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Section 1.3 Large Changes are Made of Tiny Changes


The speed of light is $2.998 \cdot 10^8 \mathrm{\frac{m}{sec}}$. This means that it travels $2.998 \cdot 10^8$ m in one second (one light-second). 

However, light travels $2.998 \cdot 10^8$ m by traveling one meter at a time. So, it takes $\frac{1}{2.998} \cdot 10^{-8} \mathrm{sec}$, which is $3.336 \cdot 10^{-9}$ (3.3 billionths) sec to travel one meter. 

Light also travels one light-second by traveling one millimeter at a time. Light therefore would travel one millimeter, which is $\frac{1}{1000}$ meters ($1 \cdot 10^{-3}$ m), in $\frac{1}{1000}$ the time that light takes to travel one meter. 

In the same vein, light would travel one micrometer $1 \cdot 10^{-6}$ m) in $\frac{1}{1000}$ the time it takes to travel one millimeter. It would therefore take approximately$3.336 \cdot 10^{-15}$ seconds for light to travel one micrometer. (Check this.)

But light nevertheless travels one light-second in one second, or one light-year in one year, or a billion light-years in a billion years—all the time traveling one micrometer at a time! It just travels a very large number of micrometers in a very large number of really small bits of time.

Exercise Set 1.3

  1. The Grand Canyon is enormous. It is 433 km long and has an average depth of 1.6 km. The US Forest Service estimates its volume at 4.17 trillion $(4.17 \cdot 10^{12}) \mathrm{m^3}$. Suppose the dump trucks that re-created the Sahara Desert (Exercise 1.1.5) are re-purposed, lining up to fill the Grand Canyon. Suppose also that they can dump one load every 30 seconds. How many years would it take to fill the Grand Canyon.
  2. The dimensions of a dollar bill are 15.5956 cm by 6.6294 cm by 0.010922 cm. The U.S. federal government spent $3.8 trillion in fiscal year 2011.

    1. How high, in km, would be a stack of dollar bills that contains the same amount of money as spent by the U.S. federal government in 2011?

    2. An Olympic swimming pool is 50 m long, 25 m wide, and 1.5 m deep. How many Olympic pools would be filled by the dollar bills spent by the U.S. federal government in 2011?

    3. A $100 bill has the same dimensions as a $1 bill. An episode of Breaking Bad shows three wooden platforms that each held a stack of $100 bills measuring approximately $1 \mathrm{m^3}$. What, approximately, would be the value of those three stacks if they contained real $100 bills?