Final Velocity (v_{f}) =

v_{f} = v_{o} + at

Conservation of Momentum for Inelastic and Completely Elastic Collisions =

m_{a}v_{ai} + m_{b}v_{bi} = m_{a}v_{af} + m_{b}v_{bf}

Celsius to Kelvin Conversion =

T_{c} = T_{k} - 273

Equation to determine the angular frequency (ω) of a spring:

ω = 2πf = √(^{k}/_{m})

where m is the mass of the object

Equation to determine the beat frequency:

f_{beat} = |f_{1} - f_{2}|

Equation to determine pressure:

P = ^{F}/_{A}

where F is the normal force and A is the area; a scalar quantity; units = Pa =^{ N}/_{m2}

Equation to determine the buoyant force on a floating object:

F_{buoy} = (V_{fluid displaced})(p_{fluid)}(g) = weight of the object

units = (kg)(^{m}/_{s2})

Speed of light in a vacuum (c) =

3.8 X 10^{8} ^{m}/_{s}

Equation to determine the peak wavelength emitted by an object at a given temperature (Wien's Displacement Law):

(λ_{peak})(T) = constant = 2.9 X 10^{-3 }m•K

Equation to determine the critical velocity (Vc) of a fluid flowing through a tube:

v_{c} = ^{NRη}/_{ρD}

where N_{R} is a given constant; η is the viscosity of the fluid, ρ is the density of the fluid, and D is the diameter of the tube

units = ^{m}/_{s}

Force of Static Friction (f_{s}) =

0 ≤ f_{s} ≤ u_{s}N

Equation to determine the buoyant force on a fully submerged object:

F_{buoy} = (V_{object submerged})(p_{fluid})(g)

units = (kg)(^{m}/_{s2})

Electrostatics F, U, E, and V grid

Equation to determine Young's Modulus (perpendicular application of force):

Y = ^{(F/A) }/_{ (∆L/L)}

it gives the change in length of a solid when a pressure is applied perpendicularly to it (compression or stretching)

Equation to determine the period (T) of a pendulum:

T = 2π√(^{L}/_{g})

Gamma decay is :

the emission of a gamma photon. The energy of the parent nucleus is lowered, but the mass and atomic number remain the same.

Equation to determine the position of minima on the screen in a double slit experiment:

dsinθ = (m + ^{1}/_{2})λ (m = 0, 1, 2, ...)

where d is the distance between the slits, θ is the angle between the center of the slits and the maxima, λ is the wavelength of the light, and m is the integer representing the order

**USE SMALL ANGLE APPROXIMATION**

**SINθ ****≈ ****TANθ**

Equation to determine the wavelength of standing waves for closed pipes:

λ = ^{4L}/_{n}

where L is the length of the closed pipe and n is the number of nodes

Coulomb's Law (F) =

F = kq_{1}q_{2 }/ r^{2}

(k = 9 X 10^{9})

[gives the force two charges exert on one another]

Final Velocity Squared (V_{f}^{2}) =

v_{f}^{2} = v_{o}^{2} + 2a(Δx)

Centripetal Force =

F = mv^{2}/r

Change in volume of a fluid or object subjected to a change in temperature (ΔV) =

ΔV = βVΔT

(β = 3α = a constant)

When PE is converted to KE completely, velocity (v) =

v = √ 2gh

^{1}/_{5} =

.2

Equation used to determine the drop in electric potential across a resistor:

V = iR

Ohm's Law

Equation to determine the total voltage drop across multiple resistors in a series:

V_{s} = V_{1} + V_{2} + V_{3} + V_{4} + ...

When n identical resistors are wired in parallel, the total resistance is given by the equation:

^{R}/_{n}

The equations for voltage, resistance, and capacitance in parallel:

V_{p} = V_{1} = V_{2} = V_{3} ...

1/R_{p} = 1/R_{1} + 1/R_{2} + 1/R_{3} ...

C_{p} = C_{1} + C_{2} + C_{3} + ...

Equation to determine the intensity of a wave:

I = ^{P}/_{A}

where P is the power and A is the surface area of the object the wave comes in contact with

Equation to determine the new sound level after the intensity of a sound is changed by some factor:

B_{f} = B_{i} + 10log(I_{f}/I_{i})

where (I_{f}/I_{i}) is the ratio of the final intensity to the initial intensity

Equation to determine the weight of any volume of a given substance:

W = ρVg

where p is the density and V is the volume of the substance

Equation to determine the change in energy of an electron due to absorption or emission of a photon:

hf = |∆E|

units = joules

Torque (t) experience by an electric dipole =

t = (qd)EsinØ

(E = electric field)

Positron decay is:

when a proton splits into a positron (β+) and a neutron. A proton is lost and the mass number remains the same.

Equation to determine the focal length in lenses where thickness cannot be ignored:

^{1}/_{f }= (n-1)(^{1}/r_{1} - ^{1}/r_{2})

where n is the index of refraction of the lens material, r_{1 }is the radius of the curvature of the first lens surface, and r_{2} is the radius of curvature of the second lens surface

Equation used to determine the increase in capacitance due to a dielectric material:

C' = CK

where C is the original capacitance and K is the dielectric constant of the material

Electric Field (E) =

E = F/q_{o} = kq_{1}/r^{2}

(k = 9 X 10^{9})

[E = the electric field produced by a source charge (q_{1}) at a chosen point in space. F is the force felt by the test charge (q_{o}) when put into the electric field.]

Total Mechanical Energy (E) =

E = PE + KE

(Constant in a conservative system)

Equation to determine the total voltage across multiple capacitors in parallel:

V_{p} = V_{1} = V_{2} = V_{3} = V_{4 }+ ...

Equation to find the degree of refraction of a light ray upon entering a new medium:

n_{1}sinθ_{1} = n_{2}sinθ_{2}

theta is always measured with respect to the perpendicular to the boundary

Equation to determine gauge pressure:

P_{g} = P - P_{atm} = (P_{o} + ρgh) - P_{atm}

units = Pa

Equation to determine the total voltage across multiple capacitors in series:

V_{s} = V_{1} + V_{2} + V_{3} + V_{4} + ...

^{1}/_{8} =

.125

Equation to determine the total voltage drop across multiple resistors in parallel:

V_{p} = V_{1} = V_{2} = V_{3 }= V_{4} ...

Center of Mass (x) =

x = (m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3}) / (m_{1} + m_{2} + m_{3})

For a plane mirror, the mirror equation becomes:

^{1}/_{o} + ^{1}/_{i }= 0

because at any time the object is at the focal point, the reflected rays will be parallels and the image will be at infinity

Displacement (Δx) =

[3 equations]

Δx = v_{o}t + ^{1}/_{2}at^{2}

Δx = v_{f}t - ^{1}/_{2}at^{2}

Δx = v/t

Equation to determine the resistance of a given resistor:

R = ^{pL}/_{A}

where p is the resistivity, R is the resistance, L is the length of the resistor, and A is the cross-sectional area of the resistor

Acceleration (a) =

a = ^{Δv}/_{Δt}

Gravitational Force =

F = Gm_{1}m_{2 }/ r^{2}

(G = 6.67 X 10^{-11})

Change in entropy of a system at a given temperature (ΔS) =

ΔS = Q/T

(Q = heat gained or lost)

(T = temperature in Kelvin)

(Entropy = how much energy is spread out)

Bernouli's equation (an expression of conservation of energy for a flowing fluid):

P_{1} + ^{1}/_{2}ρv_{1}^{2} + ρgy_{1} = P_{2} + ^{1}/_{2}ρv_{2}^{2} + ρgy_{2}

Force of Kinetic Friction (f_{k}) =

f_{k} = u_{k}N

Force (F) =

F = ma

Equation to determine Shear Modulus (parallel application of force):

S = ^{(F/A) }/ _{(x/h)}

where x is the lateral displacement and h is the vertical displacement

Kinetic Energy (KE) =

KE = ^{1}/_{2}mv^{2}

Change in length of an object subjected to a change in temperature (ΔL) =

ΔL = αLΔT

(α = a constant)

Density equation:

p = ^{m}/_{v}

a scalar quantity; units = ^{kg}/_{m3} or ^{g}/_{mL} or ^{g}/_{cm3}

Equation to determine the speed (f) of a wave =

v = fλ = ^{ω}/_{k }= ^{λ}/_{T}

Work-Energy Theorem

W_{net} = ΔKE = KE_{f} - KE_{i}

√ 2 =

1.4

Electrical potential difference between two points in space =

V_{b} - V_{a} = W_{ab}/q_{o}

(W_{ab} is the work needed to move a test charge q_{o} through an electric field from a to b)

Equation to determine the mass lost as binding energy in a nucleus:

**E = mc ^{2}**

where m is the mass of c is the speed of light

Equation to determine the frequency (f) of a pendulum:

f = ^{1}/_{T} or ^{ω}/_{2π}

^{1}/_{6} =

.167

Magnitude of the magnetic field produced by a straight current-carrying wire at a chosen point in space (B) =

B = u_{o}*i* / 2πr

(u_{o} = 4π X 10^{-7} Tesla)

Potential Energy (PE) =

PE = mgh

Efficiency =

Efficiency = W_{out} / W_{in}

= ^{(load)(load distance) }/ _{(effort)(effort distance)}

Change in the total internal energy of a system undergoing a thermodynamic process (ΔU) =

ΔU = Q - W

(Q = heat; heat flow in is positive)

(W = work done by the system)

The equations for voltage, resistance, and capacitance in series:

V_{s} = V_{1 }+ V_{2} + V_{3} ...

R_{s} = R_{1 }+ R_{2} + R_{3} ...

1/C_{s} = 1/C_{1} + 1/C_{2} + 1/C_{3}

Equation to estimate the energy of an electron with a given quantum number (n) in electron-volts (eV):

En = - ^{13.6 eV}/_{n2}

Equation to determine the perceived frequency from the Doppler Effect:

When the detector is moving toward the source, use the top sign in the top row.

When the source is moving toward the detector, use the minus sign in the bottom row.

Equation to determine the period (T) of a spring:

T= 2π√(^{m}/_{k})

Equation to determine the frequency (f) of a spring:

f = ^{1}/_{T} or ^{ω}/_{2π}

^{1}/_{7} =

.14

Work done on or by a system that undergoes a change in volume at constant pressure (W) =

W = PΔV

(P = pressure)

The Work Function is:

the minimum energy required to eject an electron and is related to the threshold frequency of a given metal:

**W = hf _{t}**

Power (P) =

P = Fv = ^{W}/_{t} = ^{energy}/_{time}

(W = work; units = Watts)

Magnetic force on a moving charge in an external magnetic field (F) =

F = qvBsinØ

(q = charge; v = velocity; B = magnitude of magnetic field)

Equation to determine the power dissipated by a resistor:

P = iV = i^{2}R = V^{2}/R

where i is the current through the resistor, V is the voltage drop across the resistor, and R is the resistance of the resistor

Equation to determine the angular frequency (ω) of a pendulum:

ω = 2πf = √(^{g}/_{L})

where L is the length of the pendulum

Units of Pressure Conversions

1 atm = 760 torr = 760 mm Hg = 101.3 kPa

Equation to determine the resultant capacitance across multiple capacitors in series:

1/C_{s} = 1/C_{1} + 1/C_{2} + 1/C_{3} + ..

Equation to determine the electric field at a point in space between the plates of a parallel plate capacitor:

E = ^{V}/_{d}

where V is the volatage applied across the plates and d is the separation between the plates

Equation to estimate the average magnitude of alternating current over time:

i_{rms} = i_{max} / √2

where i_{max} is the maximum current and i_{rms} is the average current

Equation to determine object distance (o), image distance (i), focal length (f), and magnification (m) for lenses:

^{1}/_{o }+ ^{1}/_{i} = ^{1}/_{f}

m = ^{-i}/_{o}

Conservation of Momentum for Completely Inelastic Collisions =

m_{a}v_{ai} + m_{b}v_{bi} = (m_{a} + m_{b})v_{f}

Equation to determine Bulk modulus (chage in volume due to pressure):

B =^{ (F/A) }/ _{(∆V/V)}

where V is volume

Equation of Pascal's principle for incompressible fluids in containers:

P = ^{F1}/_{A1 }= ^{F2}/_{A2}

V = A_{1}d_{1} = A_{2}d_{2}

W = F_{1}d_{1} = F_{2}d_{2}

Equation to determine the resultant resistance of multiple resistors in parallel:

1/R_{p} = 1/R_{1} + 1/R_{2} + 1/R_{3} + ...

Equation to determine the frequency of a standing wave for closed pipes:

f = ^{nv}/_{4L}

where n is the number of nodes and L is the length of the closed pipe

Work (W) =

W = FdcosØ

Equation to determine the position of maxima on the screen in a double slit experiment:

dsinθ = mλ (m = 0, 1, 2, ...)

where d is the distance between the slits, θ is the angle between the center of the slits and the maxima, λ is the wavelength of the light, and m is the integer representing the order

**USE SMALL ANGLE APPROXIMATION: **

**SINθ ≈ TANθ**

Equation to determine the magnification of a system of lenses not in contact:

M = (m_{1})(m_{2})(m_{3})...

Heat gained or lost by an object subjected to a change in phase (Q) =

Q = mL

(m = mass; L = heat of transformation)

Equation to estimate the average magnitude of AC voltage over one period:

V_{rms} = V_{max} / √2

where V_{max} is the maximum voltage and V_{rms} is the average voltage

Impulse (I) =

I = Ft = Δp = mv_{f} - mv_{i}

Electric potential (V) due to a dipole =

V = (kqd/r^{2})cosØ

(k = 9 X 10^{9})

Lenses in contact definition and equation:

a series of lenses with negligible distances between them (contact lenses). They behave as a single lens.

^{1}/_{f} = ^{1}/_{f1} + ^{1}/_{f2} + ^{1}/_{f3} + ...

P = P_{1} + P_{2} + P_{3} + ...

Air Resistance Equation

f = kv^{2}

Equation to determine the resultant capacitance across multiple capacitors in parallel:

C_{p }= C_{1 }+ C_{2} + C_{3} + C_{4} + ...

^{1}/_{9} =

.11

Equation to determine the frequency of a standing wave for strings and open pipes:

f = ^{nv}/_{2L}

where n is the number of nodes and L is the length of the string or pipe.

Equation for exponential decay:

*n* = *n*_{o}e^{-λt}

The continuity equation:

v_{1}A_{1} = v_{2}A_{2} = constant (the rate of flow)

Law of Reflection equation:

θ_{1} = θ_{2}

where θ_{1} is the incident angle and θ_{2} is the reflected angle

the angles are always measured from normal (a line perpendicular to the surface of the medium)

Weight =

Weight = mg

Equation to estimate the energy of an electron with a given quantum number (n) in joules:

E = - ^{Rh}/_{n2}

where R is Rydberg's constant and n is the quantum level

Beta decay is:

Loss of an electron (e- or β-). Emitted when a neutron in the nucleus decays into a proton and an antineutrino (β-). A neutron is lost and a proton takes its place. Mass number remains the same and atomic number increases by one.

The magnitude of a dipole moment (p) =

p = qd

(q = charge magnitude; d = distance between chrages of dipole)

Equation for the instantaneous current of an alternating current:

* i* = i

_{max}sin(2πft) = i

_{max}sin(ωt)

where ** i** is the instantaneous current, i

_{max}is the maximum current, f is the frequency, and ω is the angular frequency

Equation to determine the location of dark fringes due to single-slit diffraction with a lens:

asinθ = nλ (n = 1, 2, 3,...)

where a is the width of the slit, lambda is the wavelength of the incident wave, and theta is the angle made by the line drawn from the center of the lens to the dark fringe and the line perpendicular to the screen

Equation to determine the wavelength of standing waves for strings and open pipes:

λ = ^{2L}/_{n}

where L is the length of the string or pipe and n is the number of antinodes.

√ 3 =

1.7

Electric Potential (V) =

V = W/q_{o }= Ed = kq_{1}/r

(q_{1} = source charge; W = work; k = 9 X 10^{9})

Mechanical Advantage =

= force_{out} / force_{in}

Total electric field at a given point in space (E_{total}) =

E_{total }= E_{q1} + E_{q2} + E_{q3} ...

Equation to determine the total energy emitted by a blackbody (Stefan-Boltzmann Law):

E_{T} = σT^{4}

where σ is a constant, T is the temperature, and E_{T} is the total energy emitted per unit of area

units = ^{W}/_{m2}

Equation to determine the energy of a photon:

E = hf

where h is Planck's constant and f is the frequency of the light. Once you know the frequency, you can find the wavelength using:

λ = ^{c} / _{f}

Electrical Potential Energy (U) =

U = W = qEd = q_{1}ΔV = kq_{1}q_{2}/r

(W = work; V = electric potential; k = 9 X 10^{9})

Equation to determine the index of refraction:

n = ^{c}/_{v}

where c is the speed of light in a vacuum (3.8 X 10^{8}), v is the speed of light in the given medium, and n is the index of refraction

Velocity (v) =

v = ^{Δx}/_{Δt}

Gravity on Earth (g) =

g = ^{Gme}/_{re2}

Average Velocity (v)=

v = ^{(}^{vo + vf) }/ _{2}

Equation to determine the capacitance of a parallel plate capacitor:

C = ε_{o }(^{A}/_{d})

where ε_{o} is 8.85 X 10^{-12}, A is the area of overlap of the two plates, and d is the separation of the two plates

Equation to determine the potential energy of a spring:

PE = ^{1}/_{2}kx^{2}

Equation to determine the critical angle:

derived from Snell's Law

sinθ_{c} = n_{2} / n_{1}

Equation to determine the potential energy stored in a capacitor:

U = ^{1}/_{2}CV^{2}

where C is the capacitance of the capacitor and V is the voltage applied

Equation to find the magnification of an image for both mirrors and lenses:

m = - ^{i}/_{o}

where o is the distance of the object from the mirror and i is the distance of the image from the mirror or lens

if the absolute value of m is less than 1, the image is reduced

if the absolute value of m is greater than 1, the object is enlarged

Equation to determine the actual voltage supplied by a cell to a circuit:

V = ε_{cell} - ir_{int}

where i is the current, r_{int} is the internal resistance of the material, and ε_{cell} is the emf of the cell

Equation to determine the change in energy of an electron due to absorption or emission of a photon:

∆E = E_{f} - E_{i }

(both of these values will always be negative; if ∆E is negative, their was an emission of energy and the electron came down states)

units = joules

Equation to determine the maximum kinetic energy of an electron ejected by an incident photon:

K_{max }= hf - W

where W is the work function of the metal in question

**W = hf _{t}**

Equation to find the wavelength of a photon:

λ = ^{c}/_{f}

where c = 3 X 10^{8}

Equation to determine the resultant resistance of multiple resistors in a series:

R_{s} = R_{1} + R_{2} + R_{3} + R_{4} + ...

Isotopic notation:

immediately left to the chemical symbol, the mass number and atomic number can be read top to bottom, respectively.

Area of a piston:

πr^{2}

(the surface of a piston is circular)

Equation to determine absolute pressure:

P = P_{o} + ρgh

where P is the absolute pressure, P_{o} is the pressure at the surface, and ρgh is (density fluid above)(gravity)(height of submerged object below surface)

units = ^{N}/_{m2}

Magnitude of the magnetic field produced by a circular loop of current-carrying wire at the center of the loop (B) =

B = u_{o}*i* / 2r

(u_{o} = 4π X 10^{-7} Tesla)

Total electric current passing through a conductor per unit of time (*i*) =

*i* = Δq/Δt

(q = charge)

Equation to determine the wavelength or frequency of light traveling in air or a vacuum:

c = fλ

where c is the speed of light and is equal to 3 X 10^{8};

units = m/s

Equation to determine the capacitance of a capacitor:

C = ^{Q}/_{V}

where Q is the absolute value of the charge and V is the voltage applied

Alpha decay is:

Loss of a 4-He nucleus (2 protons and 2 neutrons). Mass number decreases by 4 and atomic number decreases by 2.

SI unit of viscosity:

(N)(^{s}/_{m2})

Electron capture is:

When a radionuclide captures an inner electron that combines with a proton to form a neutron. Atomic number is one less and mass number remains the same.

Torque (t) =

t = rFsinØ

CCW = (+)

CW = (-)

Equation to determine the object or image distance from a mirror, focal length, or radius of curvature:

^{1}/_{o }+ ^{1}/_{i} =^{ 1}/_{f} = ^{2}/_{r}

where o is the distance of the object from the mirror, i is the distance of the image from the mirror, f is the distance from the focal point to the mirror (focal length), and r is the distance between the center of curvature and the mirror (for spherical mirrors)

units = m^{-1}

Equation to determine the restoring force of a pendulum:

F = -mgsinØ

where Ø is the angle between the pendulum arm and the vertical

Conservation of Kinetic Energy =

^{1}/_{2}m_{a}v_{ai}^{2} + ^{1}/_{2}m_{b}v_{bi}^{2} = ^{1}/_{2}m_{a}v_{af}^{2} + ^{1}/_{2}m_{b}v_{bf}^{2}

(Completely elastic collisions only)

Equation to determine the sound level (B):

B = 10log(I/I_{o})

where I_{o} is 1 X 10^{-12} (the threshold of hearing)

Equation to determine the restoring force of a spring (Hooke's Law):

F = -kx

where k is the spring constant and x is the displacement of the spring from its equilibrium length

Magnetic force on a current-carrying wire in a uniform external magnetic field (F) =

F = *i*LBsinØ

(*i* = current; L = length of wire; B = magnitude of magnetic field)

Centripetal Acceleration =

a = v^{2}/r

Electric field (E) due to a dipole =

E = kqd/r^{3}

(k = 9 X 10^{9})

For spherical mirrors, what does the focal length (f) equal?

f = ^{r}/_{2}

Momentum (p) =

p = mv

Density of water:

1,000 ^{kg}/_{m3} or 1 ^{g}/_{cm3}

Heat gained or lost by a substance subjected to a change in temperature (Q) =

Q = mcΔT

(m = mass; c = specific heat)

1 Tesla = ? gauss

10^{4} gauss

(1 gauss = 1 X 10^{-4} Tesla)

Half-life is:

the amount of time required for half of a sample of radioactive nuclei to decay

**T _{1/2 }= ^{0.693}/_{λ}**