1.3 Unit conversion

It is often necessary to convert from one unit to another. For example, if you are reading a European cookbook, some quantities may be expressed in units of liters and you need to convert them to cups. Or perhaps you are reading walking directions from one location to another and you are interested in how many miles you will be walking. In this case, you may need to convert units of feet or meters to miles.

Let’s consider a simple example of how to convert units. Suppose we want to convert 80 m to kilometers. The first thing to do is to list the units you have and the units to which you want to convert. In this case, we have units in meters and we want to convert to kilometers . Next, we need to determine a conversion factor relating meters to kilometers. A conversion factor    is a ratio that expresses how many of one unit are equal to another unit. For example, there are 12 in. in 1 ft, 1609 m in 1 mi, 100 cm in 1 m, 60 s in 1 min, and so on. Refer to Appendix B for a more complete list of conversion factors. In this case, we know that there are 1000 m in 1 km. Now we can set up our unit conversion. We write the units we have and then multiply them by the conversion factor so the units cancel out, as shown:

Note that the unwanted meter unit cancels, leaving only the desired kilometer unit. You can use this method to convert between any type of unit. Now, the conversion of 80 m to kilometers is simply the use of a metric prefix, as we saw in the preceding section, so we can get the same answer just as easily by noting that

since “kilo-” means 10 ³ (see [link] ) and $1=\mathrm{-2}+3.$ However, using conversion factors is handy when converting between units that are not metric or when converting between derived units, as the following examples illustrate.

Converting nonmetric units to metric

Strategy

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,
   
   $\text{Average speed}=\frac{\text{Distance}}{\text{Time}}.$
2. Substitute the given values for distance and time:
   
   $\text{Average speed}=\frac{10\text{mi}}{20\text{min}}=0.50\frac{\text{mi}}{\text{min}}.$
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:
   
   $0.50\frac{\cancel{\text{mile}}}{\bcancel{\text{min}}}\times \frac{1609\text{m}}{1\cancel{\text{mile}}}\times \frac{1\bcancel{\text{min}}}{60\text{s}}=\frac{(0.50)(1609)}{60}\text{m/s}=13\text{m/s}.$