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Section 35.1 Say Goodbye to the Mars Climate Orbiter

The $300,000,000 NASA project called the Mars Climate Orbiter launched on December 11, 1998. Approximately 9 months later, it crashed into the surface of Mars and was declared a total loss. The culprit was a piece of hardware that gave distance in meters to a piece of software that was working in feet.

This incident stands out among the many unit conversion errors (or simply the failure to convert units at all) because of a combination of the magnitude of the project and the high capacity minds that were working on it. Most errors of this type are not nearly as catastrophic, but that does not negate the importance of understanding how to perform unit conversions.

Units are names of measurements of quantities. Here are some examples:

  • Time is measured in seconds, minutes, hours, days, and years.

  • Lengths are measured in inches, feet, miles, centimeters, meters, and kilometers.

  • Volumes are measured in tablespoons, cups, quarts, gallons, liters, and cubic meters.

  • Quantities are measured in dozens, hundreds, thousands, millions, and moles.

Activity 35.1.1. Conversion Factors.

The most important feature about these units is that they represent specific quantities that can be related to each other. For example, there are 12 inches in a foot. And there are 60 seconds in a minute. These create mathematical relationships by simply translating the words into a formula:

\begin{equation*} \begin{array}{ccccc} \text{12 inches in a foot} \amp \longrightarrow \amp 1 \text{ foot} = 12 \text{ inches} \\ \text{60 seconds in a minute} \amp \longrightarrow \amp 1 \text{ minute} = 60 \text{ seconds} \end{array} \end{equation*}

These equations can be turned into fractions that we call conversion factors. Conversion factors have the property that they are equal to the number 1. Notice that every equality of this type creates two conversion factors.

\begin{equation*} \begin{array}{ccccc} 1 \text{ foot} = 12 \text{ inches} \amp \longrightarrow \amp \frac{1 \text{ foot}}{12 \text{ inches}} \amp \text{and} \amp \frac{12 \text{ inches}}{1 \text{ foot}} \\ 1 \text{ minute} = 60 \text{ seconds} \amp \longrightarrow \amp \frac{1 \text{ minute}}{60 \text{ seconds}} \amp \text{and} \amp \frac{60 \text{ seconds}}{1 \text{ minute}} \end{array} \end{equation*}

Try it!

Write an equation that relates feet to yards, then use it to determine two conversion factors.

Solution.
\begin{equation*} \begin{array}{rl} 3 \text{ feet} = 1 \text{ yard} \amp \hspace{1cm} \longrightarrow \hspace{1cm} \frac{3 \text{ feet}}{1 \text{ yard}} = 1 \hspace{0.5cm} \text{ and } \hspace{0.5cm} 1 = \frac{1 \text{ yard}}{3 \text{ feet}} \end{array} \end{equation*}

Activity 35.1.2. Using Conversion Factors.

The importance of conversion factors being equal to the number is that multiplying by does not change the value of a number. With an appropriate choice of conversion factors, it is possible to make original unit cancel out, leaving you with the value in the new unit. Here is an example:

\begin{equation*} \begin{aligned} 8 \text{ minutes} \amp = 8 \text{ minutes} \cdot 1 \amp \eqnspacer \amp \text{Multiplying by $1$} \\ \amp = 8 \text{ minutes} \cdot \frac{60 \text{ seconds}}{1 \text{ minute}} \amp \amp \text{Conversion factor} \\ \amp = 8 \cancel{\text{ minutes}} \cdot \frac{60 \text{ seconds}}{1 \cancel{\text{ minute}}} \amp \amp \text{Cancel the units} \\ \amp = 480 \text{ seconds} \end{aligned} \end{equation*}

This example was drawn out with details for emphasis. Your presentation can be shortened.

\begin{equation*} \begin{aligned} 8 \text{ minutes} \amp = 8 \cancel{\text{ minutes}} \cdot \frac{60 \text{ seconds}}{1 \cancel{\text{ minute}}} \amp \eqnspacer \amp \text{Cancel the units} \\ \amp = 480 \text{ seconds} \end{aligned} \end{equation*}

Try it!

Convert 36 feet to yards using a conversion factor.

Solution.
\begin{equation*} \begin{aligned} 36 \text{ feet} \amp = 36 \text{ }\cancel{feet} \cdot \frac{1 \text{ yard}}{3 \text{ }\cancel{feet}} \amp \eqnspacer \amp \text{Cancel the units} \\ \amp = 12 \text{ yards} \end{aligned} \end{equation*}

Activity 35.1.3. Scientific Prefixes.

You may be aware that there are certain prefixes that apply the the unit. These prefixes are another way that scientists avoid having to write really long numbers. Here are some of the common prefixes and their meaning:

\begin{equation*} \begin{array} {|c|c|c|c|} \hline \text{Prefix} \amp \text{Symbol} \amp \text{Value as Power of 10} \amp \text{Value in Standard Form} \\ \hline \text{nano} \amp n \amp 10^{-9} \amp 0.000000001 \\ \hline \text{micro} \amp \mu \amp 10^{-6} \amp 0.000001 \\ \hline \text{milli} \amp m \amp 10^{-3} \amp 0.001 \\ \hline \text{centi} \amp c \amp 10^{-2} \amp 0.01 \\ \hline \text{deci} \amp d \amp 10^{-1} \amp 0.1 \\ \hline \text{deka} \amp da \amp 10^{1} \amp 10 \\ \hline \text{hecto} \amp h \amp 10^{2} \amp 100 \\ \hline \text{kilo} \amp k \amp 10^{3} \amp 1000 \\ \hline \text{mega} \amp M \amp 10^{6} \amp 1000000 \\ \hline \text{giga} \amp G \amp 10^{9} \amp 1000000000 \\ \hline \text{tera} \amp T \amp 10^{12} \amp 1000000000000 \\ \hline \end{array} \end{equation*}

These prefixes are placed in front of a unit to change its value. For example, a kilometer is 1000 meters and a millimeter is 0.001 meters. This gives us mathematical relationships that we can use to write conversion factors. While it is not wrong to use decimals in the conversion factors with these prefixes, it's generally considered better to use integers. So instead of using \(\frac{0.001 \text{ meters}}{1 \text{ millimeter}}\text{,}\) we would usually use \(\frac{1 \text{ meter}}{1000 \text{ millimeters}}\text{.}\)

Try it!

Write an equation that relates kilograms to grams, then use it to determine two conversion factors.

Solution.
\begin{equation*} \begin{array}{rl} 1000 \text{ grams} = 1 \text{ kilogram} \amp \hspace{0.5cm} \longrightarrow \hspace{0.5cm} \frac{1000 \text{ grams}}{1 \text{ kilogram}} = 1 \hspace{0.5cm} \text{ and } \hspace{0.5cm} 1 = \frac{1 \text{ kilogram}}{1000 \text{ grams}} \end{array} \end{equation*}