Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CAPSTONE Laboratory #1: Review of Mathematics “Mathematics is just organized reasoning.”- Richard Feynman “There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.” – Mark Twain 1. INTRODUCTION You cannot do physics and astronomy without mathematics, but the mathematics needed to make progress is usually quite modest. The mathematics you need for this program will be reviewed in the first half of this lab. The second part of the lab introduces you to the elements of mathematical reasoning. Enrico Fermi (who built the first nuclear reactor in the world at this university) used to set problems to his students such as “How many piano tuners are there in Chicago?” These problems require some knowledge (for example, the number of houses in Chicago); some estimation (for example, the length of time it takes to tune a piano) and some guess work (for example, the fraction of houses that have pianos). By writing down all the assumptions explicitly, you can see what parts of the problem involve knowledge, what parts estimation, what parts guessing and how much extra work and research you need to do to get a reasonable answer. Quite often you can make surprisingly good estimations of the solutions to very difficult problems by simplifying the problem or thinking about how knowledge that you do have sheds light on the problem. 2. NUMBERS AND GRAPHS (2.1) Evaluate the following using a calculator. (a) 3.72 x 102 + 1.6 x 10-2 + 4.4 (b) 3.1415 8.36 10 6 4.365 10 5 97485 0.0245 2.47 1010 9.4 10 2 (2.2) Using logarithms In dealing with the extremely large and extremely small numbers met in astronomy, it is often useful to use exponents or powers of ten. For example the number 10000000 is represented by the notation 107 and 0.001 by the notation 10-3. The number 1 is expressed as 100. Numbers are often expressed in “scientific notation” where the actual number, say 23456.3, is written as a number between 1 and 10 times an appropriate power of ten; i.e. this number is written as 2.34563 x 104. This notation is related to the logarithm (or log) of a number. The logarithm of a number, x, is another number, y, 1 defined so that y=10x. Since you all have calculators you have a special button that converts numbers into logarithms. The logarithm of the number 3 is about 0.5; the logarithm of the number 3000 is about 3.5. Because the sizes and physical conditions of objects in the universe are so different, we often plot the logarithms of numbers in a graph to allow us to represent different types of object in a single diagram. For instance, because the range of brightness of stars in the sky is huge they are usually expressed in a logarithmic scale with the bright star Vega being given a value of 0. A further convenience arises because multiplication of two numbers corresponds to addition of their logarithms, division of two numbers by subtraction. The rules of logarithms are: log(a x b) log(a/b) log ab = = = log a +log b log a – log b b x log a = = = 10a + b 10a -b 10a x b The rules of exponents are 10a x 10b 10a/10b (10a)b Thus 1047 x 1036 = 1083 and the logarithm of the product is 47 + 36 =83. Using your calculator write down the logarithms of the following numbers: (a) log(256453.2) = ? (b) log(0.00000123) = ? (c) What is the number whose logarithm is 4.56253? That is, when log(x) = 4.56253, what number does x represent? 2 (2.3) Plotting graphs Plotting data in a graph allows us to see if there are trends, implying perhaps some deeper physical connection, and gives us a qualitative feel for the scatter, which measures the “goodness” of the data. Being able to draw graphs to represent data is an essential skill in science. Sometimes you plot the data on a linear scale, suitable for cases in which there is a linear relationship between the two parameters (distance=velocity * time), and sometimes on a logarithmic scale, when the relationship may be more of a power law (y = x2 implies that log(y) = 2 * log(x) ). (a) Plot a graph of the following data that gives the distance of an aircraft from a fixed point as a function of time. Time 7:00 8:00 9:00 10:00 11:00 Distance (miles 750 1300 1850 2400 2950 Draw a straight line through the data and determine: (i) How fast is the aircraft traveling? (ii) What time did the aircraft take off? (b) The following data gives the luminosity of a star (how bright the star is, sun=1) against its temperature. Plot a graph of this data on a linear scale and then on a logarithmic scale. Temperature (0K) 2,500 3,200 3,850 4,450 5,150 5,700 6,400 8,700 9,600 15,200 29,000 Luminosity 0.0008 0.013 0.075 0.18 0.36 1.0 4.0 24 63 750 16,000 Log (Temperature) Log(Luminosity) 3 (i) What is the temperature? approximate relationship between luminosity and (2.4) Small angle approximation You were probably taught at school to measure angles in degrees; thus there are 360 degrees in a circle and the angles within a triangle add up to 180 degrees. You were also taught that the distance BC in a right angle triangle is given by BC = Sin( * AB = Tan( ) * AC B A C In astronomy we often deal with small angles within a triangle and it is easier to work in radians, rather than degrees. There are 2 radians in a circle; if we have a coin 1 inch in diameter at a distance of 100 inches we find that the angle between the edges of the coin is 1/100 radian. For small angle then, Sin ≈ Tan ≈ provided is measured in radians and this approximation considerably simplifies the mathematics. (a) Astronomers frequently use the arcsecond as a unit of angular measurement. There are 60 arcseconds in an arcminute and 60 arcminutes in a degree. How many arcseconds are there in a radian? (b) A unit of distance we shall use in this program is the parsec. This is the distance at which, if we looked back to earth, the orbit of the earth around the sun would subtend one arcsecond [this strange definition comes about because the distances to stars is measured using parallax, which measures how much a star appears to move in the sky because of Earth’s motion around the Sun]. 1 parsec 1 A.U. 1 arcsecond Sun 4 The radius of the orbit of the earth is called an Astronomical Unit (A.U.) and has been measured by radar as 149,597,870.691 kilometers (km). (i) How many kilometers are there in a light-year? (ii) How many light-years are there in a parsec? 3. FERMI QUESTIONS Write down your assumptions and simplifications clearly as well as your calculations. (a) How many times is your favorite song listened to in one day by the residents of Chicago? (b) How many drops of water are there in Lake Michigan? (c) How many times you expect your heart to beat in your lifetime? (d) McDonalds claims to have sold over 100 billion hamburgers. How many cows have been used to make this number of hamburgers? (e) Come up with your own version of a Fermi question, switch with your partner or neighbor, and try to figure out each others question. 5