FORMULAS

KINEMATICS

Non-Uniform Motion	d	v₁	v₂	t	a
$\Delta \vec{d} = \dfrac{\vec{v}_1 + \vec{v}_2}{2} \Delta t$	✓	✓	✓	✓	✘
$\vec{v}_2^2 = \vec{v}_1^2 + 2a \Delta\vec{d}$	✓	✓	✓	✘	✓
$\Delta \vec{d} = \vec{v}_1 \Delta t + \frac{1}{2}\vec{a} \Delta t^2$	✓	✓	✘	✓	✓
$\Delta \vec{d} = \vec{v}_2 \Delta t - \frac{1}{2}\vec{a} \Delta t^2$	✓	✘	✓	✓	✓
$\vec{v}_2 = \vec{v}_1 + \vec{a} \Delta t$	✘	✓	✓	✓	✓

Acceleration due to gravity

$g = -9.81 m/s^2$

Uniform Motion

$d = v\,t$

Components:

$\vec{v}_y = \vec{v}\sin\theta \\ \\ \vec{v}_x = \vec{v}\cos\theta$

For $\theta$ = angle of elevation. Average: Distance uses average speed if the 2 trips have the same time.

FORCES

Newton's 1^st Law Objects at rest or in constant motion remain that way unless a net force acts on it. Newton's 2^nd Law Net force is directly proportional to the product of acceleration and mass (F = m × a). Newton's 3^rd Law For every action, there is an equal and opposite reaction.

Weight = Force of Gravity

$F_g = m × g$

Friction

$F_{kinetic\ friction} = \mu_k × F_N$

$F_{static\ friction} = \mu_s × F_N$

Circular Motion

$a_c = \dfrac{v^2}{r}$

$F_c = m \dfrac{v^2}{r}$

v is tangential velocity

F_c points to circle center

WORK, ENERGY, POWER

$\text{Power} = \dfrac{Energy}{Time} = \dfrac{\Delta W}{\Delta t} = F × \vec{v}$

$\%\ \text{Efficiency} = \dfrac{Energy\ used}{Energy\ in} × 100 \%$

$E_{total} = KE_{total} + PE_{total}$

$\Delta E = KE_2 - KE_1 + PE_2 - PE_1$

$W = \Delta E$

$W = F × d × \cos\theta$

$KE = \frac{1}{2}mv^2$

$PE = m × g × h$

MOMENTUM

$J = \Delta p$

p = momentum
J = impulse

$p = m × \vec{v}$

$J = F × \Delta t$

$\tau = r × F × \sin\theta$

τ = torque

Elastic

$KE_1 = KE_2$

$p_1 = p_2$

Inelastic

$p_1 = p_2$

$m_{a\,i}v_{a\,i} + m_{b\,i}v_{b\,i} = m_{a\,f}v_{a\,f} + m_{b\,f}v_{b\,f}$

$\frac{1}{2}m_{a\,i}v_{a\,i}^2 + \frac{1}{2}m_{b\,i}v_{b\,i}^2 = \frac{1}{2}m_{a\,f}v_{a\,f}^2 + \frac{1}{2}m_{b\,f}v_{b\,f}^2$

GRAVITY

$F_g = \dfrac{Gm_1m_2}{r^2}$

G = universal gravitational constant

$g\ field = \dfrac{-GM}{r^2} = \dfrac{F}{m}$

$V\ potential = \dfrac{-GM}{r}$

$U\ potential\ energy = \dfrac{-GMm}{r}$

$U = V × m$

Kepler's 3^rd

$\dfrac{R^3}{T^2} = \dfrac{GM}{4\pi^2}$

T = period
R = mean orbit radius
M = mass of central body

Constants:

$G = 6.67 × 10^{-11} \frac{Nm^2}{kg^2}$

$r_{Earth} = 6.4 × 10^{6} m$

$m_{Earth} = 5.98 × 10^{24} kg$

$m_{Sun} = 2 × 10^{30} kg$

$m_{Moon} = 7.34 × 10^{22} kg$

LIGHT

$v = f × λ$

$\dfrac{1}{f} = \dfrac{1}{i} + \dfrac{1}{o} = \dfrac{2}{r}$

f = focal length
i = image distance
o = object distance

$M = \dfrac{h_i}{h_o} = -\dfrac{i}{o}$

+M = magnified
-M = smaller
|M| > 1 = upright
|M| < 1 = inverted
h_i = image height
h_o = object height

$n = \dfrac{c}{v}$

n = index of refraction
c = speed of light in vacuum
v = speed in other medium

$c = 3.00 × 10^{8} \tfrac{m}{s}$

$n_1 \sin\theta_1 = n_2 \sin\theta_2$

$\begin{align} Intensity & = \dfrac{Power}{Area} \\ \\ & = \dfrac{Energy}{(Area)(time)} \end{align}$

Constructive

$m\,λ = d\,\sin\theta$

d = width of slits
m = fringe order integer
λ = wavelength Destructive

$\left(m + \frac{1}{2}\right)\,λ = d\,\sin\theta$

Small Angle Approximation

$m\,λ = \dfrac{d\,y}{x}$

y = fringe distance on screen
x = screen distance from slits

Thin Films

$2d = \dfrac{λ}{n}$

n = index of refraction

$\left(m + \frac{1}{2}\right)\,λ = 2\,d\,n$

(Constructive)

$m\,λ = 2\,d\,n$

(Destructive)

$\left(m + \frac{1}{4}\right)\,λ = 2\,d\,n$

(Destructive,
antireflective)

Polarization

$I = I_0 \,\cos^2\theta$

Brewster

$\tan\theta = \dfrac{n_2}{n_1}$

Resolution

$\theta = \dfrac{1.22\,λ}{a}$

a = aperature

SOUND

$\vec{v}_{sound} = 331 \tfrac{m}{s}$

$v = f × λ$

$v = 331 + 0.6 × T$

speed of sound changes with air temp.
v in m/s, T in ˚C

$\begin{align} Intensity & = \dfrac{Power}{Area} \\ \\ & = (pressure)(\vec{v}) \end{align}$

Intensity in W/m²
Surface area sphere = $4\pi r^2$

$dB = 10\,\log\left(\dfrac{I}{I_0}\right)$

I₀ is threshold of human hearing
I₀ = 10^-12 W/m²

$\text{times louder} = 10 ^\left(\tfrac{dB_H \,-\, dB_L}{10}\right)$

db_H is higher
db_L is lower

Strings & Open Pipes

$λ = \dfrac{2L}{n}$

n = 1, 2, 3, 4, ...

$f = \dfrac{n\,v}{2L}$

Closed Pipes

$λ = \dfrac{4L}{n}$

n = 1, 3, 5, 7, ...

$f = \dfrac{n\,v}{4L}$

Frequency, Doppler

$f_{beat} = \Big|f_1 - f_2\Big|$

$f’ = f × \dfrac{\left(v \pm v_D\right)}{\left(v \mp v_s\right)}$

together: ±
apart: ∓

SIMPLE HARMONIC MOTION

Strings/Springs

$\vec{v} = \sqrt{\dfrac{F_T}{\mu}}$

$\mu$ = linear density (kg/m)
F_T = Tension (N)
L = length (m)
m = mass (kg)
v = speed (m/s)

$\mu = \dfrac{m}{L}$

All Waves

$\vec{v} = f × λ$

$f = \dfrac{1}{T}$

$E_{T} = KE + PE$

$ω = 2 \pi f$

Springs

$T = 2\pi \sqrt{\dfrac{m}{k}}$

T = period
m = mass
k = spring const.

$F = -k × x$

$KE = \frac{1}{2}m\,v^2$

$PE = \frac{1}{2}k\,x^2$

$ω^2 = \dfrac{k}{m}$

Pendulums

$T = 2\pi \sqrt{\dfrac{L}{g}}$

T = period
L = length
g = gravity

$KE = \frac{1}{2}m\,v^2$

$PE = m\,g\,h$

$ω^2 = \dfrac{g}{L}$

Wave Equations

$KE = \frac{1}{2} mω^2(x_0^2 - x^2)$

$PE = \frac{1}{2} mω^2(x^2)$

$E_{T} = \frac{1}{2} mω^2(x_0^2)$

$\vec{x}_{max} = A$

$\vec{v}_{max} = -ω \, A$

$\vec{a}_{max} = -ω^2 \, A$

$\begin{align} v & = ω \sqrt{A^2 - x^2} \\ & = \pm \sqrt{\frac{k}{m} (A^2 - x^2) } \\ & = \pm \vec{v}_{max} \sqrt{1 - \frac{x^2}{A^2}} \end{align}$

At x_min (x₀ = 0)

$\vec{x} = A \sin (ω\,t)$

$\vec{v} = -ω \, A \cos (ω\,t)$

$\vec{a} = -ω^2 \, A \sin (ω \,t)$

At x_max (x₀ = A)

$\vec{x} = A \cos (ω\,t)$

$\vec{v} = -ω \, A \sin (ω\,t)$

$\vec{a} = -ω^2 \, A \cos (ω \,t)$

v = velocity
a = acceleration
x = displacement
A = amplitude
ω = angular frequency

ELECTRIC & MAGNETIC FIELDS

$F_{moving\ charge} = q\,v\,B\,\sin\theta$

$F_{current\ wire} = I\,L\,B\,\sin\theta$

$F = \dfrac{k\,q_1\,q_2}{r^2} = \dfrac{q_1\,q_2}{4\,\pi\,ε_0\,r^2}$

$k = 8.99 × 10^9 \, \tfrac{Nm^2}{C^2}$

$φ = N\,B\,A\,\cos\theta$

φ = flux

$\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p} + \dfrac{I_s}{I_p}$

V = voltage
N = number of turns
I = current
s = secondary
p = primary

Subheader

$U = q\,V$

U = electric potential energy
V = electric potential

$V = E\,d$

d = separation distance
E = electric field

$V = \dfrac{\Delta\,U}{q} = \dfrac{k\,q}{r}$

$q = N × e × F_k$

N = # turns

$ε = \dfrac{\Delta\,p}{\Delta\,t}$

$ε = B\,L\,v$

ε = EMF, Volts
B = magnetic field
L = length
v = speed

$\begin{align} U & = q\,\Delta d \\ \\ & = q\,E\,d \\ \\ & = \dfrac{k\,q_1\,q_2}{r} \end{align}$

$E = \dfrac{F}{q} = \dfrac{k\,q}{r}$

Constants:

$q_{e^-} = -1.0602 × 10^{-19} C$

$q_{p^+} = +1.0602 × 10^{-19} C$

$m_{electron} = 9.11 × 10^{-31} kg$

$m_{proton} = 1.67 × 10^{-27} kg$

$Faraday\ k = 6.02 × 10^{23} \tfrac{C}{mol\,e^-}$

Magnetism

$B = µ_0\,i\,n$

$µ_0 = 4\,\pi × 10^{-7} \tfrac{T\,m}{A}$

i = current
r = radius Straight Wire

$B = \dfrac{µ_0\,i}{2\,\pi\,r}$

Center of Loop

$B = \dfrac{µ_0\,i\,n}{2\,r}$

Solenoid Turns

$B = \dfrac{µ_0\,i\,n}{L}$

DC & AC CIRCUITS

$i = \dfrac{\Delta\,q}{\Delta\,t}$

$V = i \, r$

$C = \dfrac{Q}{V}$

$R = \dfrac{p\,I}{A}$

Subheader

$\begin{align} P & = i\,V \\ \\ & = \dfrac{V^2}{R} \\ \\ & = i^2\,R \end{align}$

$C = \dfrac{k\,E_0\,A}{d}$

k = constant
A = area
d = distance

$E_0 = 8.85 × 10^{-12} \tfrac{C^2}{m^2\,N}$

$\begin{align} E & = \frac{1}{2}Q\,V \\ \\ & = \frac{1}{2}C\,V^2 \\ \\ & = \frac{1}{2}\frac{Q^2}{C} \end{align}$

$Power = \dfrac{Energy}{time}$

Series Circuit

$R_T = R_1 + R_2 + R_3 + \, ...$

$V_T = V_1 + V_2 + V_3 + \, ...$

$i_T = i_1 = i_2 = i_3 = \, ...$

$\dfrac{1}{C_{eq}} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dfrac{1}{C_3} + \, ...$

Parallel Circuit

$\dfrac{1}{R_T} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3} + \, ...$

$V_T = V_1 = V_2 = V_3 = \, ...$

$i_T = i_1 + i_2 + i_3 + \, ...$

$C_{eq} = C_1 + C_2 + C_3 + \, ...$

THERMODYNAMICS

1^st Law Thermodynamics

$\Delta U = Q - W$

+Q = heat added to system
-Q = removed from system
+W = work done by system (out)
-W = work done on system (in)

$W = Q_{in} - Q_{out}$

Specific Heat

$C = m × c$

c = specific heat = specific heat capacity
C = heat capacity
Q = heat energy

$Q = m × c × \Delta t$

$Q = m × L$

(for phase changes)
L = latent heat

$\begin{align} \text{molar heat capacity} & = c × molar\ mass \\ & = \dfrac{Q}{n × \Delta t} \end{align}$

Carnot Heat Efficiency

$Efficiency_{carnot} = \left(1 - \dfrac{T_L}{T_H}\right) × 100\%$

$Work = Q_H - Q_L$

H = hot, in
L = cold, out

QUANTUM

$E = h × f$

$c = f × λ$

$E = m × c^2$

$\frac{1}{2}mv^2 = qV$

$1 eV = 1.602 × 10^{-19} J$

$h = 6.63 × 10^{-34} Js$

h = Planck's constant

$KE = hf - φ$

$φ = h × f_0$

$hf = φ - qV$

$λ = \dfrac{h \, c}{E} = \dfrac{h}{p} = \dfrac{h}{m \, v}$

Relativity

$L = L_O \sqrt{ 1 - \dfrac{v^2}{c^2} }$

$t = \dfrac{t_0}{ \sqrt{ 1 - \dfrac{v^2}{c^2} } }$

CONSTANTS

Quantity	Symbol	Value
Speed of light in a vacuum	c	2.9979 × 10⁸ m/s (exact)
Planck's constant	h	6.6260 × 10^–34 J·s
Universal gas constant	R	0.08205 L·atm/(mol·K) 8.3145 J/(mol·K) 8.3145 m³·Pa/(mol·K) 8.3145 L·kPa/(mol·K) 0.083145 L·bar/(mol·K) 62.364 L·Torr/(mol·K)
Avogadro constant	N_A	6.0221 × 10²³ formula units/mol
Mass of an electron	m_e	9.10938 × 10^–28 g 5.4858 × 10^–4 amu
Mass of a proton	m_p	1.67262 × 10⁻²⁴ g 1.00727 amu
Mass of a neutron	m_n	1.674927 × 10⁻²⁴ g 1.00866 amu
Charge on an electron	e	–1.60217 × 10^–19 C
Boltzmann constant	k	1.3806 × 10^–23 J/K
Faraday constant	F	96485 C/mol of electrons
Rydberg constant	R_H	1.09737 × 10⁷ m^–1 2.179872 × 10^–18 J 3.289842 × 10¹⁵ s^–1
Electric constant	ε₀	8.85418 × 10^–12 F/m 8.85418 × 10^–¹² C²/(J·m)

CONVERSIONS

Length
1 in = 2.54 cm (exact)
1 Å = 1 × 10^–10 m
1 mi = 5280 ft
1 mi = 1609.3 m
1 ft = 0.3048 m
1 m = 1.094 yd
1 km = 0.621 miles(mi)

Volume
1 mL = 1 cm³
1 L = 1.0567 qt
1 gal = 3.7854 L
1 m³ = 1000 L

Mass
1 lb = 453.6 g
1 g = 6.022 × 10²³ amu
1 kg = 2.20 lb
1 lb = 16 oz
1 oz = 28.35 g
1 ton = 2000 lb
1 metric ton = 1 Mg

Energy
1 cal = 4.184 J
1 L·atm = 101.325 J
1 eV = 1.602 × 10^–19 J
1 J = 1 kg·m²/s²1 J = 1 m³·Pa
1 J = 1 L·kPa
1 J = 1 C·V

Pressure
1 atm = 760 Torr
1 atm = 760 mmHg
1 atm = 101325 Pa
1 atm = 14.7 psi
1 atm = 1.01325 bar
1 bar = 100 kPa
1 Torr = 1 mmHg

Temperature
T_K = T_C + 273.15
T_C= (5/9)(T_F – 32)
T_F= 32 + (9/5)T_C

MAGNITUDES

10^x	Prefix	Symbol
10¹⁵	peta	P
10¹²	tera	T
10⁹	giga	G
10⁶	mega	M
10³	kilo	k
10²	hecto	h
10¹	deca	da
10⁻¹	deci	d
10⁻²	centi	c
10⁻³	milli	m
10⁻⁶	micro	µ
10⁻⁹	nano	n
10⁻¹²	pico	p
10⁻¹⁵	femto	f

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FORMULAS

KINEMATICS

FORCES

WORK, ENERGY, POWER

MOMENTUM

GRAVITY

LIGHT

SOUND

SIMPLE HARMONIC MOTION

ELECTRIC & MAGNETIC FIELDS

DC & AC CIRCUITS

THERMODYNAMICS

QUANTUM

CONSTANTS

CONVERSIONS

MAGNITUDES

10^x

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Symbol

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FAVORITES

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CLASS OPTIONS

FORMULAS

KINEMATICS

FORCES

WORK, ENERGY, POWER

MOMENTUM

GRAVITY

LIGHT

SOUND

SIMPLE HARMONIC MOTION

ELECTRIC & MAGNETIC FIELDS

DC & AC CIRCUITS

THERMODYNAMICS

QUANTUM

CONSTANTS

CONVERSIONS

MAGNITUDES

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