Homework #1: Solutions

# Homework #1: Solutions

1. ### Numbers Big and Small (12 points)

Let's get a little practice using scientific notation. All the necessary information is contained in the handouts. This is simply to familiarize yourselves with this type of notation - you will never do complicated calculations.
1. Write the number 232,898.232 in scientific notation with only 3 significant digits.

To write this number in scientific notation we need to figure out how many spaces to move the decimal point to the left until we have only one digit to the left of the decimal point. Each space we move the decimal is one power of ten (for example 20 = 2x10 = 2x101). So to obtain the number 2.32898232 we needed to move the decimal 5 times, or 5 powers of 10. This number to 3 significant digits is 2.33. We round the third digit up or down accordingly. So finally, the answer is:

2.33 x 105

2. Write the number 7.321 x 10-5 as a normal string of digits.

In the first problem you can see that multiplying by a positive power of ten indicates how many decimal places to the right one must move the decimal point to obtain a normal string of digits (we went to the left to get the scientific notation, it's opposite the direction to go to obtain the normal string). So multiplying by a negative power of ten moves the decimal to the left by that power. And so we have

0.00007321

3. What is (6 x 10274)/(3 x 10268)? Do not use a calculator. Show your work. What's the name of this number (i.e., how many million, billion, trillion, etc. is this)?

First off 6/3 = 2. That's easy enough. So now we are left with 10274/10268. Use the rules of multiplication and division of exponents. When dividing you subtract the exponents. So 274 - 268 = 6. Thus the answer is

2 x 106
= 2 Million

2. ### Galactic Messages (20 points)

To help you become familiar with the idea that a light-year is a unit of distance, and what it tells you...

Let's assume you live on the planet Pong where the only means of communication is by bicycle messenger. All bicycle messengers travel 60 km/hour (this would be pretty fast on earth, but hours on Pong aren't the same). There are only 10 hours in a Pongian day, but 100 days in a Pongian year. Now suppose your parents live three bicycle-days away (1 bicycle-day = the distance a messenger can travel in a day, analogous to 'light-year'). You shouldn't need a calculator to do this question! Use scientific notation to help you.
1. How far away (in km) do they live? (Hint: How far is a bicycle-day?)

If they live 3 bicycle-days away, the distance to your parents is the distance that a bicycle messenger can travel in 3 Pongian days. In one Pongian day, or 10 hours, a bicycle travelling 60 km/hr can travel 60 km/hr x 10 hr = 600 km. So in 3 Pongian days, it can travel 3x600=1800 km. So 3 bicycle-days is a distance of 1800 km. (One bicycle-day is a distance of 600 km).

2. If they win the lottery, how long before you can hear about it? (Hint: No calculation should be required!)

Since the only means of communication is by bicycle, the time until you get the news is (assuming a bicycle was dispatched immediately and you were at home to meet it when it arrived) the time it takes a bicycle to travel the distance to your home. They live 3 bicycle-days away, so it takes 3 days for a bicycle to reach you. (The long way to do this question is to take distance d=1800 km and time t=d/speed so t = 1800 km / 60 km/hr = 30 hours = 3 days of 10 hours each).

Now assume that your civilization has somehow colonized other planets in your solar system, but the speed of communication is still the same. You get the news that a child is born on the planet Quong which is 6x1011 meters away (~4 AU, or the approximate distance between Earth and Jupiter at closest approach).
1. How far is a bicycle-year (in meters or km)? (Hint: This is just a conversion of units)

A bicycle-year is the distance a bicycle can travel in 1 Pongian year. There are 100 days in a Pongian year, and 10 hours per Pongian day, so there are 1000 hours in a Pongian year. A bicycle can travel 60 km/hr, so in a year it can go 60 km/hr x 1000 hr = 60 000 km = 6x104 km.

2. How far away is Quong in bicycle-years? (Use scientific notation to do the calculation)

The distance to Quong is 6x1011 m, but there are 1000m = 103m in a km, so the distance to Quong is 6x1011-3 = 6x108 km. There are 6x104 km in a bicycle-year (since there are 1000 hr/year, and a bicycle can travel 60 km/h), so the distance to Quong in bicycle-years is:

6x108km / 6x104km/bicycle-year = 108-4= 104.

Quong is 10 000 bicycle-years away!

3. How old is the child when you get the news? (Hint: No calculation should be required!)

Since Quong is 10 000 bicycle-years away, even if the news was sent immediately, you will not get it for 10 000 Pongian years (the speed of communication is still that of a bicycle). The child will by then be 10 000 years old. Or dead, depending on the life span of Pongians, which none of us knows!

Note that in our solar system, Jupiter is over 30 light-minutes from Earth, so even with a MUCH faster speed of communication, there is a significant lag! Since the nearest stars are several light-years away, interstellar communication and travel are impractical without faster-than-light technology (which violates physics as we understand it).

3. ### The Universe We Live In (8 points)

1. What is the difference between our solar system, our galaxy, and the universe, and how do they relate to one another?

Our solar system is made up of the Sun (the nearest star), and bodies (like the planets) which are gravitationally bound to it. It is much smaller than the distance to the nearest star. Our galaxy is filled with many stars (~100 billion) and their planets, as well as gas and dust. The universe is the cosmos, which is everything there is. It includes billions of galaxies.

2. Do people in other cultures on earth see the Big Dipper (even if they call it something else)? What about people on planets circling other stars?

People in other cultures on earth will still see the Big Dipper, as long as they live far enough north for it to be above the horizon, because the pattern of stars is the same from all over Earth. People on planets circling a very nearby star might see the same pattern of stars if their star were very close to us compared to the stars of the Big Dipper. But the stars making up the Big Dipper will not appear in the same pattern from the viewpoint of very distant stars. For example, if you were on a planet circling one of the stars in the Big Dipper, you clearly wouldn't see this asterism as we do!

For specificity, the stars making up the Big Dipper are between 68 and 210 light years away (see Ask A Scientist Astronomy Archive for the distances of each star), and the very nearest star is over 4 light years away, so civilizations on planets circling most other stars will not see the same pattern from the Big Dipper stars. They may well see the same stars, but they won't look like a Dipper.

4. ### Sky Motion (20 points)

In the following question, remember that the stars appear to move in circles around the celestial poles, which are the projections into space of the Earth's north and south pole!
1. Where would you see the NCP if you were standing at the Earth's equator?

The north celestial pole is the projection into space of the Earth's north pole. If I'm on the equator, then the north celestial pole is a point in space beyond my northern horizon. If you don't get this, stick a skewer in an orange. The direction that the skewer points is the north celestial pole. Now imagine yourself on the equator, and figure out which direction the skewer is pointing.

2. Where would you see the SCP if you were standing at the Earth's South Pole?

If you're standing at the South Pole, then the South Celestial Pole lies directly overhead.

3. Now imagine you're at the South Pole in winter. You're cold in your parka, but do you see the stars rise and set each day?

Imagine you're on the earth surrounded by a giant clear sphere. As the earth rotates about the axis going through the north and south poles, the celestial sphere appears to rotate in the opposite direction about the same axis. This means the stars move in circles around the celestial poles (see the diagram here).

Since the South Celestial Pole is directly overhead, and the stars move in circles about this point, the stars are moving parallel to the horizon. They never rise or set.

4. Now imagine you're at the equator looking South. You've taken off your parka. Do the stars rise to the right or the left of the southern point on the horizon (ie. where is East)? Do they rise straight up, or at an angle with the horizon?

If you're at the equator looking South, then East is on your left. The stars rise East (left) of the southern point on the horizon (and set west). Since the stars move in circles about the SCP, which lies on your southern horizon, you can see HALF-circles (the other half is below your horizon). Therefore the stars rise straight up from the horizon.

5. Assume it is nighttime and clear, and that you are standing on the equator looking west. Sketch the star trails that you would see. Include a horizon, compass direction, and the North or South Celestial Pole if they are relevant. Samples for some other situations are displayed in the star trail figures on p. 16 of your text.

You would see stars setting straight down into the horizon.

5. ### Nature of Science - Doing Science (25 points)

For this question, you're going to be a scientist!
Newton's Theory: Newton's theory of gravity predicts that all objects experience the same acceleration due to gravity. This means that if you drop any two objects from the same height, the theory predicts that they will reach the floor at the same time, regardless of their mass/weight.
In a few minutes, you will take your textbook for this class in one hand, a flat piece of paper in the other, and drop them from an equal height.
1. Prediction: Based on your experience, what do you think will happen (ie which, if any, will hit the floor first)?

Based on experience, I expect that the book will hit the floor first, and the paper will drift to the floor more slowly.

2. Prediction: What does Newton's theory of gravity predict will happen? Does your prediction match the prediction of Newton's theory?

Newton's theory of gravity predicts that they will hit the floor at the same time (see the description above).

3. Experiment: Drop the book (take out the CD first!) and the paper! Describe the results of the experiment.

The book hit the floor with a thud. The paper drifted to the floor more slowly, and hit later.

4. Outcome: Do the results of the test match your predictions of parts (a) and (b)? ie Do the results support (not prove/disprove) your prediction and/or Newton's theory?

The results of the test match the prediction I made based on my everyday experience. They did not match the predictions of Newton's theory of gravity.

5. Can you think of any reason that this experiment does not completely test the predictions of Newton's theory? How might you modify the experiment to more completely test the predictions of the theory?

The experiment doesn't take into account the effects of the air. The paper, being lighter, is more affected by the air resistance than the book. To modify the experiment, place the paper on top of the book, and then drop them. The paper never leaves the book.

6. ### Nature of Science - Designing an Experiment (15 points)

You're all familiar with the horoscopes in the daily paper! Basically, astrology says that human events are influenced by the apparent positions of the Sun, Moon, and planets among the stars in our sky, particularly at the time of one's birth. After all, the Sun's position in the sky determines the seasons, and so the times of planting, harvesting, warmth, cold, daylight, and darkness. Why shouldn't the other heavenly bodies that move among the stars affect our lives as well?

Design a scientific test of astrology. In other words, design an experiment whose results will either support the assertions of astrology, or conflict with them.

1. Clearly define the methods you would use in your test, outlining all the steps involved (1 paragraph). You will not be required to carry out the experiment, but it should be one that is feasible.

I will not go into detail here, but Lab Assignment #1 contains one possible experiment, as well as a brief description of a second experiment. There are many others! One other possibility is to have your subjects write down a description of their day, then yourself compare their description with their horoscope. Another which has been done is to ask astrologers to match personality profiles with astrological signs. We'll discuss this more after you've done the lab assignment.

2. Include a description of how you would evaluate the results (1 paragraph).

Here I'm hoping that you'd consider possible biases and other sources of error, and that you'd compare your results with those you'd expect based on random chance.

3. You do not need to carry out the experiment you have designed. If you did carry out the test, do you think it would confirm the tenets of astrology, or conflict with them?

This is a personal opinion! See if it changes after you've done Lab Assignment #1.