Mathematics (P)review

Transcription

1 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 = 2 x 10 3 The star Proxima Centauri is 2.34 x miles away from Earth. If we could travel in a spaceship at 5 x 10 3 miles/hour, it would take over 5.34 x 10 5 years to get there.

2 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 10 5 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 = 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

3 Useful Powers of Ten Ten (10) = 10 1 Hundred (100) = 10 2 Thousand (1000) = 10 3 Million (1,000,000) = 10 6 Billion (1,000,000,000) = 10 9 Deca- Hecto- Kilo- Mega- Giga- Average distance between the Earth & Sun 93 million miles = 93 x 10 6 miles or 93 Mega-miles

4 Useful Powers of Ten Tenth (0.1) = 10-1 Hundredth (0.01) = 10-2 Thousandth (0.001) = 10-3 Millionth ( ) = 10-6 Billionth ( ) = 10-9 Deci- Centi- Milli- Micro- Nano- Average thickness of human hair 75 x 10-6 meters = 75 micrometers

5 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)

6 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 Mass Time Temperature Electric Current Amount of a Substance Intensity of Light Base Unit meter (m) kilogram (kg) second (s) Kelvin (K) Ampere (A) Mole (mol) Candela (cd)

7 Extension of Base Units (within metric system) Conversion Factor Gigameter (Gm) 10 9 m Megameter (Mm) 10 6 m Kilometer (km) 1000 m Hectometer (Hm) 100 m Decameter (Dm) 10 m Base Unit Meter (m) 1 m Decimeter (dm) Centimeter (cm) Millimeter (mm) Micrometer (µm) Nanometer (nm) 0.1 m 0.01 m m 10-6 m 10-9 m

8 Extension of Base Units (within metric system) Gigagram (Gg) Megagram (Mg) Kilogram (kg) Hectogram (Hg) Decagram (Dg) Conversion Factor 10 9 g 10 6 g 1000 g 100 g 10 g Base Unit Gram (g) 1 g Decigram (dg) Centigram (cg) Milligram (mg) Microgram (µg) Nanogram (ng) 0.1 g 0.01 g g 10-6 g 10-9 g

9 Extension of Base Units (within metric system) Gigabanana (Gbn) Megabanana (Mbn) Kilobanana (kbn) Hectobanana (Hbn) Decabanana (Dbn) Conversion Factor 10 9 bn 10 6 bn 1000 bn 100 bn 10 bn Base Unit Banana (bn) 1 bn Decibanana (dbn) Centibanana (cbn) Millibanana (mbn) Microbanana (µbn) Nanobanana (nbn) 0.1 bn 0.01 bn bn 10-6 bn 10-9 bn

10 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 = 1,200,000 cm

11 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 = x miles 1.6 km 1 miles Cross multiply: (4) (1) km = 1.6 x miles Solve for unknown: x = (4) (1.6) = 2.5 miles

12 Using a Ruler Measure the length of the double ended arrow. cm cm Length of the arrow = cm Read the ruler to the nearest millimeter (0.1 cm) Correct answers will have leeway of 1 mm (0.1 cm)

13 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

14 Physical Properties: Other Units of Measure Velocity (v) how fast something is moving Base Unit: meters-per-second (m/s) velocity = distance time Acceleration (a) rate something changes velocity Base Unit: meters-per-square-second (m/s 2 ) a = velocity time Force (F) push or a pull on an object Base Unit: Newton (N) F = mass! Luminosity (L) light energy emitted over time Base Unit: Watts (W) acceleration L = Energy time

15 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 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

16 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

17 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.

18 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 km = 5.45 x x 10 8 km Sirius is 545,000 times further from Earth than the Sun

19 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

20 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) Reality (Earth - Sun Distance) Reality (Earth - Sirius Distance) Model 1 meter = 8.17 x km = 5.45 x x 10 8 km (Earth - Sirius Distance) Model = 545,000 meters (545 km)

21 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!

22 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 cm Olympus Mons, Mars Set up proportion: 1 cm = 60 km 9.3 cm x km Measure the longest width Width = 9.3 cm Complete the proportion and solve x = 60 x 9.3 = 558 km