speed = distance / time

Used when only the magnitude of motion matters.

velocity = displacement / time

Velocity includes direction, unlike speed.

a = (v - u) / t

Shows how quickly velocity changes.

v = u + at

Finds final velocity for constant acceleration.

s = ut + 1/2 at^2

Finds displacement under constant acceleration.

v^2 = u^2 + 2as

Useful when time is not given.

average velocity = (u + v) / 2

Valid for motion with uniform acceleration.

s = average velocity x t

Useful after finding average velocity in uniformly accelerated motion.

u_x = u cos(theta)

Separates the launch velocity along the horizontal direction.

u_y = u sin(theta)

Separates the launch velocity along the vertical direction.

T = 2u sin(theta) / g

Total time the projectile stays in air when launched and landing at the same level.

H = u^2 sin^2(theta) / 2g

Highest vertical point reached by the projectile.

R = u^2 sin(2theta) / g

Horizontal distance traveled when start and end levels are equal.

y = x tan(theta) - gx^2 / (2u^2 cos^2(theta))

Describes the curved path of the projectile.

theta = 45°

For a projectile launched on level ground, the range is maximum at 45 degrees.

F = ma

Net force equals mass times acceleration.

W = mg

Force due to gravity acting on a body.

p = mv

Quantity of motion possessed by a body.

J = F Delta t = Delta p

Impulse equals change in momentum.

W = Fs cos(theta)

Work done by a force over displacement.

KE = 1/2 mv^2

Energy possessed due to motion.

PE = mgh

Energy due to position in a gravitational field.

E = KE + PE

Total of kinetic and potential energy.

P = W / t

Rate of doing work.

P = Fv

Useful when a force causes motion with constant velocity in the same direction.

efficiency = useful output / total input x 100%

Measures how effectively a system converts energy.

F = Gm1m2 / r^2

Attractive force between two masses.

g = GM / R^2

Gravitational acceleration at the surface of a planet.

U = -Gm1m2 / r

Potential energy of a two-body gravitational system.

v_e = sqrt(2GM / R)

Minimum speed needed to leave a planet without further propulsion.

v_o = sqrt(GM / R)

Speed for a stable circular orbit near a planet.

omega = theta / t

Angular displacement per unit time.

v = r omega

Connects angular motion to tangential speed.

a_c = v^2 / r = r omega^2

Acceleration directed toward the center of circular motion.

F_c = mv^2 / r

Net inward force required for circular motion.

f = 1 / T

Frequency is the number of revolutions or cycles per second.

P = F / A

Force per unit area.

rho = m / V

Mass per unit volume.

relative density = density of substance / density of water

Compares a substance’s density to water.

P = h rho g

Pressure due to a liquid column.

F_b = rho V g

Upthrust equals weight of displaced fluid.

A1v1 = A2v2

For incompressible flow, flow rate remains constant.

Q = Av

Volume of fluid crossing a section each second.

Q = mc Delta T

Heat needed to raise or lower temperature.

Q = mL

Heat used during change of state without changing temperature.

Delta L = alpha L0 Delta T

Change in length due to temperature.

PV = nRT

Relates pressure, volume, temperature, and moles of a gas.

W = P Delta V

For constant pressure processes.

eta = W / Q_h

Fraction of heat input converted to useful work.

Delta Q = Delta U + Delta W

Heat supplied changes internal energy and can also do work.

v = f lambda

Basic relation between frequency, wavelength, and speed.

f = 1 / T

Number of oscillations per second.

T = 1 / f

Time taken for one full oscillation.

distance = vt / 2

Distance to a reflecting surface using echo time.

I = P / A

Power carried by a wave per unit area.

f' = f (v + v_o) / (v - v_s)

Observed frequency changes when source or observer moves.

v approx 331 + 0.6T

Approximate speed of sound in air when temperature T is in degree Celsius.

c = f lambda

For electromagnetic waves in vacuum.

n = c / v

Shows how much light slows down in a medium.

n1 sin(theta1) = n2 sin(theta2)

Relates incident and refracted rays between two media.

1/f = 1/v + 1/u

Relates object distance, image distance, and focal length for mirrors.

1/f = 1/v - 1/u

Relates object and image positions for thin lenses.

m = h_i / h_o = v / u

Compares image size with object size.

P = 1/f

Lens power in diopters when focal length is in meters.

sin(C) = 1 / n

Applies when light travels from a denser to a rarer medium.

I = Q / t

Rate of flow of electric charge.

V = IR

Potential difference across a conductor equals current times resistance.

R = rho L / A

Resistance depends on material, length, and cross-sectional area.

P = VI = I^2R = V^2 / R

Rate of electrical energy conversion.

E = Pt = VIt

Energy consumed over time.

R_s = R1 + R2 + R3 + ...

Equivalent resistance adds directly in series.

1/R_p = 1/R1 + 1/R2 + 1/R3 + ...

Equivalent resistance for parallel branches.

Q = It

Charge passing through a conductor in a given time.

F = kq1q2 / r^2

Force between two point charges.

E = F / q

Force per unit positive test charge.

V = W / q

Work done per unit charge.

C = Q / V

Charge stored per unit potential difference.

F = qvB sin(theta)

Force on a charge moving inside a magnetic field.

F = BIL sin(theta)

Magnetic force on a wire carrying current.

B = mu0 I / 2pi r

Field strength at distance r from a straight conductor.

Phi = BA cos(theta)

Amount of magnetic field passing through an area.

emf = - Delta Phi / Delta t

Induced emf equals rate of change of magnetic flux.

F = q(E + v x B)

Total force on a moving charge in electric and magnetic fields.

a = -omega^2 x

Acceleration in SHM is proportional to displacement and directed toward the mean position.

omega = 2pi f = 2pi / T

Links angular speed with frequency and time period.

T = 2pi sqrt(m / k)

Time period of oscillation for a mass on a spring.

T = 2pi sqrt(l / g)

Valid for small oscillations of a simple pendulum.

E = hf

Energy of a single photon.

E = mc^2

Mass can be converted to energy and vice versa.

lambda = h / p

Matter particles also show wave behavior.

N = N0 e^(-lambda t)

Number of undecayed nuclei after time t.

T_(1/2) = 0.693 / lambda

Time needed for half of a radioactive sample to decay.

hf = phi + KE_max

Incoming photon energy is used to overcome work function and provide kinetic energy.