# Physics Equations ```1
Chapter 2 – Intro to Motion
 aav = vf - vi
Δt
 xf = xi + vt
Chapter 3 – Acceleration

vf = vi + at – velocity, time, acceleration

vav = 1/2(vi + vf) – initial, final, and average velocity

xf = xi + 1/2(vi + vf)t – position, time, velocity

xf = xi + vit + 1/2at2 – position, time, acceleration

vf2 = vi2 + 2aΔx – velocity, position, acceleration
Chapter 4 – Motion in 2-D

Position-Time Equations for Projectiles
o xf = xi + vx,it
o yf = yi + vy,it – 1/2gt2
Substituting these results into the position-time equations yields the following:

Position-Time Equations for Projectiles Launched at an Angle θ
o xf = xi + (vi cos θ)t
o yf = yi + (vi sin θ)t – 1/2gt2
Similarly, substituting into the velocity-time equations gives

Velocity-Time Equations for Projectiles Launched at an Angle θ
o vx,f = vi cos θ
o vy,f = vi sin θ – gt

Range Equation
o R = vi2 sin 2 θ
g
2
Chapter 5 – Newton’s Laws of Motion

Force
o F = ma

Sum of Forces
o ΣF = ma

Weight
o W = mg

Hooke’s Law
o F = kx

Kinetic Friction
o fk = μkN

Maximum Force of Static Friction
o Fs,max = μsN
Chapter 6 – Work and Energy

Definition of Work
o W = Fd

Definition of Work ( with force and displacement at an angle θ)
o W = Fd cos θ

Definition of Kinetic Energy
o KE = 1/2mv2

Definition of Gravitational Potential Energy
o PEgravity = mgh

Definition of Spring Potential Energy
o PEspring = 1/2kx2

Definition of Power
o P=W
t
3

Work-Energy Theorem
o Wtotal = 1/2mvf2 – 1/2mvi2
Chapter 7 – Linear Momentum and Collisions

Definition of Momentum
o p = mv

Magnitude of the Momentum
o p = mv

Total Momentum
o ptotal = p1 + p2 + …

Definition of Impulse
o I = FΔt

Magnitude of the Impulse
o I = FΔt

Momentum Impulse Theorem
o I = FΔt = Δp

Conservation of Momentum
o p f = pi
Chapter 8 – Rotational Motion and Equilibrium

Definition of Angular Position
o θ = angle measured from reference line

Definition of Average Angular Velocity
o ωav = Δθ
Δt

Tangential Speed of a Rotating Object
o vt = rω

Definition of Average Angular Acceleration
o αav = Δω
Δt
4

Tangential Acceleration of a Rotating Object
o at = rα

Moment of Inertia (for a single object of mass at a distance)
o I = mr2

Moment of Inertia (for a collection of objects)
o I = m1r12 + m2r22 + m3r32 + …

Moments of Inertia for a Hoop and a Disk
o Ihoop = mr2
o Idisk = 1/2mr2

Rotational Kinetic Energy
o KErot = 1/2Iω2

Angular Momentum
o L = Iω

Definition of Torque (for a tangential force)
o τ = rF

Definition of Torque (for a nontangential force)
o τ = rF sin θ

Newton’s Second Law for Rotational Motion
o Στ = Iα

Center of Mass for Two Objects
o xcm = m1x1 + m2x2
m1 + m2
Chapter 9 – Gravity and Circular Motion

Newton’s Law of Universal Gravitation
o F = G m1m2
r2
5

Centripetal Acceleration
o acp = v2
r

Newton’s Second Law for Circular Motion
o fcp = macp

Kepler’s First Law
o Planets follow elliptical orbits, with the Sun at one focus point of the
ellipse.

Kepler’s Second Law
o As a planet moves in its orbit, it sweeps out an equal amount of area
in an equal amount of time.

Kepler’s Third Law
o The period, T, of a planet increases as its distance from the Sun, r,
raised to the 3/2 power. That is, T = (constant)r3/2.
Chapter 10 – Temperature and Heat

The Zeroth Law of Thermodynamics
o Objects in contact with one another are in thermal equilibrium if they
have the same temperature. Nothing else matters.

Conversion between Degrees Celsius and Degrees Fahrenheit
o TF = 9/5TC + 32

Conversion between Degrees Fahrenheit and Degrees Celsius
o TC = 5/9(TF – 32)

Conversion between a Celsius Temperature and a Kelvin Temperature
o T = TC + 273.15

Definition of Coefficient of Thermal Expansion
o ΔL = αLiΔT
 The Mechanical Equivalent of Heat
o One calorie of heat is the equivalent of 4.186 J of mechanical work.
6
 1 cal = 4.186 J
 1 kcal = 4186 J

Definition of Specific Heat Capacity
o c= Q
mΔT

Definition of Pressure
o P=F
A

Heat Required to Change from One Phase to Another
o Q = mL
Chapter 11 – Thermodynamics

First Law of Thermodynamics
o ΔE = Q – W

Efficiency of a Heat Engine
o e=W
Qh

Second Law of Thermodynamics
o When objects of different temperatures are brought into contact, the
flow of thermal energy is always from the higher-temperature object
to the lower-temperature object.

Carnot’s Theorem of Maximum Efficiency
o emax = 1 – Tc
Th

Definition of Entropy Change
o ΔS = Q
T

Third Law of Thermodynamics
o It is impossible to lower the temperature of an object all the way to
absolute zero.
7
Chapter 12 – Gases, Liquids, and Solids

Ideal Gas Equation
o P = k NT
V

Boltzmann Constant
o k = 1.38 × 10-23 J/K

Ideal Gas Equation (alternative form)
o PV = NkT

o Na = 6.022 × 1023 molecules/ mol


The number of molecules is dimensionless
Universal Gas Constant
o R = Nak
 = 8.31 J/(mol ∙ K)

Ideal Gas Equation (in terms of moles)
o PV = nRT

Definition of Density
o ρ=m
v

Dependence of Pressure on Depth
o P = Patmospheric + ρgh

Dependence of Pressure on Depth
o P2 = P1 + ρgh

Equation of Continuity
o A1v1 = A2v2
Chapter 13 – Oscillations and Waves

Definition of Period
o period = time required for one cycle of a periodic motion
8


T = period
Definition of Frequency
o frequency = 1
period
 f= 1
T

Period of a Mass on a Spring
o T = 2π

Period of a Pendulum
o T = 2π

m
k
L
g
Definition of a Wavelength
o λ = wavelength
 = distance over which a wave repeats

Speed of a Wave
o v = λf

First Harmonic of a Standing Wave on a String
o λ1 = 2L
o fi = v
2L
Chapter 14 – Sound

Speed of Sound in Air (at room temperature, 20 °C)
o v = 343 m/s (approximately 1000 ft/s or 770 mi/h)

Definition of Beat Frequency
o Beat frequency = absolute value of the difference in frequency
 fbeat = | f1 – f2 |

Standing Waves in a Column of Air Closed at One End
o Fundamental frequency: f1 = v
4L
9
o Harmonic frequencies and wavelengths:
fn = nf1 = n v
4L
for n = 1, 3, 5, . . .
λn = λ1 = 4L
n n

Standing Waves in a Column of Air Closed at Both Ends
o Fundamental frequency: f1 = v
2L
o Harmonic frequencies and wavelengths:
fn = nf1 = n v
2L
for n = 1, 2, 3, . . .
λn = λ1 = 2L
n n

Doppler Effect for a Moving Source
o fobserver =
fsource
1 ± vsource
vsound
o + sign for source moving away from the observer
o – sign for source moving toward the observer

Doppler Effect for a Moving Observer
o fobserver = fsource 1 ± vobserver
vsound
o + sign for observer moving toward the source
o – sign for observer moving away from the source

Definition of Intensity
o intensity = power
area
o I=P
A
10

Intensity of a Sound from a Point Source at a Distance r
o intensity = power
area of a sphere
o I=P
4πr2
Chapter 15 – The Properties of Light

Speed of Light in a Vacuum
o c = 3.00 × 108 m/s
Chapter 16 – Reflection and Mirrors

Law of Reflection
o angle of reflection = angle of incidence
o θr = θi

Image Distance for a Plane Mirror
o image distance = – (object distance)
o di = – do

Image Height for a Plane Mirror
o image height = object height
o hi = ho

Focal Length for a Concave Mirror of Radius R
o focal length = 1/2 × radius of curvature
o f = 1/2 R

Focal Length for a Convex Mirror of Radius R
o focal length = –1/2 × radius of curvature
o f = –1/2 R
11

The Mirror Equation
o
1
+
Object distance
1
image distance
=
1
focal length
o 1+1 =1
do di

f
Focal Length
o f is positive for concave mirrors
o f is negative for convex mirrors

Image Distance
o di is positive for images in front of a mirror (real images)
o di is negative for images behind a mirror (virtual images)

Object Distance
o do is positive for real objects

Magnification
o magnification = image height = – image distance
object height
object distance
o m = hi = – di
ho

do
Sign Convention for Magnification
o m is positive for upright images
o m is negative for inverted images
Chapter 17 – Refraction and Lenses

Definition of the Index of Refraction
o speed of light in a material = speed of light in a vacuum
index of the refraction of the material
o v=c
n

Snell’s Law
o
in material 1
in material 2
index of refraction × sine of angle = index of refraction × sine of angle
12
o n1 sin θ1 = n2 sin θ2

Critical Angle for Total Internal Reflection
o critical angle = inverse sine index of refraction of new material
index of refraction of original material
-1
o θc = sin n2
n1
Chapter 18 – Interference and Diffraction

Condition for Constructive Interference
o path-length difference
length of path 2 – length of path 1 = integer × wavelength
o l2 – l1 = mλ
 m = 0, 1, 2, . . .

Condition for Destructive Interference
o path-length difference
length of path 2 – length of path 1 = (integer + 1/2) × wavelength
o l2 – l1 = (m + 1/2)λ
 m = 0, 1, 2, . . .

Conditions for Bright Fringes in a Two-Slit Experiment
o path-length difference = integer × wavelength
o d sin θ = mλ


m = 0, ±1, ±2, . . .
Linear Distance from the Central Bright Fringe
o y = L tan θ

Condition for Destructive Interference in a Thin Film
o Effective path length = integer × wavelength
o 2nt = mλ
13


m = 1, 2, 3, . . .
Condition for Constructive Interference in a Thin Film
o Effective path length = (integer + 1/2) × wavelength
o 2nt = (m + 1/2)λ


m = 0, 1, 2, . . .
Conditions for Dark Fringes in Single-Slit Interference
o path-length difference for slit = integer × wavelength
o W sin θ = mλ


m = ±1, ±2, ±3, . . .
First Dark Fringe for the Diffraction Pattern of a Circular Opening
o sine of angle to first dark fringe = 1.22 × wavelength of light
diameter of opening
o sin θ = 1.22 λ
d

Constructive Interference by a Diffraction Grating
o path-length difference = integer × wavelength
o d sin θ = mλ
 m = 0, ±1, ±2, . . .
Chapter 19 – Electric Charges and Forces

Magnitude of an Electron’s Charge
o e = 1.60 × 10-19 C

Coulomb’s Law for the Magnitude of the Electrostatic Force between Point Charges
o electrostatic force = k magnitude of charge q1 × magnitude of charge q2
(distance between charges)2
o F = k |q1| |q2|
r2
14
Chapter 20 – Electric Fields and Electric Energy

Definition of the Magnitude of the Electric Field
o If a small positive test charge q0 experiences a force of magnitude F at a
given location, the magnitude of the electric field E at that location is
Magnitude of electric field = force on positive charge
amount of charge
o E=F
q0

The Electric Force due to an Electric Field
o A charge q in an electric field E experiences a force F given by
force = amount of charge × electric field
o F = qE

Magnitude of the Electric Field due to a Point Charge
o Magnitude of electric field = k | charge |
(distance)2
o E=k|q|
r2

Rules for Drawing Electric Field Lines
o Electric field lines point in the direction of the electric field vector E at
every point
o Electric field lines start at positive (+) charges or at infinity
o Electric field lines end at negative (–) charges or at infinity
o Electric field lines are closer together where E has a greater magnitude
o The number of electric field lines entering or leaving a charge is
proportional to the magnitude of the charge

Definition of Electric Potential
o change in electric potential = change in electric potential energy
charge
o Δ V = Δ PE
q
15

Connection between the Electric Field and the Electric Potential
o electric field = – change in electric potential
distance
o E=– ΔV
d

Electric Potential for a Point Charge
o electric potential = k charge
distance
o V=k q
r

Electric Potential Energy for Point Charges q and q0
o PE = q0V = k q0q
r

Definition of Capacitance
o capacitance =
charge
electric potential difference
o C=Q
V
Chapter 21 – Electric Current and Electric Circuits

Definition of Electric Current
o electric current = amount of charge
amount of time
o I=ΔQ
Δt

Work Done by a Battery
o work done by a battery = amount of charge × emf of battery
o W = (Δ Q) ε

Ohm’s Law
o applied voltage = current × resistance
o V = IR
16

Definition of Resistivity
o Resistance = resistivity ×
length of wire
cross-sectional area
o R=ρ L
A

Equivalent Resistance for Resistors in Series
o equivalent resistance = resistance 1 + resistance 2 + resistance 3 + . . .
o Req = R1 + R2 + R3 + . . .

Equivalent Resistance for Resistors in Parallel
o
1
1
1
1
+...
equivalent resistance = resistance 1 + resistance 2 + resistance 3
o 1
1 1 1
Req = R1 + R2 + R3

Electric Power
o power = current × voltage
o P = IV
Chapter 22 – Magnetism and Magnetic Fields

Magnetic Field Right-Hand Rule
o To find the direction of the magnetic field due to a current-carrying wire,
point the thumb of your right hand along the wire in the direction of the
current, I. Your fingers will then curl around the wire in the direction of
the magnetic field.

Magnetic Field for a Long, Straight Wire
o magnetic field = permeability of free space × current
2π × radial distance from wire
o B = μ0 I
2πr
17

Magnetic Field of a Solenoid
o Magnetic field = permeability of free space × number of loops × current
length of solenoid
o B = μ0 N I
L

Magnitude of the Magnetic Force
o force = | charge | × velocity × magnetic field × sin θ
o f = | q | vB sin θ

Magnetic Force Right-Hand Rule
o To find the direction of the magnetic force on a moving positive charge,
start by pointing the fingers of your right hand in the direction of the
velocity, v. Now curl your fingers in the direction of B, through the
smallest possible angle. Your thumb points in the direction of F. If the
charge is negative, the force points opposite to the direction of your
thumb.

Magnetic Force on a Current-Carrying Wire
o force = current × length × magnetic field × sin θ
o F = ILB sin θ
Chapter 23 – Electromagnetic Induction

Definition of Magnetic Flux
o magnetic flux = magnitude of the magnetic field × area × cos θ
o Φ = BA cos θ

o induced emf = – (number of loops) × (rate of change of magnetic flux)
o ε=–N ΔΦ
Δt

Transformer Equation
o
voltage in primary coil = turns in primary coil
voltage in secondary cell turns in secondary coil
18
o Vp = Np
Vs

Ns
Transformer Equation (with current and voltage)
o current in secondary coil = voltage in primary coil
current in primary coil
voltage in secondary coil
o Is = Vp
Ip Vs
Chapter 24 – Quantum Physics

Wien’s Displacement Law
o peak frequency of emitted radiation = constant × temperature
o fpeak = (5.88 × 1010 Hz/K) T

Quantized Energy
o energy = quantum number × Planck’s constant × frequency
o E = nhf
o n = 0, 1, 2, 3, . . .

Planck’s Constant
o h = 6.63 × 10-34 J ∙ s

Basic Unit (Quantum) of Energy
o quantum of energy = hf

Energy of a Photon of Frequency
o energy of a photon = Planck’s constant × frequency
o E = hf

Cutoff Frequency
o f0 = W0
h

de Broglie Wavelength
o wavelength = Planck’s constant
momentum
19
o λ=h
p

The Heisenberg Uncertainty Principle (for Momentum and Position)
o uncertainty in y momentum × uncertainty in y position ≥ Planck’s constant
2π
o (Δpy) (Δy) ≥ h
2π
Chapter 25 – Atomic Physics

Orbital Radius of an Electron in a Hydrogen Atom
o orbital radius of an electron = (5.29 × 10-11 m) × (quantum number)2
o rn = (5.29 × 10-11 m) n2


n = 1, 2, 3, …
Energy of a Hydrogen Atom
o total energy of a hydrogen atom = – (13.6 eV) ×
o en = – (13.6 eV) 1
n2
 n = 1, 2, 3, …
Chapter 26 – Nuclear Physics

Definition of the Atomic Mass Unit
o 1 u = 1.660539 × 10-27 kg

Mass-Energy Equivalence
o energy = mass × (speed of light)2
o E = mc2

Elapsed Time Based on Activity
o time = half-life × ln initial activity rate
0.693
current activity rate
o t = T1/2 ln Rinitial
0.693 Rcurrent
1
(quantum number)2
20
Chapter 27 – Relativity

Time Dilation
o dilated time interval =
proper time
1 – (speed of clock)2
(speed of light)2
o Δt =
Δt0
1 – v2
c2

Length Contraction
o contracted length = proper length × 1 – (speed of object)2
(speed of light) 2
o L = L0 1 – v2
c2

Relativistic Energy
o relativistic energy = rest mass × (speed of light)2
1 – (speed of object)2
(speed of light)2
o E = m0c2
1 – v2
c2

Rest Energy
o rest energy = rest mass × (speed of light)2
o E0 = m0c2
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