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Section 1.1 Large and Small Quantities


We will investigate quantities that in our everyday experience are so large or so small that they are hard to envision. Then we will see how the ideas of large and small are intimately related. 

You will need to change units of measure in many of this chapter’s exercises. This is because units of measure are often presented in an appropriate magnitude to state measures in small numbers of them. However, scientists often have to relate two quantities that are expressed in different units, so they must change from one unit to another. Dealing with large and small quantities also often requires that we use scientific notation.

Astronomical distances are measured in light-years, or the distance that light travels in one year. But a light year is expressed in terms of meters. To calculate the distance light can travel in one year in meters, we need to know the speed at which light travels, in meters per second, and the number of seconds in a year.

The speed of light in a vacuum is exactly $299\,792\,458 \mathrm{\frac{meters}{sec}}$. We can say that this number is exact because in 1983 the meter was defined as $\dfrac{1}{299\,792\,458}$ the distance that light travels in one second.

The second as a unit of time is defined as the duration of $9\,192\,631\,770$ cycles between two energy levels of the cesium 133 atom. The number of seconds in a year as we normally think of a year is less exact, because one year (the time for earth to revolve once around the sun) differs slightly from one revolution to the next. For our purposes, we will take one year to be 365 days, one day to be 24 hours, one hour to be 60 minutes, and one minute to be 60 seconds. Therefore, the distance that light travels in one year is
$$\mathrm{60 \frac{sec}{min} \cdot 60 \frac{min}{hr} \cdot 24 \frac{hr}{day} \cdot 365 \frac{days}{yr} \cdot 299,792,458 \frac{meters}{sec}}$$.

Type this product into GC, using "*" for multiplication. You will see something resembling Figure 1.1.1. The part "e+15" means "$\times 10^{15}$". GC displayed its result to 13 significant digits. Your number of significant digits is determined by the menu selection in File/Document Properties.


Figure 1.1.1. GC's display of calculating $60\cdot 60\cdot 24\cdot 365\cdot 299792458$

The exact value of this product is 9 454 254 955 488 000 meters, or about 9.5 million billion meters. This is a lot of meters.

How many trips to the moon and back is one light-year? The average distance from earth to moon is 384 400 km, or $384\,400\,000$ meters. Use GC to calculate the number of round trips to the moon that is equivalent to $9\,454\,254\,955\,488\,000$ meters (one light-year).


Figure 1.1.2. Calculating the number of round trips to the moon that is equivalent to one light-year.

One light-year is more than 12 million round trips to the moon. Put another way, a pulse of light would make 12 round trips to the moon in less than one-millionth second. But one light-year is also a small distance. One light-year is about $\dfrac{4}{17}$ the distance to Proxima Centauri, the nearest star to Earth outside our solar system. It is approximately $\dfrac{1}{112}$ the width of the Milky Way. One light-year is approximately $\dfrac{1}{45\,000\,000\,000}$ (one forty-five-billionth) the distance from earth to the edge of the visible universe. The point is that large quantities are large compared to smaller quantities, but they can be very small compared to much larger quantities.

Useful Units

The International System of units (SI) is standard around the world, including the United States. The U.S. also uses a common system of units that is based on the old English system. Here are some facts about units that you can use in the following exercises.

Common Abbreviations

Unit

Abbreviation

 Unit

Abbreviation

inch

in

light year

l.y.

foot

ft

second

sec

yard

yd

minute

min

centimeter

cm

year

yr

meter

m

century

cen

kilometer

km

 

 

 

Common Conversions

Unit

Conversion

Unit

Conversion

1 in

2.54 cm

12 in

1 ft

3 ft

1 yd

100 cm

1 m

1 000 m

1 km

1 l.y.

9.454 x 1012 km

1 min

60 sec

1 hr

60 min

1 day

24 hr

1 yr

365 day

1 cen

100 yr

1 eon

1 000 yr

Large Units

Speed of light = $299\,792\,458$ meters per second

Average distance from Earth to Sun = $149\,597\,870.7$ km

Distance from earth to edge of visible universe = $4.6 \cdot 10^{10}$ light-years

Exercise Set 1.1

Use GC for your computations. Enter “*” for multiplication, “/” for division, shift-6 (“^”) for an exponent, “→” to exit an exponent or to move the next term.

  1. About how many minutes will it take a pulse of light to travel from the sun to the earth?

  2. The figure below is a video that starts with a graph of $y=x^{1000}$ in a standard coordinate system. It then shows the vertical axis being stretched so that the top of the graph window shows a y-value of 6e-64 (which means $6 \cdot 10^{-64}$, or 6 divided by $10^{64}$).

    Use a ruler to measure the distance from 0 to $6 \cdot 10^{-64}$ on the vertical axis on your screen. Approximately how far away, in light years, is 1 from the horizontal axis after the stretching stopped? Interpret your answer in units of (1) trips to the moon; (2) trips to the Sun, (3) trips across the Milky Way;  and (4) trips from earth to the edge of the visible universe.

  1. We tend to think of light surrounding us, like air. But light travels, always. Bill is standing 2 meters from his mirror. Approximately how many seconds will it take a pulse of light to bounce off his forehead, hit the mirror, and return back to his eye.

  2. Silver Ants can run at 50 $\mathrm{\frac{cm}{sec}}$ . If a Super Silver Ant was capable of running at a constant speed from San Diego to New York City (2 760 mi), how many days would it take?

  3. The Sahara Desert has an area of approximately 9 400 400 $\mathrm {km}^2$. While estimates of its average depth vary, they center around 150 m. One $\mathrm{cm}^3$ holds approximately 8 000 grains of sand.

    1. Approximately how many grains of sand are in the Sahara Desert?
      - Express your answer in millions of grains of sand.
      - Express your answer in billions of grains of sand.

    2. What fraction of the Sahara is made by 1 grain of sand?

    3. A small dump truck can carry approximately 20.5 $\mathrm{m}^3$ of sand. Suppose a long line of dump trucks were to dump a load of sand every 30 seconds. How many years would it take to re-create the Sahara Desert?

  4. Recall that a second is defined as the duration of $9\,192\,631\,770$ cycles between two energy levels of the cesium 133 atom.

    1. How many cycles between two energy levels of the cesium 133 atom make a duration of one year?
    2. What is the speed of light, measured in meters per cycle of a cesium 133 atom?

  5. (From ABC News, July 8, 2010). Maybe the odds of winning the lottery would be a lot better if Joan Ginther would stop buying all the good tickets.

    You may have heard of the Las Vegas resident, whom you probably want sitting next to you when an asteroid, shot by aliens, is aimed at your planet. Joan recently cashed in a winning $10 million scratch-off ticket, making the lucky woman a four-time lottery winner.


    • 1993: $5.4 million (paid in yearly installments). Probability: 1/15,800,000.
    • 2006: $2 million (lump-sum payoff). Probability: 1/1,028,338.
    • 2008: $3 million (lump-sum payoff). Probability: 1/909,000.
    • 2010: $10 million (lump-sum payoff). Probability: 1/1,200,000 (end quote from ABC).

    The probability of two or more independent events all happening is the product of their individual probabilities. Compare the probability of Joan winning all four lotteries with the probability of picking, at random, a grain of sand pained red that is hidden somewhere within the Sahara desert.