Useful Things

Eqn #

Eqn

Units

Calculates

1.1

v = d/t

m/s

velocity

1.2

t = d/v

s

time

1.3

a = v/t

m/s2

acceleration

1.4

v = at

m/s

velocity

1.5

vf = vi + at

m/s

final velocity

1.6

vave = (vi + vf) / 2

m/s

average velocity

1.7

d = ½ at2

m

distance

1.8

v = gt

m/s

velocity

1.9

d = ½ gt2

m

distance

1.10

F = (Gm1m2)/r2

newtons

gravitational force

1.11

x = vixt + ½ axt2

m

x position

1.12

vfx = vix +axt

m/s

final x velocity

1.13

y = viyt + ½ayt2

m

Y position

1.14

vfy = viy +gt

m/s

final y velocity

1.15

p2 = a3

yr, A.U.

Kepler’s 3rd Law

1.16

v = (GM/r)½

km/s

orbital velocity

1.17

v = (2GM/r)½

km/s

escape velocity

1.18

F = ma

newtons

force to accelerate a mass

2.1

TE = PE + KE + …

J, joules

total energy

2.2

KE = ½ mv2

J, joules

kinetic energy

2.3

PE = mgh

J, joules

potential energy

2.4

PE1 + KE1 = PE2 + KE2

J, joules

conservation of energy

2.5

v2 = 2g(H-y)

m/s

roller coaster velocity

2.6

Fc = mv2/r

newtons

centripetal force

2.7

h ≥ 2.5r

m

height for roller coaster to safely loop-d-loop

2.8

p = mv

kg m/s

momentum

2.9

p = Ft

kg m/s

momentum

2.10

L = mvr

kg • m2/s

angular momentum

3.1

C = (F-32)*5/9

degrees

Centigrade to Fahrenheit

3.2

F = (C*9/5) + 32

degrees

Fahrenheit to Centigrade

3.3

C = K -273

degrees

Kelvin to Centigrade

3.4

K = C +273

none

Centigrade to Kelvin

3.5

F = (K-273)*9/5 + 32

degrees

Kelvin to Fahrenheit

3.6

K = (F-32)*5/9 + 273

none

Fahrenheit to Kelvin

3.7

ΔL = α L ΔT

mm

change in length heated bar

3.8

Q = c m ΔT

calories

heat

3.9

Q = mHf

calories

heat to melt solid

3.10

Q = mHv

calories

heat to vaporize liquid

3.11

H = (k t A T)/d

calories

conductive heat

4.1

P = F/A

pascals

fluid pressure

4.2

P = ρgh

pascals

pressure with depth

4.3

Fb = ρfVg

newtons

buoyant force

4.4

PV = constant

pascals, m3

Boyle’s Law

4.5

V = constant * T

m3

Charles/Gay-Lussac’s Law

4.6

PV = nRT

pascals, m3

Ideal Gas Law

4.7

P T

pascals, °C

pressure proportional to temperature

4.8

P 1 / V

pascals, m3

pressure inversely proportional to volume

5.1

i = r

degrees

Law of Reflection

5.2

n = c/v

none

index of refraction

5.3

​sin i/sin r = v1/v2= n2/n1

none

Snell’s Law

6.1

f = c/λ

hertz

frequency of light

7.1

T = 2,897,000/ λmax

K

temperature of a star

7.2

E = hf

joules

energy of EM wave

7.3

E = hc/ λ

joules

energy as function of wavelength

7.4

t = to / sqrt (1-(v2/c2))

s

time in two reference frames

7.5

ɣ = 1/(sqrt(1-(v2/c2))

none

gamma

7.6

t = ɣto

s

time in two reference frames

Letters Used in Equations


A.U. = astronomical Unit = 1.5 x 10
8 km

 

A = area

 

a = acceleration

 

a = semi-major axis of orbiting body

 

c = speed of light = 3 x 108 m/s

 

c = specific heat

 

d = distance, thickness

 

EM = electromagnetic

 

EMS = electromagnetic spectrum

 

F = force

 

f = frequency of EM radiation

 

Fc = centripetal force

 

Fg = gravitational force

 

G = universal gravitational constant = 6.67 x 10-11

Nm2/kg2

 

g = acceleration of gravity = 9.8 m/s2

 

H = maximum height of roller coaster

 

H = heat flow

 

Hf = heat of fusion (melting)

 

Hv = heat of vaporization (evaporating)

 

h = height

 

h = Planck Constant = 6.636 x 10-34 Js

 

i = angle of incidence

 

k =  thermal conductivity

 

KE = kinetic energy

 

ΔL = change in length

 

L = angular momentum

 

PE = potential energy

 

M, m = mass

 

n = number of moles of a substances (n x 6.022 x

1023 molecules)

 

n = refractive index

 

P = pressure

 

p = period of an orbiting body

 

p = momentum

 

Q = heat

 

R = the universal gas constant = 8.31 J/molK

 

r = distance, radius

 

r = angle of reflection

 

TE = total energy

 

T = temperature

 

ΔT = change in temperature

 

t = time

 

to = time in alternative reference frame

 

V = volume

 

v = velocity

 

vf = final velocity

 

vfx = final velocity in x direction

 

vfy = final velocity in y direction

 

vi = initial velocity

 

vix = initial velocity in x direction

 

viy = initial velocity in y direction

 

y = instantaneous height of roller coaster

 

yr = year

 

Greek Letters


ɣ (gamma) = factor in special relativity

 

λ (lambda) = wavelength

 

λmax = wavelength of max emission

 

ρ (rho) = density

 

ρf =fluid density

 

 

Units Conversion factors

 

Distance: nm = nanometer = 1 x 10-9 m

 

Energy: calorie = 1cal = 4.18 J

 

Energy: joule = 1 J = kg*m2/ s2

 

Force: newton = 1 N = kg*m/s2

 

Frequency: hertz = 1 Hz = 1/s

 

Pressure: pascal = 1 Pa = 1 N/m2

 

 

Scientific Notation

Signigicant Figures

Any number can be written as the product of two numbers, an integer and 10 to some power. This is especially useful for very small and very large numbers, such as

0.00000043 = 4.3 x 10-7

or

299,873,000 = 2.99873 x 108


1 = 100

10 = 101

100 = 102

1,000 = 103

1,000,000 = 106 (a million)

1,000,000,000 = 109 (a billion)


0.1 = 10-1

0.01 = 10-2

0.001 = 10-3

0.000001 = 10-6 (a millionth)

0.000000001 = 10-9 (a billionth)


To multiply using scientific notation multiple the integer terms and add the exponents of the 10x terms.

(2.3 x 103) x (1.5 x 104) = 3.4 x 107


To divide using scientific notation divide the integer terms and subtract the exponents of the 10x terms.

(5 x 105) / (2 x 103) = 2.5 x 102


To raise to a power using scientific notation raise the integer terms to the indicated power and multiple the exponent of the 10x terms by the power.

(3 x 103)4 = (3 x 3 x 3 x 3 x103*4) = 81 x 1012 = 8.1 x 1013


To find the xth root of a number using scientific notation, find the root of the integer term and divide the exponent of the 10x terms by the power.

(27 x 109)1/3 = 3 x 103

(note that a fractional exponent such as 1/3 means to take the 3rd root).

Have you ever had a problem such as this:


X = 292/2.1


and turned in an answer like this:


X = 139.047619


If you look at the two numbers in the original problem, 292 and 2.1, each has only two or three digits. We have no information that the numbers aren’t really:


292.000000 or 292.439 or 291.56238


and 2.10000000, or 2.11, or 2.09876542345.


In other words we don’t know how accurately known each number really is. We say that 292 has 3 significant digits, and 2.1 has two significant digits.


Scientists have agreed that answers should only be given to the same number of significant digits as the least well-known number in any operation. So for this problem there are only two digits in 2.1 so the answer can only have two significant digits:

X = 140


Notice that writing 139 implies that we know the last digit is a 9, and not an 8 or a 0. We don’t know that, so we round up to 140, with the 0 at the end not counting as a significant digit.


Every time you do a calculation look to see how many digits there are in the number with the least significant digits, and make your answer match that precision.


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