138 equations · variable meanings · SI units · printable PDF

AP Physics 1 Formula Sheet & Complete Equation Library 2027

This page includes every current exam-reference equation plus the essential definitions, derived relationships, graph relationships and common model formulas used across AP Physics 1. Every formula card now defines each symbol and gives its SI unit, so the equation, variable meaning and unit convention stay together.

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Quick answer

What is on the AP Physics 1 formula sheet?

This formula resource contains 52 current exam-reference entries plus 86 clearly labeled course/derived relationships, for 138 searchable relationships total. Every formula card explains variables, SI units, meaning and use conditions.

Key facts

52 exam-reference entries.
86 course/derived relationships.
138/138 cards include variable names and SI units.

Verified for 2026–27Updated August 31, 2026How this page is verified

Does this include all AP Physics 1 formulas?

Yes for practical AP Physics 1 study: the library contains all 52 current exam-reference entries plus 86 essential course definitions and derived/model relationships. The second group is intentionally labeled because many useful relationships are not printed as separate lines on the exam reference sheet.

52exam-reference entries

86course & derived formulas

138searchable relationships

138/138formula cards include variable names & SI units

Interactive formula library

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Search by formula, symbol, variable name, SI unit, concept or use. Filter by course unit and by whether the relationship is printed on the exam reference information or is an essential course/derived formula.

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138formulas shown

Exam-reference information

Constants & conversions

4 formulas in this group.

Constants & conversionsExam reference

G = 6.67 \times 10^{-11}\,\mathrm{m^3/(kg\,s^2)} = 6.67 \times 10^{-11}\,\mathrm{N\,m^2/kg^2}

G = 6.67 x 10^-11 m^3/(kg s^2) = 6.67 x 10^-11 N m^2/kg^2

Universal gravitational constant

Use: Universal gravitation and gravitational potential energy.

Watch: Use center-to-center separation r.

Variables & SI units

Guniversal gravitational constantN·m²/kg²

Constants & conversionsExam reference

1\,\mathrm{atm} = 1.0 \times 10^5\,\mathrm{N/m^2} = 1.0 \times 10^5\,\mathrm{Pa}

1 atm = 1.0 x 10^5 N/m^2 = 1.0 x 10^5 Pa

Atmospheric pressure

Use: Pressure comparisons and fluid problems.

Watch: Keep absolute pressure and gauge pressure distinct.

Variables & SI units

atmstandard atmosphere1 atm = 1.0×10⁵ Pa

Papascal, SI pressure unitN/m²

Nnewton, SI force unitkg·m/s²

Constants & conversionsExam reference

g = 9.8\,\mathrm{m/s^2}

g = 9.8 m/s^2

Acceleration due to gravity

Use: Near-Earth mechanics.

Watch: Choose the axis before assigning a sign to g.

Variables & SI units

gacceleration due to gravitym/s²

Constants & conversionsExam reference

g = 9.8\,\mathrm{N/kg}

g = 9.8 N/kg

Gravitational field strength

Use: Weight and gravitational-field interpretation near Earth.

Watch: Same numerical magnitude as 9.8 m/s^2, but different units and meaning.

Variables & SI units

ggravitational field strength near EarthN/kg

Exam-reference information

Unit 1 · Kinematics

3 formulas in this group.

Unit 1 · KinematicsExam reference

v_x = v_{x0} + a_x t

v_x = v_x0 + a_x t

Constant-acceleration velocity

Use: Connect velocity, acceleration and time.

Watch: Only valid when a_x is constant.

Variables & SI units

v_x0initial x-component of velocitym/s

v_xx-component of velocitym/s

a_xx-component of accelerationm/s²

ttimes

Unit 1 · KinematicsExam reference

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

x = x_0 + v_x0 t + 1/2 a_x t^2

Constant-acceleration position

Use: Connect position/displacement with time.

Watch: Keep x and y components separate in 2D motion.

Variables & SI units

v_x0initial x-component of velocitym/s

a_xx-component of accelerationm/s²

x_0initial x-coordinatem

xposition coordinatem

ttimes

Unit 1 · KinematicsExam reference

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

v_x^2 = v_x0^2 + 2 a_x (x - x_0)

Time-free kinematics

Use: Use when time is absent from the useful known/unknown set.

Watch: Squaring can hide direction information; restore the physical sign afterward.

Variables & SI units

v_x0initial x-component of velocitym/s

v_xx-component of velocitym/s

a_xx-component of accelerationm/s²

x_0initial x-coordinatem

xposition coordinatem

Exam-reference information

Unit 2 · Force & Translational Dynamics

6 formulas in this group.

Unit 2 · Force & Translational DynamicsExam reference

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

x_cm(vector) = sum(m_i x_i(vector)) / sum(m_i)

Center-of-mass position

Use: Locate the center of mass of a system.

Watch: All coordinates/vectors must use the same reference frame.

Variables & SI units

x_cmcenter-of-mass positionm

x_iposition of object im

m_imass of object ikg

Unit 2 · Force & Translational DynamicsExam reference

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

a_sys(vector) = sum F(vector) / m_sys = F_net(vector) / m_sys

System acceleration

Use: Apply Newton’s second law to a defined system.

Watch: Include only forces external to that chosen system.

Variables & SI units

a_sysacceleration of the chosen systemm/s²

ΣFvector sum of external forcesN

m_systotal system masskg

F_netnet external forceN

Unit 2 · Force & Translational DynamicsExam reference

\left|\vec{F}_g\right| = G\frac{m_1m_2}{r^2}

|F_g| = G m_1 m_2 / r^2

Universal gravitational force

Use: Find the gravitational interaction between separated masses.

Watch: r is center-to-center separation.

Variables & SI units

F_ggravitational-force magnitudeN

Guniversal gravitational constantN·m²/kg²

m_1mass of object 1kg

m_2mass of object 2kg

rcenter-to-center separationm

Unit 2 · Force & Translational DynamicsExam reference

\left|\vec{F}_f\right| \le \left|\mu\vec{F}_N\right|

|F_f| <= |mu F_N|

Friction relationship

Use: Model static or kinetic friction with the appropriate coefficient.

Watch: Static friction is not automatically equal to mu_s F_N.

Variables & SI units

F_ffriction forceN

F_Nnormal forceN

μcoefficient of frictiondimensionless

Unit 2 · Force & Translational DynamicsExam reference

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

F_s(vector) = -k Delta x(vector)

Ideal spring force

Use: Relate ideal-spring force to the change in spring length measured from relaxed length.

Watch: The sign gives force direction; the object-spring system equilibrium can differ from relaxed length when other forces such as gravity act.

Variables & SI units

Δxchange in spring length from relaxed length (extension/compression)m

F_sspring forceN

kspring constantN/m

Unit 2 · Force & Translational DynamicsExam reference

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

a_c = v^2 / r

Centripetal acceleration

Use: Find inward acceleration for curved or circular motion.

Watch: Centripetal force is not an extra force; identify the real inward net force.

Variables & SI units

a_ccentripetal accelerationm/s²

rradius / separationm

vspeed / velocity magnitudem/s

Exam-reference information

Unit 3 · Work, Energy & Power

8 formulas in this group.

Unit 3 · Work, Energy & PowerExam reference

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

K = 1/2 m v^2

Translational kinetic energy

Use: Energy associated with translational speed.

Watch: Kinetic energy is scalar and nonnegative.

Variables & SI units

mmasskg

vspeed / velocity magnitudem/s

Kkinetic energyJ

Unit 3 · Work, Energy & PowerExam reference

W = F_{\parallel}d = Fd\cos\theta

W = F_parallel d = F d cos(theta)

Work by a constant force

Use: Calculate energy transfer by a force through a displacement.

Watch: Use only the force component parallel to displacement.

Variables & SI units

F_parallelforce component parallel to displacement/velocityN

θanglerad

Fforce magnitudeN

ddistance / displacement / axis separationm

WworkJ

Unit 3 · Work, Energy & PowerExam reference

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

Delta K = sum W_i = sum F_parallel,i d_i

Work-energy theorem

Use: Connect net work to change in kinetic energy.

Watch: Choose the system boundary consistently.

Variables & SI units

ΔKchange in kinetic energyJ

ΣW_inet work / sum of worksJ

F_∥,iforce component parallel to displacement for force iN

d_idisplacement associated with force im

Unit 3 · Work, Energy & PowerExam reference

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

U_s = 1/2 k (Delta x)^2

Elastic potential energy

Use: Energy stored by an ideal spring as it is stretched or compressed from the spring’s equilibrium/relaxed length used by the elastic-energy model.

Watch: Do not automatically equate this spring-length reference with the attached object–spring system equilibrium position when other forces shift the system equilibrium.

Variables & SI units

Δxspring stretch/compression from the spring’s equilibrium/relaxed length used by the elastic-energy model; do not confuse this with a shifted object–spring system equilibrium positionm

U_selastic potential energyJ

kspring constantN/m

Unit 3 · Work, Energy & PowerExam reference

U_G = -G\frac{m_1m_2}{r}

U_G = -G m_1 m_2 / r

Universal gravitational potential energy

Use: Use when gravitational field varies appreciably with r.

Watch: Zero reference is at infinite separation in this model.

Variables & SI units

U_Ggravitational potential energyJ

Guniversal gravitational constantN·m²/kg²

m_1mass of object 1kg

m_2mass of object 2kg

rcenter-to-center separationm

Unit 3 · Work, Energy & PowerExam reference

\Delta U_g = mg\,\Delta y

Delta U_g = m g Delta y

Near-Earth gravitational potential change

Use: Use for vertical changes where g is effectively constant.

Watch: Select the vertical coordinate consistently.

Variables & SI units

ΔU_gchange in gravitational potential energyJ

Δychange in vertical positionm

ggravitational acceleration / field strengthm/s² (equiv. N/kg)

mmasskg

Unit 3 · Work, Energy & PowerExam reference

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

P_avg = W / Delta t = Delta E / Delta t

Average power

Use: Rate of work or energy transfer over an interval.

Watch: Power is a rate, not an amount of energy.

Variables & SI units

Δtelapsed times

ΔEchange in energyJ

P_avgaverage powerW

WworkJ

Unit 3 · Work, Energy & PowerExam reference

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

P_inst = F_parallel v = F v cos(theta)

Instantaneous mechanical power

Use: Find power at an instant from force and speed.

Watch: Use the force component parallel to velocity.

Variables & SI units

F_parallelforce component parallel to displacement/velocityN

P_instinstantaneous powerW

θanglerad

Fforce magnitudeN

vspeed / velocity magnitudem/s

Exam-reference information

Unit 4 · Linear Momentum

4 formulas in this group.

Unit 4 · Linear MomentumExam reference

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

p(vector) = m v(vector)

Linear momentum

Use: Describe translational momentum.

Watch: Momentum is a vector; retain direction.

Variables & SI units

mmasskg

vspeed / velocity magnitudem/s

plinear momentumkg·m/s

Unit 4 · Linear MomentumExam reference

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

F_net(vector) = Delta p / Delta t = m Delta v / Delta t = m a

Momentum form of Newton’s second law

Use: Connect net force to momentum change over time.

Watch: Use net external force for a chosen system.

Variables & SI units

Δvchange in velocitym/s

Δtelapsed times

Δpchange in momentumkg·m/s

F_netnet forceN

mmasskg

aaccelerationm/s²

Unit 4 · Linear MomentumExam reference

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

J(vector) = F_avg(vector) Delta t = Delta p(vector)

Impulse-momentum theorem

Use: Connect force-time information to momentum change.

Watch: Area under a force-time graph is impulse.

Variables & SI units

Δtelapsed times

Δpchange in momentumkg·m/s

F_avgaverage forceN

JimpulseN·s (kg·m/s)

Unit 4 · Linear MomentumExam reference

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

v_cm(vector) = sum p_i / sum m_i = sum(m_i v_i) / sum m_i

Center-of-mass velocity

Use: Find the velocity of a multi-object system’s center of mass.

Watch: Vector velocities must retain direction/sign.

Variables & SI units

v_cmcenter-of-mass velocitym/s

m_imass of object ikg

v_iinitial velocitym/s

p_imomentum of object ikg·m/s

Exam-reference information

Units 5–6 · Rotation & Rotating Systems

15 formulas in this group.

Units 5–6 · Rotation & Rotating SystemsExam reference

\omega = \omega_0 + \alpha t

omega = omega_0 + alpha t

Constant-angular-acceleration velocity

Use: Rotational analogue of v = v0 + at.

Watch: alpha must be constant.

Variables & SI units

ω_0initial angular velocityrad/s

ωangular velocity / angular speedrad/s

αangular accelerationrad/s²

ttimes

Units 5–6 · Rotation & Rotating SystemsExam reference

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

theta = theta_0 + omega_0 t + 1/2 alpha t^2

Constant-angular-acceleration position

Use: Relate angular displacement to time.

Watch: Use radians when linking angular and linear quantities.

Variables & SI units

ω_0initial angular velocityrad/s

θ_0initial angular positionrad

αangular accelerationrad/s²

θanglerad

ttimes

Units 5–6 · Rotation & Rotating SystemsExam reference

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

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

Time-free rotational kinematics

Use: Use when time is not needed.

Watch: Sign convention still matters.

Variables & SI units

ω_0initial angular velocityrad/s

θ_0initial angular positionrad

ωangular velocity / angular speedrad/s

αangular accelerationrad/s²

θanglerad

Units 5–6 · Rotation & Rotating SystemsExam reference

v = r\omega

v = r omega

Tangential speed

Use: Connect linear speed to angular speed.

Watch: r is perpendicular distance from the rotation axis.

Variables & SI units

ωangular velocity / angular speedrad/s

rradius / separationm

vspeed / velocity magnitudem/s

Units 5–6 · Rotation & Rotating SystemsExam reference

a_T = r\alpha

a_T = r alpha

Tangential acceleration

Use: Connect tangential and angular acceleration.

Watch: This does not include radial/centripetal acceleration.

Variables & SI units

αangular accelerationrad/s²

a_Ttangential accelerationm/s²

rradius / separationm

Units 5–6 · Rotation & Rotating SystemsExam reference

\tau = r_{\perp}F = rF\sin\theta

tau = r_perp F = r F sin(theta)

Torque

Use: Quantify a force’s rotational effect about an axis.

Watch: theta is the angle between r and F; line of action matters.

Variables & SI units

τtorque magnitudeN·m

r_⊥perpendicular lever armm

rdistance from axis to point of force applicationm

Fforce magnitudeN

θangle between position vector and forcerad

Units 5–6 · Rotation & Rotating SystemsExam reference

I = \sum m_i r_i^2

I = sum m_i r_i^2

Rotational inertia

Use: Compute rotational inertia for point masses.

Watch: Mass farther from the axis contributes with r^2.

Variables & SI units

m_imass of object ikg

r_idistance of mass i from rotation axism

Irotational inertiakg·m²

Units 5–6 · Rotation & Rotating SystemsExam reference

I^\prime = I_{cm} + Md^2

I' = I_cm + M d^2

Parallel-axis theorem

Use: Shift rotational inertia to a parallel axis.

Watch: d is the separation between the center-of-mass axis and new axis.

Variables & SI units

I'rotational inertia about shifted axiskg·m²

I_cmrotational inertia about parallel center-of-mass axiskg·m²

Mtotal masskg

ddistance between the two parallel axesm

Units 5–6 · Rotation & Rotating SystemsExam reference

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

alpha_sys = sum tau / I_sys = tau_net / I_sys

Rotational Newton’s second law

Use: Connect net external torque and angular acceleration.

Watch: Torque and rotational inertia must be evaluated about the same axis.

Variables & SI units

α_sysangular acceleration of systemrad/s²

Στsum of external torquesN·m

I_sysrotational inertia of systemkg·m²

τ_netnet external torqueN·m

Units 5–6 · Rotation & Rotating SystemsExam reference

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

K_rot = 1/2 I omega^2

Rotational kinetic energy

Use: Energy associated with rigid-body rotation.

Watch: Rolling can have translational and rotational kinetic energy simultaneously.

Variables & SI units

ωangular velocity / angular speedrad/s

K_rotrotational kinetic energyJ

Irotational inertiakg·m²

Units 5–6 · Rotation & Rotating SystemsExam reference

W = \tau\,\Delta\theta

W = tau Delta theta

Rotational work

Use: Work done by a constant torque through an angle.

Watch: Use Delta theta in radians.

Variables & SI units

Wrotational workJ

τtorqueN·m

Δθangular displacementrad

Units 5–6 · Rotation & Rotating SystemsExam reference

L = I\omega

L = I omega

Rigid-body angular momentum

Use: Angular momentum of a rigid rotating system about an axis.

Watch: Conservation requires negligible net external torque about that axis.

Variables & SI units

ωangular velocity / angular speedrad/s

Langular momentumkg·m²/s

Irotational inertiakg·m²

Units 5–6 · Rotation & Rotating SystemsExam reference

L = rmv\sin\theta

L = r m v sin(theta)

Particle angular momentum

Use: Angular momentum of a moving particle about an origin/axis.

Watch: Use the component perpendicular to r.

Variables & SI units

θanglerad

mmasskg

rradius / separationm

vspeed / velocity magnitudem/s

Langular momentumkg·m²/s

Units 5–6 · Rotation & Rotating SystemsExam reference

\Delta L = \tau\,\Delta t

Delta L = tau Delta t

Angular impulse

Use: Connect torque-time to angular-momentum change.

Watch: Use net external torque for a system.

Variables & SI units

ΔLchange in angular momentumkg·m²/s

τnet torque during intervalN·m

Δttime intervals

Units 5–6 · Rotation & Rotating SystemsExam reference

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

Delta x_cm = r Delta theta

Rolling displacement relation

Use: Connect center-of-mass translation to rotation.

Watch: Requires rolling without slipping.

Variables & SI units

Δx_cmcenter-of-mass displacementm

rrolling radiusm

Δθangular displacementrad

Exam-reference information

Unit 7 · Oscillations

5 formulas in this group.

Unit 7 · OscillationsExam reference

T = \frac{1}{f}

T = 1/f

Period-frequency relation

Use: Convert frequency to period or vice versa.

Watch: Do not confuse f with angular frequency.

Variables & SI units

Tperiods

ffrequencyHz (s⁻¹)

Unit 7 · OscillationsExam reference

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

T_s = 2 pi sqrt(m/k)

Mass-spring period

Use: Ideal mass-spring SHM period.

Watch: Ideal period is independent of amplitude.

Variables & SI units

T_smass-spring periods

πpi, mathematical constantdimensionless

mmasskg

kspring constantN/m

Unit 7 · OscillationsExam reference

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

T_p = 2 pi sqrt(l/g)

Simple-pendulum period

Use: Small-angle pendulum SHM.

Watch: Large angles violate the small-angle approximation.

Variables & SI units

T_ppendulum periods

πpi, mathematical constantdimensionless

ggravitational acceleration / field strengthm/s² (equiv. N/kg)

llengthm

Unit 7 · OscillationsExam reference

x = A\cos(2\pi f t)

x = A cos(2 pi f t)

SHM position · cosine form

Use: Position versus time when the phase matches a cosine start.

Watch: Select the phase form that matches initial conditions.

Variables & SI units

xdisplacement from equilibriumm

Aamplitudem

ffrequencyHz (s⁻¹)

ttimes

πpi, mathematical constantdimensionless

Unit 7 · OscillationsExam reference

x = A\sin(2\pi f t)

x = A sin(2 pi f t)

SHM position · sine form

Use: Position versus time for an alternate starting phase.

Watch: Sine and cosine forms differ by phase, not by physics.

Variables & SI units

xdisplacement from equilibriumm

Aamplitudem

ffrequencyHz (s⁻¹)

ttimes

πpi, mathematical constantdimensionless

Exam-reference information

Unit 8 · Fluids

7 formulas in this group.

Unit 8 · FluidsExam reference

\rho = \frac{m}{V}

rho = m/V

Density

Use: Mass per unit volume.

Watch: Track volume-unit conversions carefully.

Variables & SI units

ρdensitykg/m³

mmasskg

Vvolumem³

Unit 8 · FluidsExam reference

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

P = F_perp/A

Pressure

Use: Normal force distributed over an area.

Watch: Pressure is scalar even though force is vector.

Variables & SI units

PpressurePa

F_⊥force normal to the surfaceN

Asurface aream²

Unit 8 · FluidsExam reference

P = P_0 + \rho gh

P = P_0 + rho g h

Absolute hydrostatic pressure

Use: Pressure at a depth in a static fluid.

Watch: h is vertical depth.

Variables & SI units

P_0reference/surface pressurePa

ρdensitykg/m³

ggravitational acceleration / field strengthm/s² (equiv. N/kg)

hheight / depthm

PpressurePa

Unit 8 · FluidsExam reference

P_{gauge} = \rho gh

P_gauge = rho g h

Gauge pressure

Use: Pressure above a reference/surface pressure.

Watch: Do not add P_0 twice.

Variables & SI units

P_gaugegauge pressurePa

ρdensitykg/m³

ggravitational acceleration / field strengthm/s² (equiv. N/kg)

hheight / depthm

Unit 8 · FluidsExam reference

F_b = \rho Vg

F_b = rho V g

Buoyant force

Use: Archimedes principle using displaced-fluid volume.

Watch: V is displaced-fluid volume, not automatically the whole object volume.

Variables & SI units

F_bbuoyant forceN

ρfluid densitykg/m³

Vdisplaced fluid volumem³

ggravitational acceleration magnitudem/s²

Unit 8 · FluidsExam reference

A_1v_1 = A_2v_2

A_1 v_1 = A_2 v_2

Continuity

Use: Steady incompressible flow through changing area.

Watch: Smaller cross-sectional area means greater speed under the model assumptions.

Variables & SI units

v_1flow speed at point 1m/s

v_2flow speed at point 2m/s

A_1cross-sectional area 1m²

A_2cross-sectional area 2m²

Unit 8 · FluidsExam reference

P_1 + \rho gy_1 + \frac{1}{2}\rho v_1^2 = P_2 + \rho gy_2 + \frac{1}{2}\rho v_2^2

P_1 + rho g y_1 + 1/2 rho v_1^2 = P_2 + rho g y_2 + 1/2 rho v_2^2

Bernoulli equation

Use: Connect pressure, height and speed along a streamline.

Watch: Use only for the stated ideal-flow assumptions.

Variables & SI units

P_1pressure at point 1Pa

P_2pressure at point 2Pa

ρfluid densitykg/m³

ggravitational acceleration magnitudem/s²

y_1vertical position at point 1m

y_2vertical position at point 2m

v_1flow speed at point 1m/s

v_2flow speed at point 2m/s

Course formula library

Math, vectors & SI relationships

8 formulas in this group.

Math, vectors & SI relationshipsCourse / derived

1\,\mathrm{N}=1\,\mathrm{kg\,m/s^2}

1 N = 1 kg m/s^2

Newton in base SI units

Use: Check force dimensions and unit conversions.

Watch: A newton is a derived unit, not a base unit.

Variables & SI units

Nnewtonkg·m/s²

kgkilogramkg

mmetrem

sseconds

Math, vectors & SI relationshipsCourse / derived

1\,\mathrm{J}=1\,\mathrm{N\,m}

1 J = 1 N m

Joule relationship

Use: Connect work/energy to force-distance units.

Watch: N·m used for energy is called a joule; torque also has N·m dimensions but is not energy.

Variables & SI units

JjouleN·m

Nnewtonkg·m/s²

mmetrem

Math, vectors & SI relationshipsCourse / derived

1\,\mathrm{W}=1\,\mathrm{J/s}

1 W = 1 J/s

Watt relationship

Use: Connect power to energy transfer rate.

Watch: Power is energy per time, not total energy.

Variables & SI units

WwattJ/s

JjouleN·m

sseconds

Math, vectors & SI relationshipsCourse / derived

1\,\mathrm{Pa}=1\,\mathrm{N/m^2}

1 Pa = 1 N/m^2

Pascal relationship

Use: Convert pressure units.

Watch: Pressure uses force divided by area.

Variables & SI units

PapascalN/m²

Nnewtonkg·m/s²

mmetrem

Math, vectors & SI relationshipsCourse / derived

\theta_{rad}=\theta_{deg}\frac{\pi}{180}

theta_rad = theta_deg pi/180

Degrees to radians

Use: Convert angles before using arc-length and angular-motion relationships.

Watch: Rotational linear relations such as s=r theta require radians.

Variables & SI units

θ_radangle in radiansrad

θ_degangle in degrees° (non-SI)

πpi, mathematical constantdimensionless

Math, vectors & SI relationshipsCourse / derived

1\,\mathrm{rev}=2\pi\,\mathrm{rad}

1 rev = 2 pi rad

Revolution-radian conversion

Use: Convert cycles/revolutions to angular displacement.

Watch: Do not mix rev and rad inside v=r omega.

Variables & SI units

revrevolution2π rad

radradiandimensionless SI angle unit

πpi, mathematical constantdimensionless

Math, vectors & SI relationshipsCourse / derived

A_x=A\cos\theta

A_x = A cos(theta)

Vector x-component

Use: Resolve a vector when theta is measured from +x.

Watch: If the angle is measured from another axis, component assignment changes.

Variables & SI units

A_xx-component of vector Asame unit as A

Amagnitude of vector Acontext dependent

θangle measured from +x directionrad

Math, vectors & SI relationshipsCourse / derived

A_y=A\sin\theta

A_y = A sin(theta)

Vector y-component

Use: Resolve a vector when theta is measured from +x.

Watch: Assign the sign from direction after finding the magnitude.

Variables & SI units

A_yy-component of vector Asame unit as A

Amagnitude of vector Acontext dependent

θangle measured from +x directionrad

Course formula library

Unit 1 · Kinematics · essential derived formulas

14 formulas in this group.

Unit 1 · Kinematics · essential derived formulasCourse / derived

\bar{v}=\frac{\Delta x}{\Delta t}

v_avg = Delta x / Delta t

Average velocity

Use: Definition from displacement over elapsed time.

Watch: Average velocity uses displacement, not total distance.

Variables & SI units

Δxchange in position / displacementm

Δtelapsed times

v_avgaverage velocitym/s

Unit 1 · Kinematics · essential derived formulasCourse / derived

\text{average speed}=\frac{\text{total distance}}{\Delta t}

average speed = total distance / Delta t

Average speed

Use: Scalar rate based on path length.

Watch: Do not replace distance with displacement if the object reverses direction.

Variables & SI units

v_avg(speed)average speedm/s

distancetotal distance traveledm

Δtelapsed times

Unit 1 · Kinematics · essential derived formulasCourse / derived

\bar{a}=\frac{\Delta v}{\Delta t}

a_avg = Delta v / Delta t

Average acceleration

Use: Definition from velocity change over time.

Watch: Velocity change is signed/vector; speed change alone is not enough.

Variables & SI units

Δvchange in velocitym/s

Δtelapsed times

a_avgaverage accelerationm/s²

Unit 1 · Kinematics · essential derived formulasCourse / derived

v_{avg}=\frac{v_i+v_f}{2}\quad(\text{constant }a)

v_avg = (v_i + v_f)/2 for constant acceleration

Average velocity for constant acceleration

Use: Shortcut when acceleration is constant.

Watch: Not valid for arbitrary nonconstant acceleration.

Variables & SI units

v_avgaverage velocitym/s

v_iinitial velocitym/s

v_ffinal velocitym/s

Unit 1 · Kinematics · essential derived formulasCourse / derived

\Delta x=v_{avg}\Delta t\quad(\text{constant }a)

Delta x = v_avg Delta t for constant acceleration

Displacement from average velocity

Use: Combine with the constant-a average velocity relation.

Watch: Use only with the appropriate average velocity.

Variables & SI units

Δxchange in position / displacementm

Δtelapsed times

v_avgaverage velocitym/s

Unit 1 · Kinematics · essential derived formulasCourse / derived

v_{0x}=v_0\cos\theta

v_0x = v_0 cos(theta)

Projectile horizontal launch component

Use: Resolve launch velocity into x-component.

Watch: Angle convention matters.

Variables & SI units

θanglerad

v_0xinitial horizontal velocity componentm/s

v_0initial speedm/s

Unit 1 · Kinematics · essential derived formulasCourse / derived

v_{0y}=v_0\sin\theta

v_0y = v_0 sin(theta)

Projectile vertical launch component

Use: Resolve launch velocity into y-component.

Watch: Assign the vertical sign from your axis choice.

Variables & SI units

θanglerad

v_0yinitial vertical velocity componentm/s

v_0initial speedm/s

Unit 1 · Kinematics · essential derived formulasCourse / derived

x=x_0+v_{0x}t\quad(a_x=0)

x = x_0 + v_0x t when a_x=0

Horizontal projectile motion

Use: Use when horizontal acceleration is negligible.

Watch: Horizontal and vertical motions share the same time.

Variables & SI units

v_0xinitial horizontal velocity componentm/s

a_xx-component of accelerationm/s²

x_0initial x-coordinatem

xposition coordinatem

ttimes

Unit 1 · Kinematics · essential derived formulasCourse / derived

y=y_0+v_{0y}t-\frac{1}{2}gt^2

y = y_0 + v_0y t - 1/2 g t^2

Vertical projectile motion

Use: Use with +y upward and negligible air resistance.

Watch: If you choose downward positive, signs change.

Variables & SI units

v_0yinitial vertical velocity componentm/s

y_0initial y-coordinatem

ggravitational acceleration / field strengthm/s² (equiv. N/kg)

yvertical positionm

ttimes

Unit 1 · Kinematics · essential derived formulasCourse / derived

v_y=v_{0y}-gt

v_y = v_0y - g t

Projectile vertical velocity

Use: Use with +y upward.

Watch: At the top, v_y=0 but acceleration is still -g.

Variables & SI units

v_0yinitial vertical velocity componentm/s

v_yvertical velocity componentm/s

ggravitational acceleration / field strengthm/s² (equiv. N/kg)

ttimes

Unit 1 · Kinematics · essential derived formulasCourse / derived

v_{A/B}=v_{A/C}-v_{B/C}

v_A/B = v_A/C - v_B/C

Relative velocity

Use: Connect velocities measured in different inertial frames.

Watch: Keep one sign convention for the whole chain.

Variables & SI units

v_A/Bvelocity of A relative to Bm/s

v_A/Cvelocity of A relative to Cm/s

v_B/Cvelocity of B relative to Cm/s

Unit 1 · Kinematics · essential derived formulasCourse / derived

T=\frac{2v_0\sin\theta}{g}\quad(\text{same launch/landing height})

T = 2 v_0 sin(theta)/g for same launch and landing height

Projectile time of flight

Use: Shortcut for symmetric projectile motion.

Watch: Not valid when launch and landing heights differ.

Variables & SI units

Ttime of flights

v_0launch speedm/s

θlaunch anglerad

ggravitational acceleration magnitudem/s²

Unit 1 · Kinematics · essential derived formulasCourse / derived

R=\frac{v_0^2\sin(2\theta)}{g}\quad(\text{same launch/landing height})

R = v_0^2 sin(2 theta)/g for same launch and landing height

Projectile range

Use: Shortcut for ideal same-level launch/landing.

Watch: Not valid for different vertical endpoints or air resistance.

Variables & SI units

Rhorizontal rangem

v_0launch speedm/s

θlaunch anglerad

ggravitational acceleration magnitudem/s²

Unit 1 · Kinematics · essential derived formulasCourse / derived

h_{max}=\frac{v_0^2\sin^2\theta}{2g}

h_max = v_0^2 sin^2(theta)/(2g)

Maximum height above launch point

Use: Derived from v_y^2 relation at the top.

Watch: This is height above the launch level, not necessarily above the ground.

Variables & SI units

h_maxmaximum height above launch pointm

v_0launch speedm/s

θlaunch anglerad

ggravitational acceleration magnitudem/s²

Course formula library

Unit 2 · Forces · essential derived formulas

8 formulas in this group.

Unit 2 · Forces · essential derived formulasCourse / derived

\sum\vec{F}=m\vec{a}

sum F(vector) = m a(vector)

Newton’s second law

Use: Core force model for an object/system of constant mass.

Watch: Draw the free-body diagram before writing component equations.

Variables & SI units

ΣFvector sum of forcesN

mmasskg

aaccelerationm/s²

Unit 2 · Forces · essential derived formulasCourse / derived

F_g=mg

F_g = m g

Near-Earth weight magnitude

Use: Weight near Earth when g is effectively constant.

Watch: Weight is a force; mass is not weight.

Variables & SI units

F_gweight / gravitational force magnitudeN

mmasskg

ggravitational field strengthN/kg (equiv. m/s²)

Unit 2 · Forces · essential derived formulasCourse / derived

f_k=\mu_k F_N

f_k = mu_k F_N

Kinetic friction magnitude

Use: Sliding-contact friction model.

Watch: Direction opposes relative sliding motion.

Variables & SI units

f_kkinetic-friction force magnitudeN

μ_kcoefficient of kinetic frictiondimensionless

F_Nnormal forceN

Unit 2 · Forces · essential derived formulasCourse / derived

f_{s,max}=\mu_s F_N

f_s,max = mu_s F_N

Maximum static friction

Use: Threshold before slipping begins.

Watch: Actual static friction can be anywhere from zero up to this maximum.

Variables & SI units

f_s,maxmaximum static-friction forceN

μ_scoefficient of static frictiondimensionless

F_Nnormal forceN

Unit 2 · Forces · essential derived formulasCourse / derived

F_{radial}=m\frac{v^2}{r}

F_radial = m v^2/r

Required inward net force

Use: Net radial force required for circular motion.

Watch: Do not add a separate “centripetal force” to the free-body diagram.

Variables & SI units

F_radialnet radial/inward forceN

mmasskg

rradius / separationm

vspeed / velocity magnitudem/s

Unit 2 · Forces · essential derived formulasCourse / derived

g(r)=\frac{GM}{r^2}

g(r) = G M/r^2

Gravitational field magnitude at radius r

Use: Field produced by a spherical mass M outside it.

Watch: r is measured from the center of the source mass.

Variables & SI units

g(r)gravitational field magnitude at radius rN/kg (equiv. m/s²)

Guniversal gravitational constantN·m²/kg²

Msource/central masskg

rdistance from source centerm

Unit 2 · Forces · essential derived formulasCourse / derived

v=\frac{2\pi r}{T}

v = 2 pi r/T

Speed in uniform circular motion

Use: Relate period to path length per revolution.

Watch: Use the radius of the circular path.

Variables & SI units

πpi, mathematical constantdimensionless

rradius / separationm

vspeed / velocity magnitudem/s

Tperiods

Unit 2 · Forces · essential derived formulasCourse / derived

a_c=\frac{4\pi^2r}{T^2}

a_c = 4 pi^2 r/T^2

Centripetal acceleration from period

Use: Combine v=2pi r/T with a_c=v^2/r.

Watch: Only for uniform circular speed.

Variables & SI units

a_ccentripetal accelerationm/s²

πpi, mathematical constantdimensionless

rradius / separationm

Tperiods

Course formula library

Unit 3 · Energy · essential derived formulas

7 formulas in this group.

Unit 3 · Energy · essential derived formulasCourse / derived

E_{mech}=K+U

E_mech = K + U

Mechanical energy

Use: Combine kinetic and relevant potential energies.

Watch: Which potential energies belong depends on the chosen system.

Variables & SI units

E_mechmechanical energyJ

Kkinetic energyJ

Upotential energyJ

Unit 3 · Energy · essential derived formulasCourse / derived

K_i+U_i=K_f+U_f\quad(\text{isolated mechanical model})

K_i + U_i = K_f + U_f for an isolated mechanical model

Mechanical-energy conservation

Use: Use when no energy is transferred out of the modeled mechanical system.

Watch: Friction can convert mechanical energy to thermal energy depending on system choice.

Variables & SI units

K_iinitial kinetic energyJ

U_iinitial potential energyJ

K_ffinal kinetic energyJ

U_ffinal potential energyJ

Unit 3 · Energy · essential derived formulasCourse / derived

K_i+U_i+W_{ext}=K_f+U_f

K_i + U_i + W_ext = K_f + U_f

Mechanical energy with external work

Use: Account for work transferring energy across the chosen system boundary.

Watch: Be explicit about what is inside the system.

Variables & SI units

W_extexternal workJ

K_iinitial kinetic energyJ

K_ffinal kinetic energyJ

U_iinitial potential energyJ

U_ffinal potential energyJ

Unit 3 · Energy · essential derived formulasCourse / derived

W_{nc}=\Delta E_{mech}

W_nc = Delta E_mech

Nonconservative work and mechanical energy

Use: Useful bookkeeping when nonconservative work changes mechanical energy.

Watch: Sign depends on whether nonconservative work adds or removes mechanical energy.

Variables & SI units

W_ncnonconservative workJ

ΔE_mechchange in mechanical energyJ

Unit 3 · Energy · essential derived formulasCourse / derived

W_s=-\Delta U_s

W_s = -Delta U_s

Work done by an ideal spring

Use: Conservative spring work is opposite the change in spring potential energy.

Watch: Distinguish work done by the spring from work done on the spring.

Variables & SI units

W_swork done by ideal springJ

ΔU_schange in elastic potential energyJ

Unit 3 · Energy · essential derived formulasCourse / derived

W_g=-\Delta U_g

W_g = -Delta U_g

Work done by gravity

Use: Conservative gravitational work is opposite gravitational potential-energy change.

Watch: The sign depends on initial and final vertical positions.

Variables & SI units

ΔU_gchange in gravitational potential energyJ

W_gwork done by gravityJ

Unit 3 · Energy · essential derived formulasCourse / derived

\eta=\frac{E_{useful}}{E_{input}}\times100\%

efficiency = useful energy output / energy input x 100%

Efficiency

Use: Compare useful output to input energy (or power).

Watch: Efficiency cannot exceed 100% for an ordinary passive device.

Variables & SI units

ηefficiencydimensionless (often %)

E_usefuluseful output energyJ

E_inputinput energyJ

Course formula library

Unit 4 · Momentum · essential derived formulas

6 formulas in this group.

Unit 4 · Momentum · essential derived formulasCourse / derived

\vec{p}_{sys}=\sum_i\vec{p}_i

p_sys(vector) = sum p_i(vector)

System momentum

Use: Add vector momenta of all objects in the chosen system.

Watch: Direction/sign matters.

Variables & SI units

p_systotal system momentumkg·m/s

p_imomentum of object ikg·m/s

Unit 4 · Momentum · essential derived formulasCourse / derived

\sum\vec{p}_i=\sum\vec{p}_f\quad(J_{ext}\approx0)

sum p_initial = sum p_final when external impulse is negligible

Momentum conservation

Use: Use when net external impulse on the system is negligible.

Watch: Momentum is conserved for the system, not necessarily each object.

Variables & SI units

Σp_initialtotal initial momentumkg·m/s

Σp_finaltotal final momentumkg·m/s

Unit 4 · Momentum · essential derived formulasCourse / derived

J=\text{signed area under the }F\text{-}t\text{ graph}

J = signed area under the force-time graph

Impulse from a force-time graph

Use: Use graph area to find momentum change.

Watch: Area below the time axis contributes negative impulse.

Variables & SI units

JimpulseN·s (kg·m/s)

Fforce on graphN

ttime on graphs

Unit 4 · Momentum · essential derived formulasCourse / derived

v_f=\frac{m_1v_{1i}+m_2v_{2i}}{m_1+m_2}\quad(\text{stick together})

v_f = (m_1 v_1i + m_2 v_2i)/(m_1+m_2) for a perfectly inelastic collision

Perfectly inelastic collision speed

Use: Shortcut after two objects stick together.

Watch: This assumes external impulse is negligible.

Variables & SI units

v_fcommon final velocity after stickingm/s

m_1mass of object 1kg

v_1iinitial velocity of object 1m/s

m_2mass of object 2kg

v_2iinitial velocity of object 2m/s

Unit 4 · Momentum · essential derived formulasCourse / derived

\frac12m_1v_{1i}^2+\frac12m_2v_{2i}^2=\frac12m_1v_{1f}^2+\frac12m_2v_{2f}^2\quad(\text{elastic})

total K_initial = total K_final for an elastic collision

Elastic-collision kinetic energy

Use: Use together with momentum conservation for ideal elastic collisions.

Watch: Momentum conservation alone does not imply kinetic-energy conservation.

Variables & SI units

K_initialinitial total kinetic energyJ

K_finalfinal total kinetic energyJ

Unit 4 · Momentum · essential derived formulasCourse / derived

\Delta K_{collision}=K_f-K_i

Delta K_collision = K_f - K_i

Collision kinetic-energy change

Use: Quantify kinetic-energy loss or gain during a collision.

Watch: Negative Delta K is common for inelastic collisions.

Variables & SI units

ΔK_collisioncollision kinetic-energy changeJ

K_iinitial kinetic energyJ

K_ffinal kinetic energyJ

Course formula library

Unit 5 · Torque & rotational dynamics · essential derived formulas

15 formulas in this group.

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

\Delta\theta=\theta_f-\theta_i

Delta theta = theta_f - theta_i

Angular displacement

Use: Rotational analogue of linear displacement.

Watch: Use radians in linear-angular conversions.

Variables & SI units

Δθangular displacementrad

θ_iinitial angular positionrad

θ_ffinal angular positionrad

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

\omega_{avg}=\frac{\Delta\theta}{\Delta t}

omega_avg = Delta theta / Delta t

Average angular velocity

Use: Angular displacement per time.

Watch: Angular direction/sign follows the chosen rotational convention.

Variables & SI units

Δθangular displacementrad

ω_avgaverage angular velocityrad/s

Δtelapsed times

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

\alpha_{avg}=\frac{\Delta\omega}{\Delta t}

alpha_avg = Delta omega / Delta t

Average angular acceleration

Use: Angular-velocity change per time.

Watch: Positive/negative depends on rotational sign convention.

Variables & SI units

Δωchange in angular velocityrad/s

α_avgaverage angular accelerationrad/s²

Δtelapsed times

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

s=r\theta

s = r theta

Arc length

Use: Connect angular displacement and distance along a circle.

Watch: theta must be in radians.

Variables & SI units

θanglerad

rradius / separationm

sarc lengthm

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

\omega=2\pi f=\frac{2\pi}{T}

omega = 2 pi f = 2 pi/T

Angular speed from frequency or period

Use: Convert cycle rate to radians per second.

Watch: Do not confuse ordinary frequency f with angular speed omega.

Variables & SI units

ωangular velocity / angular speedrad/s

πpi, mathematical constantdimensionless

Tperiods

ffrequencyHz (s⁻¹)

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

a_c=\omega^2r

a_c = omega^2 r

Centripetal acceleration from angular speed

Use: Equivalent to v^2/r when v=r omega.

Watch: This is radial, not tangential, acceleration.

Variables & SI units

ωangular velocity / angular speedrad/s

a_ccentripetal accelerationm/s²

rradius / separationm

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

F_{radial}=m\omega^2r

F_radial = m omega^2 r

Required radial net force from angular speed

Use: Net inward force for uniform circular motion.

Watch: Identify which real forces provide it.

Variables & SI units

F_radialnet radial/inward forceN

ωangular velocity / angular speedrad/s

mmasskg

rradius / separationm

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

\sum\tau=0\quad(\text{rotational equilibrium})

sum tau = 0 for rotational equilibrium

Rotational equilibrium condition

Use: Use when angular acceleration is zero.

Watch: Choose one axis and one sign convention for every torque.

Variables & SI units

Στsum of torques about the chosen axisN·m

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

\sum\vec{F}=0\quad(\text{translational equilibrium})

sum F(vector) = 0 for translational equilibrium

Translational equilibrium condition

Use: Use with torque equilibrium for a static rigid body.

Watch: Zero net force alone does not guarantee zero net torque.

Variables & SI units

ΣFvector sum of external forcesN

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

I_{hoop}=MR^2

I_hoop = M R^2

Thin hoop rotational inertia

Use: Common ideal rigid-body model about its symmetry axis.

Watch: Use only for the stated hoop/ring geometry and axis.

Variables & SI units

I_hooprotational inertia of thin hoopkg·m²

Mmass of body / central bodykg

Rradius / rangem

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

I_{disk}=\frac12MR^2

I_disk = 1/2 M R^2

Solid disk/cylinder rotational inertia

Use: Common ideal model about central symmetry axis.

Watch: Axis and mass distribution matter.

Variables & SI units

I_diskrotational inertia of solid disk/cylinderkg·m²

Mmass of body / central bodykg

Rradius / rangem

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

I_{sphere}=\frac25MR^2

I_sphere = 2/5 M R^2

Solid sphere rotational inertia

Use: Common ideal model about a diameter.

Watch: Do not use for a hollow sphere.

Variables & SI units

I_sphererotational inertia of solid spherekg·m²

Mmass of body / central bodykg

Rradius / rangem

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

I_{shell}=\frac23MR^2

I_shell = 2/3 M R^2

Thin spherical shell rotational inertia

Use: Common ideal shell model about a diameter.

Watch: Use only for a thin shell.

Variables & SI units

I_shellrotational inertia of thin spherical shellkg·m²

Mmass of body / central bodykg

Rradius / rangem

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

I_{rod,cm}=\frac{1}{12}ML^2

I_rod,cm = 1/12 M L^2

Thin rod about center

Use: Common ideal rod model with axis through center perpendicular to rod.

Watch: Different axes require a different I or parallel-axis theorem.

Variables & SI units

I_rod,cmrotational inertia of thin rod about centerkg·m²

Mrod masskg

Lrod lengthm

Unit 5 · Torque & rotational dynamics · essential derived formulasCourse / derived

I_{rod,end}=\frac13ML^2

I_rod,end = 1/3 M L^2

Thin rod about one end

Use: Common ideal rod model with axis through one end perpendicular to rod.

Watch: Do not confuse with the center-axis value.

Variables & SI units

I_rod,endrotational inertia of thin rod about one endkg·m²

Mrod masskg

Lrod lengthm

Course formula library

Unit 6 · Energy & momentum of rotating systems · essential derived formulas

8 formulas in this group.

Unit 6 · Energy & momentum of rotating systems · essential derived formulasCourse / derived

K_{rolling}=\frac12Mv_{cm}^2+\frac12I_{cm}\omega^2

K_rolling = 1/2 M v_cm^2 + 1/2 I_cm omega^2

Total rolling kinetic energy

Use: Add translational and rotational kinetic energy for a rolling rigid body.

Watch: Requires correct I about the center-of-mass axis.

Variables & SI units

K_rollingtotal rolling kinetic energyJ

ωangular velocity / angular speedrad/s

v_cmcenter-of-mass velocitym/s

I_cmrotational inertia about center-of-mass axiskg·m²

Mmass of body / central bodykg

Unit 6 · Energy & momentum of rotating systems · essential derived formulasCourse / derived

v_{cm}=R\omega\quad(\text{no slip})

v_cm = R omega for rolling without slipping

Rolling speed condition

Use: Link center-of-mass speed to angular speed.

Watch: Not valid during slipping.

Variables & SI units

v_cmcenter-of-mass speedm/s

Rrolling radiusm

ωangular speedrad/s

Unit 6 · Energy & momentum of rotating systems · essential derived formulasCourse / derived

a_{cm}=R\alpha\quad(\text{no slip})

a_cm = R alpha for rolling without slipping

Rolling acceleration condition

Use: Link center-of-mass and angular acceleration.

Watch: Not valid during slipping.

Variables & SI units

a_cmcenter-of-mass accelerationm/s²

Rrolling radiusm

αangular accelerationrad/s²

Unit 6 · Energy & momentum of rotating systems · essential derived formulasCourse / derived

\sum L_i=\sum L_f\quad(\tau_{ext}\approx0)

sum L_initial = sum L_final when external torque is negligible

Angular-momentum conservation

Use: Use about an axis where net external torque is negligible.

Watch: Angular momentum depends on the chosen origin/axis.

Variables & SI units

ΣL_initialtotal initial angular momentumkg·m²/s

ΣL_finaltotal final angular momentumkg·m²/s

Unit 6 · Energy & momentum of rotating systems · essential derived formulasCourse / derived

v_{orbit}=\sqrt{\frac{GM}{r}}

v_orbit = sqrt(G M/r)

Circular-orbit speed

Use: Speed for a small satellite in a circular orbit about mass M.

Watch: r is orbital radius from the center of M.

Variables & SI units

v_orbitcircular-orbit speedm/s

Guniversal gravitational constantN·m²/kg²

Mmass of body / central bodykg

rradius / separationm

Unit 6 · Energy & momentum of rotating systems · essential derived formulasCourse / derived

T_{orbit}=2\pi\sqrt{\frac{r^3}{GM}}

T_orbit = 2 pi sqrt(r^3/(G M))

Circular-orbit period

Use: Period of a circular orbit about mass M.

Watch: Derived for circular orbit.

Variables & SI units

T_orbitorbital periods

πpi, mathematical constantdimensionless

Guniversal gravitational constantN·m²/kg²

Mmass of body / central bodykg

rradius / separationm

Unit 6 · Energy & momentum of rotating systems · essential derived formulasCourse / derived

K_{orbit}=\frac{GMm}{2r}

K_orbit = G M m/(2r)

Circular-orbit kinetic energy

Use: Kinetic energy in a circular gravitational orbit.

Watch: Specific to a circular orbit.

Variables & SI units

K_orbitorbital kinetic energyJ

Guniversal gravitational constantN·m²/kg²

Mmass of body / central bodykg

mmasskg

Unit 6 · Energy & momentum of rotating systems · essential derived formulasCourse / derived

E_{orbit}=-\frac{GMm}{2r}

E_orbit = -G M m/(2r)

Circular-orbit total mechanical energy

Use: Total K+U for a circular gravitational orbit.

Watch: Negative total energy indicates a bound orbit.

Variables & SI units

E_orbittotal orbital mechanical energyJ

Guniversal gravitational constantN·m²/kg²

Mmass of body / central bodykg

mmasskg

Course formula library

Unit 7 · Oscillations · essential derived formulas

9 formulas in this group.

Unit 7 · Oscillations · essential derived formulasCourse / derived

\omega_{SHM}=2\pi f=\frac{2\pi}{T}

omega_SHM = 2 pi f = 2 pi/T

Angular frequency

Use: Convert between SHM period/frequency and angular frequency.

Watch: Angular frequency is not the same quantity as ordinary frequency.

Variables & SI units

ω_SHMangular frequency of SHMrad/s

πpi, mathematical constantdimensionless

Tperiods

ffrequencyHz (s⁻¹)

Unit 7 · Oscillations · essential derived formulasCourse / derived

\omega_s=\sqrt{\frac{k}{m}}

omega_s = sqrt(k/m)

Mass-spring angular frequency

Use: Ideal spring-mass oscillator.

Watch: Assumes ideal spring and SHM.

Variables & SI units

ω_smass-spring angular frequencyrad/s

mmasskg

kspring constantN/m

Unit 7 · Oscillations · essential derived formulasCourse / derived

\omega_p=\sqrt{\frac{g}{\ell}}

omega_p = sqrt(g/l)

Small-angle pendulum angular frequency

Use: Ideal small-angle pendulum.

Watch: Large amplitudes break the approximation.

Variables & SI units

ω_ppendulum angular frequencyrad/s

ggravitational acceleration / field strengthm/s² (equiv. N/kg)

llengthm

Unit 7 · Oscillations · essential derived formulasCourse / derived

a=-\frac{k}{m}x=-\omega^2x

a = -(k/m)x = -omega^2 x

SHM acceleration-position relation

Use: Restoring acceleration is proportional and opposite to displacement.

Watch: x is measured from equilibrium.

Variables & SI units

ωangular velocity / angular speedrad/s

xposition coordinatem

mmasskg

aaccelerationm/s²

kspring constantN/m

Unit 7 · Oscillations · essential derived formulasCourse / derived

v_{max}=\omega A

v_max = omega A

Maximum SHM speed

Use: Maximum speed occurs at equilibrium.

Watch: A is amplitude from equilibrium.

Variables & SI units

v_maxmaximum SHM speedm/s

ωangular frequencyrad/s

Aamplitudem

Unit 7 · Oscillations · essential derived formulasCourse / derived

a_{max}=\omega^2A

a_max = omega^2 A

Maximum SHM acceleration

Use: Maximum acceleration magnitude occurs at maximum displacement.

Watch: Direction is toward equilibrium.

Variables & SI units

a_maxmaximum SHM accelerationm/s²

ωangular frequencyrad/s

Aamplitudem

Unit 7 · Oscillations · essential derived formulasCourse / derived

E_{SHM}=\frac12kA^2

E_SHM = 1/2 k A^2

Total ideal spring oscillator energy

Use: Total mechanical energy in ideal spring SHM.

Watch: Assumes no dissipative energy transfer.

Variables & SI units

E_SHMtotal oscillator energyJ

kspring constantN/m

Aamplitudem

Unit 7 · Oscillations · essential derived formulasCourse / derived

K=\frac12k(A^2-x^2)

K = 1/2 k (A^2 - x^2)

SHM kinetic energy vs position

Use: From total energy minus spring potential energy.

Watch: Works for the ideal spring oscillator model.

Variables & SI units

Kkinetic energyJ

kspring constantN/m

Aamplitudem

xdisplacement from equilibriumm

Unit 7 · Oscillations · essential derived formulasCourse / derived

v=\omega\sqrt{A^2-x^2}

v = omega sqrt(A^2 - x^2)

SHM speed vs position

Use: Relate speed to position without time.

Watch: This gives speed magnitude; direction requires motion context.

Variables & SI units

vspeed at displacement xm/s

ωangular frequencyrad/s

Aamplitudem

xdisplacement from equilibriumm

Course formula library

Unit 8 · Fluids · essential derived formulas

11 formulas in this group.

Unit 8 · Fluids · essential derived formulasCourse / derived

Q=\frac{\Delta V}{\Delta t}

Q = Delta V/Delta t

Volume flow rate

Use: Volume of fluid passing per unit time.

Watch: Do not confuse volume V with speed v.

Variables & SI units

Qvolume flow ratem³/s

ΔVvolume transferredm³

Δtelapsed times

Unit 8 · Fluids · essential derived formulasCourse / derived

Q=Av

Q = A v

Volume flow rate in a uniform cross section

Use: Connect cross-sectional area and average flow speed.

Watch: Use area perpendicular to flow.

Variables & SI units

Qvolume flow ratem³/s

Across-sectional aream²

vaverage flow speedm/s

Unit 8 · Fluids · essential derived formulasCourse / derived

F=PA

F = P A

Force from uniform pressure

Use: Rearrangement of P=F_perp/A.

Watch: Force is normal to the surface for pressure loading.

Variables & SI units

Fnormal forceN

PpressurePa

Aaream²

Unit 8 · Fluids · essential derived formulasCourse / derived

\frac{F_1}{A_1}=\frac{F_2}{A_2}\quad(\text{Pascal model})

F_1/A_1 = F_2/A_2 in an ideal hydraulic system

Pascal principle / hydraulic force

Use: Pressure transmitted through an ideal confined fluid.

Watch: Height differences or other pressure terms may need inclusion.

Variables & SI units

F_1force on piston 1N

A_1piston area 1m²

F_2force on piston 2N

A_2piston area 2m²

Unit 8 · Fluids · essential derived formulasCourse / derived

W_{app}=mg-F_b

W_app = m g - F_b

Apparent weight in a fluid

Use: Scale/support reading when buoyant force acts upward.

Watch: Sign/direction depends on the force arrangement.

Variables & SI units

W_appapparent weight / support readingN

mobject masskg

ggravitational acceleration magnitudem/s²

F_bbuoyant forceN

Unit 8 · Fluids · essential derived formulasCourse / derived

F_b=mg\quad(\text{floating equilibrium})

F_b = m g for floating equilibrium

Floating condition

Use: Vertical equilibrium for a freely floating object.

Watch: Only the submerged portion displaces fluid.

Variables & SI units

F_bbuoyant forceN

mobject masskg

ggravitational acceleration magnitudem/s²

Unit 8 · Fluids · essential derived formulasCourse / derived

\frac{V_{sub}}{V_{obj}}=\frac{\rho_{obj}}{\rho_{fluid}}\quad(\text{floating})

V_sub/V_obj = rho_obj/rho_fluid for a floating object

Fraction submerged

Use: Derived from buoyancy equilibrium for a uniform floating object.

Watch: Assumes the object floats without other vertical forces.

Variables & SI units

V_subsubmerged volumem³

V_objtotal object volumem³

ρ_objobject densitykg/m³

ρ_fluidfluid densitykg/m³

Unit 8 · Fluids · essential derived formulasCourse / derived

\Delta P=\rho g\,\Delta h

Delta P = rho g Delta h

Hydrostatic pressure difference

Use: Pressure change between two depths in the same static fluid.

Watch: Use vertical depth difference.

Variables & SI units

ΔPpressure differencePa

ρfluid densitykg/m³

ggravitational acceleration magnitudem/s²

Δhvertical depth differencem

Unit 8 · Fluids · essential derived formulasCourse / derived

\dot{m}=\rho Q=\rho Av

mass flow rate = rho Q = rho A v

Mass flow rate

Use: Convert volume flow rate to mass per time for constant density.

Watch: Density must be consistent with the flow model.

Variables & SI units

ṁmass flow ratekg/s

ρfluid densitykg/m³

Qvolume flow ratem³/s

Across-sectional aream²

vflow speedm/s

Unit 8 · Fluids · essential derived formulasCourse / derived

v=\sqrt{2g\Delta h}\quad(\text{Torricelli, ideal})

v = sqrt(2 g Delta h) for ideal Torricelli outflow

Torricelli outflow speed

Use: Derived from Bernoulli for a large reservoir and small outlet.

Watch: Requires the ideal assumptions and appropriate pressure endpoints.

Variables & SI units

videal outflow speedm/s

ggravitational acceleration magnitudem/s²

Δhheight difference between surface and outletm

Unit 8 · Fluids · essential derived formulasCourse / derived

\sum Q_{in}=\sum Q_{out}\quad(\text{steady incompressible flow})

sum Q_in = sum Q_out for steady incompressible flow

Continuity at a junction

Use: Apply conservation of volume flow across multiple branches.

Watch: For compressible flow, mass conservation is the safer general statement.

Variables & SI units

Q_inincoming branch flow ratem³/s

Q_outoutgoing branch flow ratem³/s

Reference tables

Prefixes, trig values, geometry and conventions

These tables support the equation library and reduce unit, angle and geometry mistakes.

Metric prefixes

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

Common-angle trig

θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	1/2	√3/2	√3/3
37°	3/5	4/5	3/4
45°	√2/2	√2/2	1
53°	4/5	3/5	4/3
60°	√3/2	1/2	√3
90°	1	0	undefined

Common symbol & unit guide

Quantity	Common symbols	SI unit
Position / length	x, y, r, h, d, ℓ	m
Velocity	v, v_x, v_cm	m/s
Acceleration	a, a_c, a_T	m/s²
Force	F, F_N, F_g	N
Energy / work	K, U, E, W	J
Power	P	W
Momentum / impulse	p, J	kg·m/s; N·s
Angle	θ, Δθ	rad
Angular speed	ω	rad/s
Angular acceleration	α	rad/s²
Torque	τ	N·m
Rotational inertia	I	kg·m²
Pressure	P	Pa
Density	ρ	kg/m³
Volume flow rate	Q	m³/s

The same symbol can represent different quantities in different contexts (for example P can mean power or pressure). Each individual formula card gives the context-specific meaning and unit.

Geometry & right triangles

Rectangle: A = bh
Triangle: A = ½bh
Circle: A = πr², C = 2πr, s = rθ
Rectangular solid: V = ℓwh
Cylinder: V = πr²ℓ, S = 2πrℓ + 2πr²
Sphere: V = 4πr³/3, S = 4πr²
Right triangle: a²+b²=c²; sinθ=opposite/hypotenuse; cosθ=adjacent/hypotenuse; tanθ=opposite/adjacent

Exam-model conventions

Treat the stated frame as inertial unless told otherwise.
Neglect air resistance unless the prompt says it matters.
Treat springs and strings as ideal unless stated otherwise.
Treat fluids as ideal and pipes as completely filled unless stated otherwise.

Why are there more formulas here than on the exam reference?

The exam reference sheet prints frequently used relationships, but AP Physics 1 also uses definitions, conservation statements, graph relationships, special-case results and relationships derived from the printed equations. Those are included here as Course / derived formulas rather than being mislabeled as exam-provided.

ImportantA rearrangement or special-case equation is useful only when its assumptions match the problem. Model selection comes before substitution.

Fast model-selection sequence

Choose the system and axes.
Identify the model: kinematics, forces, energy, momentum, rotation, oscillation or fluid.
Check assumptions.
Select or derive the relationship.
Check dimensions, sign and limiting behavior.

AP Physics 1 formula-sheet questions students ask

Are all formulas on this page printed on exam day?

No. The cards marked Exam reference correspond to the current provided reference entries. Cards marked Course / derived are useful definitions, derived relationships and common models that may need to be recalled or derived from physics principles.

Why are the equations rendered differently now?

The equation library uses native browser MathML, so fractions, roots, subscripts, superscripts, vectors and Greek symbols render as mathematical notation without depending on an external math service.

Do I need to memorize the 52 exam-reference entries?

The provided reference information reduces memorization, but you still need to understand meaning, applicability, graph relationships, system choices and derivations.

What should I memorize?

Prioritize definitions, model conditions, sign conventions, graph meanings, conservation conditions and the relationships that are not supplied as separate reference lines.

Review all 8 units Use formulas in virtual labs Apply them in FRQs See 2027 exam format