Unit Conversion

Converting Nonmetric Units to Metric
The distance from the university to home is 10 mi and it usually takes 20 min to drive this distance. Calculate the average speed in meters per second (m/s). (Note: Average speed is distance traveled divided by time of travel.)

Strategy
First we calculate the average speed using the given units, then we can get the average speed into the desired units by picking the correct conversion factors and multiplying by them. The correct conversion factors are those that cancel the unwanted units and leave the desired units in their place. In this case, we want to convert miles to meters, so we need to know the fact that there are 1609 m in 1 mi. We also want to convert minutes to seconds, so we use the conversion of 60 s in 1 min.

Solution

1. Calculate average speed. Average speed is distance traveled divided by time of travel. (Take this definition as a given for now. Average speed and other motion concepts are covered in later chapters.) In equation form,
   [latex]\text{Average speed}=\frac{\text{Distance}}{\text{Time}}.[/latex]
2. Substitute the given values for distance and time:
   [latex]\text{Average speed}=\frac{10\phantom{\rule{0.2em}{0ex}}\text{mi}}{20\phantom{\rule{0.2em}{0ex}}\text{min}}=0.50\phantom{\rule{0.2em}{0ex}}\frac{\text{mi}}{\text{min}}.[/latex]
3. Convert miles per minute to meters per second by multiplying by the conversion factor that cancels miles and leave meters, and also by the conversion factor that cancels minutes and leave seconds:
   [latex]0.50\phantom{\rule{0.2em}{0ex}}\frac{\overline{)\text{mile}}}{\overline{)\text{min}}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1609\phantom{\rule{0.2em}{0ex}}\text{m}}{1\phantom{\rule{0.2em}{0ex}}\overline{)\text{mile}}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1\phantom{\rule{0.2em}{0ex}}\overline{)\text{min}}}{60\phantom{\rule{0.2em}{0ex}}\text{s}}=\frac{\left(0.50\right)\left(1609\right)}{60}\phantom{\rule{0.2em}{0ex}}\text{m/s}=13\phantom{\rule{0.2em}{0ex}}\text{m/s}.[/latex]

Significance
Check the answer in the following ways:

1. Be sure the units in the unit conversion cancel correctly. If the unit conversion factor was written upside down, the units do not cancel correctly in the equation. We see the “miles” in the numerator in 0.50 mi/min cancels the “mile” in the denominator in the first conversion factor. Also, the “min” in the denominator in 0.50 mi/min cancels the “min” in the numerator in the second conversion factor.
2. Check that the units of the final answer are the desired units. The problem asked us to solve for average speed in units of meters per second and, after the cancellations, the only units left are a meter (m) in the numerator and a second (s) in the denominator, so we have indeed obtained these units.

Converting between Metric Units
The density of iron is [latex]7.86\phantom{\rule{0.2em}{0ex}}{\text{g/cm}}^{3}[/latex] under standard conditions. Convert this to kg/m³.

Strategy
We need to convert grams to kilograms and cubic centimeters to cubic meters. The conversion factors we need are [latex]1\phantom{\rule{0.2em}{0ex}}\text{kg}={10}^{3}\text{g}[/latex] and [latex]1\phantom{\rule{0.2em}{0ex}}\text{cm}={10}^{-2}\text{m}\text{.}[/latex] However, we are dealing with cubic centimeters [latex]{\text{(cm}}^{3}=\text{cm}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{cm),}[/latex] so we have to use the second conversion factor three times (that is, we need to cube it). The idea is still to multiply by the conversion factors in such a way that they cancel the units we want to get rid of and introduce the units we want to keep.

Solution

Significance
Remember, it’s always important to check the answer.

1. Be sure to cancel the units in the unit conversion correctly. We see that the gram (“g”) in the numerator in 7.86 g/cm³ cancels the “g” in the denominator in the first conversion factor. Also, the three factors of “cm” in the denominator in 7.86 g/cm³ cancel with the three factors of “cm” in the numerator that we get by cubing the second conversion factor.
2. Check that the units of the final answer are the desired units. The problem asked for us to convert to kilograms per cubic meter. After the cancellations just described, we see the only units we have left are “kg” in the numerator and three factors of “m” in the denominator (that is, one factor of “m” cubed, or “m³”). Therefore, the units on the final answer are correct.