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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