Download Mathematics (P)review

Mathematics (P)review
The star Proxima Centauri is 23,400,000,000,000 miles
away from Earth. If we could travel in a spaceship at 5000
miles/hour, it would take over 534,000 years to get there.
Scientific Notation eliminates all unnecessary place holders
by making use of powers of 10
E.g.
2000 = 2 x 103
The star Proxima Centauri is 2.34 x 1013 miles away from
Earth. If we could travel in a spaceship at 5 x 103
miles/hour, it would take over 5.34 x 105 years to get there.
How to Write a Number in Scientific Notation
Scientific notation is a mantissa multiplied by the appropriate power of ten (10)
534,000 = 5.34 x 105
Step 1: Find the mantissa by moving the decimal so that there is
only one digit to the left (eliminate unnecessary digits).
7,180,000 = 7.18 x 106
0.00942 = 9.42 x 10-3
Step 2: Find the appropriate power of 10:
• when moved to the left, power = + number of places moved
• when moved to the right, power = - number of places moved
Useful Powers of Ten
Ten (10) = 101
Deca-
Hundred (100) = 102
Hecto-
Thousand (1000) = 103
Kilo-
Million (1,000,000) = 106
Mega-
Billion (1,000,000,000) = 109
Giga-
Average distance between the Earth & Sun
93 million miles = 93 x 106 miles or 93 Mega-miles
Useful Powers of Ten
Tenth (0.1) = 10-1
Deci-
Hundredth (0.01) = 10-2
Centi-
Thousandth (0.001) = 10-3
Milli-
Millionth (0.000001) = 10-6
Micro-
Billionth (0.000000001) = 10-9
Nano-
Average thickness of human hair
75 x 10-6 meters = 75 micrometers
Measurement Systems
Every measurement consists of a value and a unit
It is 215 miles from Boston to New York City
Having one system of units allows everyone to be on the same page
Requirements for any measurement system include:
- A standardized basis
- Easy to convert measurements within the system
The metric system is both the scientific standard and the world
standard (including U.S. though the British Imperial System is
use in everyday experience)
The Metric System
Seven basic properties of nature are identified within the SI System and
each has an assigned base unit.
Basic Physical Property
Distance
Base Unit
meter (m)
Mass
kilogram (kg)
Time
second (s)
Temperature
Kelvin (K)
Electric Current
Ampere (A)
Amount of a Substance
Mole (mol)
Intensity of Light
Candela (cd)
Extension of Base Units
(within metric system)
Base Unit
Conversion Factor
Gigameter (Gm)
109 m
Megameter (Mm)
106 m
Kilometer (km)
1000 m
Hectometer (Hm)
100 m
Decameter (Dm)
10 m
Meter (m)
1m
Decimeter (dm)
0.1 m
Centimeter (cm)
0.01 m
Millimeter (mm)
0.001 m
Micrometer (µm)
10-6 m
10-9 m
Nanometer (nm)
Extension of Base Units
(within metric system)
Base Unit
Conversion Factor
Gigagram (Gg)
109 g
Megagram (Mg)
106 g
Kilogram (kg)
1000 g
Hectogram (Hg)
100 g
Decagram (Dg)
10 g
Gram (g)
1g
Decigram (dg)
0.1 g
Centigram (cg)
0.01 g
Milligram (mg)
0.001 g
Microgram (µg)
10-6 g
10-9 g
Nanogram (ng)
Extension of Base Units
(within metric system)
Base Unit
Conversion Factor
Gigabanana (Gbn)
109 bn
Megabanana (Mbn)
106 bn
Kilobanana (kbn)
1000 bn
Hectobanana (Hbn)
100 bn
Decabanana (Dbn)
10 bn
Banana (bn)
1 bn
Decibanana (dbn)
0.1 bn
Centibanana (cbn)
0.01 bn
Millibanana (mbn)
0.001 bn
Microbanana (µbn)
10-6 bn
10-9 bn
Nanobanana (nbn)
Conversion of Units - Metric
Multiply the given value by the ratio of the conversion factor (CF) of the
given unit to the CF of the desired unit
Given Value
x
CF of given unit
CF of desired unit
Problem: Convert 12 km to centimeters
12 km
x
1000
0.01
= 1,200,000 cm
Conversion Between Different Systems
Problem: Convert 4 km to miles
• Set up a proportion using the proper conversion factors:
1 mile = 1.6 km
4 km
1.6 km
=
x miles
1 miles
• Cross multiply:
(4) · (1) km = 1.6 · x miles
• Solve for unknown:
x = (4) ÷ (1.6) = 2.5 miles
Using a Ruler
Measure the length of the double ended arrow.
cm
1
2
3
4
5
6
11
7
8
12
9
10
11
12
13
cm
Length of the arrow = 11.73576498 cm
Read the ruler to the nearest millimeter (0.1 cm)
Correct answers will have leeway of 1 mm (0.1 cm)
13
14
Using the Protractor
Step 1: Line up the bottom of the protractor with one leg of the angle.
Step 2: Line up the center of the protractor with the vertex of the angle.
Step 3: Read the appropriate scale:
- Inner scale for angles that open up clockwise (∠AOB = 149°)
- Outer scale for angles that open up counterclockwise (∠COB = 31°)
B
A
O
C
Physical Properties: Other Units of Measure
Velocity (v) – how fast something is moving
Base Unit: meters-per-second (m/s)
distance
velocity =
time
velocity
Acceleration (a) – rate something changes velocity a =
time
2
Base Unit: meters-per-square-second (m/s )
Force (F) – push or a pull on an object
Base Unit: Newton (N)
F = mass ! acceleration
Luminosity (L) – light energy emitted over time
Base Unit: Watts (W)
Energy
L=
time
Astronomical Measurements
Astronomers use more accessible dimensions to visualize larger scales
Astronomical Unit: the average Earth-Sun distance (150 million km)
- distances within star-planet & star-star systems
Light-year (LY): the distance light travels in a year (9.5 trillion km)
- nearest star is Proxima Centauri @ 4.3 LY
- “solar neighborhood” ~ few thousand LY
- diameter of our Galaxy ~ 100,000 LY
- nearest major galaxy (M31) ~ 3 Million LY
- “diameter” of observable universe ~ 93 Billion LY
On small scales, the distance to an object in light-years provides the
“look-back” time
Comparing Measurements
Two measurements can be compared by using subtraction
or ratios
Compare the size of the Earth (Diameter = 12,756 km)
to the size of Venus (Diameter = 12,118 km)
Diameter of Earth - Diameter of Venus = 12,756 - 12,118 = 638 km
(The Earth is 638 km larger than Venus)
For many cases in astronomy, subtraction is not very useful
Comparing Measurements
Compare the distance between Earth and the Sun
(150,000,000 km) to the distance from Earth to the star,
Sirius (81,700,000,000,000 km)
(Earth-Sirius distance) - (Earth-Sun Distance)
(81,700,000,000,000 km) - (150,000,000 km) = 81,699,850,000,000 km
Sirius is 81,699,850,000,000 km further from Earth
than the Sun.
Ratios
We use a ratio to see “how many times larger” one
measurement is compared to another
(Distance from Earth to Sirius)
(Distance from Earth to the Sun)
81,700,000,000,000 km
150,000,000 km
=
8.17 x 1013 km
1.50 x 108 km
= 5.45 x 105
Sirius is 545,000 times further from Earth than the Sun
Proportions
When two ratios are equal, they are said to be in proportion.
Astronomers use proportions to develop accurate models or
to determine the true sizes of objects from photographs.
Griffith Observatory
Rose Center, AMNH
Using Proportions to Make a Scale Model
A model Earth and a model Sun are separated by 1 meter. How far
away do we put a “star” to represent the distance to Sirius?
Solve by setting the ratio of the model equal to the ratio of the real situation
(Earth - Sirius Distance)Model
(Earth - Sun Distance)Model
(Earth - Sirius Distance)Model
1 meter
(Earth - Sirius Distance)Model
=
=
(Earth - Sirius Distance)Reality
(Earth - Sun Distance)Reality
8.17 x 1013 km
1.50 x 108 km
= 5.45 x 105
= 545,000 meters (545 km)
Proportions
When a model Earth and a model Sun are separated by 1 meter,
then a model star needs to be put 545 kilometers away!
Proportions
When the scale of an image is known, then it can be used to
find the dimensions of objects
Find the longest width of the base of Olympus
Mons if the scale of the image is: 1 cm = 60 km
Set up proportion:
9
8
7
6
5
1 cm
9.3 cm
=
60 km
x km
4
3
Measure the longest width
2
Width = 9.3 cm
1
cm
Complete the proportion and solve
Olympus Mons, Mars
x = 60 x 9.3 = 558 km