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/s |
acceleration |

1.4 |
v = at |
m/s |
velocity |

1.5 |
v |
m/s |
final velocity |

1.6 |
v |
m/s |
average velocity |

1.7 |
d = ½ at |
m |
distance |

1.8 |
v = gt |
m/s |
velocity |

1.9 |
d = ½ gt |
m |
distance |

1.10 |
F = (Gm |
newtons |
gravitational force |

1.11 |
x = v |
m |
x position |

1.12 |
v |
m/s |
final x velocity |

1.13 |
y = v |
m |
Y position |

1.14 |
v |
m/s |
final y velocity |

1.15 |
p |
yr, A.U. |
Kepler’s 3 |

1.16 |
v = (GM/r) |
km/s |
orbital velocity |

1.17 |
v = (2GM/r) |
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 mv |
J, joules |
kinetic energy |

2.3 |
PE = mgh |
J, joules |
potential energy |

2.4 |
PE |
J, joules |
conservation of energy |

2.5 |
v |
m/s |
roller coaster velocity |

2.6 |
F |
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 • m |
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 = mH |
calories |
heat to melt solid |

3.10 |
Q = mH |
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 |
F |
newtons |
buoyant force |

4.4 |
PV = constant |
pascals, m |
Boyle’s Law |

4.5 |
V = constant * T |
m |
Charles/Gay-Lussac’s Law |

4.6 |
PV = nRT |
pascals, m |
Ideal Gas Law |

4.7 |
P ∝ T |
pascals, °C |
pressure proportional to temperature |

4.8 |
P ∝1 / V |
pascals, m |
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 = v |
none |
Snell’s Law |

6.1 |
f = c/λ |
hertz |
frequency of light |

7.1 |
T = 2,897,000/ λ |
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 = t |
s |
time in two reference frames |

7.5 |
ɣ = 1/(sqrt(1-(v |
none |
gamma |

7.6 |
t = ɣt |
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 10^{8} 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}

Nm^{2}/kg^{2}

g = acceleration of gravity = 9.8 m/s^{2}

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

10^{23} 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

t_{o} = time in alternative reference frame

V = volume

v = velocity

v_{f} = final velocity

v_{fx} = final velocity in x direction

v_{fy} = final velocity in y direction

v_{i} = initial velocity

v_{ix} = initial velocity in x direction

v_{iy} = initial velocity in y direction

y = instantaneous height of roller coaster

yr = year

Greek Letters

ɣ (gamma) = factor in special relativity

λ (lambda) = wavelength

λ_{max} = wavelength of max emission

ρ (rho) = density

ρ_{f} =fluid density

Units Conversion factors

Distance: nm = nanometer = 1 x 10^{-9} m

Energy: calorie = 1cal = 4.18 J

Energy: joule = 1 J = kg*m^{2}/ s^{2}

Force: newton = 1 N = kg*m/s^{2}

Frequency: hertz = 1 Hz = 1/s

Pressure: pascal = 1 Pa = 1 N/m^{2}

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 10^{8}

__ __

1 = 10^{0}

10 = 10^{1}

100 = 10^{2}

1,000 = 10^{3}

1,000,000 = 10^{6 }(a million)

1,000,000,000 = 10^{9} (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 10^{x} terms.

(2.3 x 10^{3}) x (1.5 x 10^{4}) = 3.4 x 10^{7}

To __divide__ using scientific notation divide the integer terms and __subtract the exponents__ of the 10^{x} terms.

(5 x 10^{5}) / (2 x 10^{3}) = 2.5 x 10^{2}

To __raise to a power__ using scientific notation raise the integer terms to the indicated power and __multiple the exponent__ of the 10^{x}
terms by the power.

(3 x 10^{3})^{4} = (3 x 3 x 3 x 3 x10^{3}*^{4}) = 81 x 10^{12} = 8.1 x 10^{13}

^{ }

To __find the xth root__ of a number using scientific notation, find the root of the integer term and __divide the exponent__ of the 10^{x}
terms by the power.

(27 x 10^{9})^{1/3 }= 3 x 10^{3}

(note that a fractional exponent such as ^{1/3} means to take the 3^{rd} 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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