widens: the same volume flows past every point each second, so a narrower pipe means faster flow.

Watch out: Halving the diameter and thinking the speed doubles. Area depends on the diameter squared, so the speed goes up 4 times.

Learn it: 8.4 Fluids and Conservation Laws

Bernoulli's equation

\displaystyle 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

What the symbols mean

$P_1, P_2$: pressure at two points in the flow; Unit: Pa
$\rho$: fluid density; Unit: kg/m³
$g$: gravitational field strength; Unit: m/s²
$y_1, y_2$: heights of the two points; Unit: m
$v_1, v_2$: fluid speeds at the two points; Unit: m/s

Use it when: Use it for flowing fluids: it's conservation of energy per unit volume. Faster flow at the same height means lower pressure.

Watch out: Using it without continuity. Usually you need continuity first to find the second speed.

Learn it: 8.4 Fluids and Conservation Laws

Geometry and trigonometry

The shapes and right-triangle rules every physics booklet ends with. You'll mostly use them to find areas for pressure, volumes for density, and components of vectors.

Area of a rectangle and a triangle

\displaystyle A = bh \qquad A = \frac{1}{2}bh

What the symbols mean

$A$: area; Unit: m²
$b$: base; Unit: m
$h$: height, measured at a right angle to the base; Unit: m

Use it when: Use these for the area under a graph: the area under a velocity–time graph is displacement, and the area under a force–time graph is impulse.

Watch out: Forgetting the $\tfrac{1}{2}$ for a triangular piece of a graph. Split odd shapes into rectangles and triangles and add them.

Learn it: 1.3 Representing Motion · 4.2 Change in Momentum and Impulse

Area and circumference of a circle

\displaystyle A = \pi r^2 \qquad C = 2\pi r

What the symbols mean

$A$: area; Unit: m²
$C$: circumference, the distance once around; Unit: m
$r$: radius; Unit: m

Use it when: Use $A = \pi r^2$ for the cross-section of a pipe in the continuity equation, and $C = 2\pi r$ for the distance covered in one lap of circular motion, so $v = 2\pi r/T$ .

Watch out: Plugging in the diameter. Problems often give a pipe's diameter; halve it first.

Learn it: 8.4 Fluids and Conservation Laws · 2.9 Circular Motion

Arc length

\displaystyle s = r\theta

What the symbols mean

$s$: arc length, the distance along the circle; Unit: m
$r$: radius; Unit: m
$\theta$: angle swept out, in radians; Unit: rad

Use it when: Use it to turn an angle turned into a distance traveled along a circle, for example the length of string unwound from a spool.

Watch out: Using degrees. The angle must be in radians: 180° is π rad.

Learn it: 5.1 Rotational Kinematics · 5.2 Connecting Linear and Rotational Motion

Volume of a box and a cylinder

\displaystyle V = \ell wh \qquad V = \pi r^2 \ell

What the symbols mean

$V$: volume; Unit: m³
$\ell$: length; Unit: m
$w$: width; Unit: m
$h$: height; Unit: m
$r$: radius of the cylinder; Unit: m

Use it when: Use these to find the volume of an object for its density or for the buoyant force when it's underwater.

Watch out: Mixing centimeters and meters. Convert lengths to meters before you multiply: 1 cm³ is 10⁻⁶ m³, not 10⁻² m³.

Learn it: 8.1 Internal Structure and Density · 8.3 Fluids and Newton’s Laws

Surface area of a cylinder

\displaystyle S = 2\pi r\ell + 2\pi r^2

What the symbols mean

$S$: total surface area: the curved side plus both ends; Unit: m²
$r$: radius; Unit: m
$\ell$: length of the cylinder; Unit: m

Use it when: Use it when a force or pressure acts over the outside of a can or pipe. For pressure on just one end, you only want the end's area, πr².

Watch out: Using the whole surface area when the problem only cares about one face, like the bottom of a container.

Learn it: 8.2 Pressure

Volume and surface area of a sphere

\displaystyle V = \frac{4}{3}\pi r^3 \qquad S = 4\pi r^2

What the symbols mean

$V$: volume; Unit: m³
$S$: surface area; Unit: m²
$r$: radius; Unit: m

Use it when: Use the volume for the buoyant force on a ball or balloon, or for the mass of a planet from its density.

Watch out: Doubling the radius doesn't double the volume: it multiplies it by 8, because r is cubed.

Learn it: 8.1 Internal Structure and Density · 8.3 Fluids and Newton’s Laws

Pythagorean theorem

\displaystyle a^2 + b^2 = c^2

What the symbols mean

$a$: the side opposite the angle θ; Unit: same as the sides
$b$: the side next to the angle θ; Unit: same as the sides
$c$: the hypotenuse, the longest side; Unit: same as the sides

Use it when: Use it to find the size of a vector from its two perpendicular components, like a velocity from its horizontal and vertical parts.

Watch out: Adding the components instead: 3 m/s east and 4 m/s north is 5 m/s, not 7 m/s.

Learn it: 1.5 Vectors and Motion in Two Dimensions · 1.1 Scalars and Vectors in One Dimension

Sine, cosine and tangent

\displaystyle \sin\theta = \frac{a}{c} \qquad \cos\theta = \frac{b}{c} \qquad \tan\theta = \frac{a}{b}

What the symbols mean

$\theta$: the angle you're working from; Unit: ° or rad
$a$: the side opposite θ; Unit: same as the sides
$b$: the side next to θ; Unit: same as the sides
$c$: the hypotenuse; Unit: same as the sides

Use it when: Use them to split a vector into components: a force $F$ at angle $\theta$ above the horizontal has a horizontal part $F\cos\theta$ and a vertical part $F\sin\theta$ .

Watch out: Grabbing cos for the horizontal part every time. It's cos only when θ is measured from the horizontal; on a ramp, the part of gravity along the slope is mg sin θ.

Learn it: 1.5 Vectors and Motion in Two Dimensions · 2.2 Forces and Free-Body Diagrams

Not on the sheet: know these

The exam expects you to know these without being given them.

Weight near Earth's surface
$\displaystyle F_g = mg$
The sheet only has the general law of gravitation. Near the surface, the force of gravity on an object is just its mass times 9.8 N/kg.
Learn it: 2.6 Gravitational Force
Forces on a ramp
$\displaystyle F_\parallel = mg\sin\theta \qquad F_N = mg\cos\theta$
On a frictionless ramp at angle θ, gravity's pull along the slope is mg sin θ and the normal force is mg cos θ, if nothing else pushes into the ramp.
Learn it: 2.2 Forces and Free-Body Diagrams · 2.5 Newton’s Second Law
Newton's third law
Forces come in pairs: equal in size, opposite in direction, acting on two different objects. Because they act on different objects, they never cancel each other.
Learn it: 2.3 Newton’s Third Law
Conservation of energy with outside work
$\displaystyle K_i + U_i + W_{other} = K_f + U_f$
The sheet lists each kind of energy but not how to put them together. Total energy stays the same unless an outside force (like friction) does work on the system.
Learn it: 3.4 Conservation of Energy
Conservation of momentum
$\displaystyle \sum \vec{p}_i = \sum \vec{p}_f$
With no net outside force, a system's total momentum stays the same. In an elastic collision kinetic energy is conserved too; in an inelastic one it isn't.
Learn it: 4.3 Conservation of Linear Momentum · 4.4 Elastic and Inelastic Collisions
Projectiles: two motions at once
Horizontal and vertical motion are independent. With no air resistance, $v_x$ stays constant and $a_y = -9.8\ \text{m/s}^2$ ; at the top of the path, $v_y = 0$ but the speed isn't zero.
Learn it: 1.5 Vectors and Motion in Two Dimensions
Reading motion graphs
Slope of position–time is velocity, slope of velocity–time is acceleration, and the area under velocity–time is displacement. Free-response questions ask for these all the time.
Learn it: 1.3 Representing Motion
Orbital speed
$\displaystyle v = \sqrt{\frac{GM}{r}}$
Set gravity equal to the centripetal force, $GMm/r^2 = mv^2/r$ , to get it. The satellite's own mass cancels out.
Learn it: 6.6 Motion of Orbiting Satellites · 2.9 Circular Motion
Energy in simple harmonic motion
$\displaystyle E = \frac{1}{2}kA^2$
The total energy of a spring-mass oscillator. Speed is greatest at equilibrium, where all the energy is kinetic, and zero at the ends, where it's all spring energy.
Learn it: 7.4 Energy of Simple Harmonic Oscillators