## Unit Conversions Example: A Short Drive Home

Contents

Suppose that you drive the 10.0 km from your university to home in 20.0 min. Calculate your average speed

- in kilometers per hour (km/h) and
- 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 factor and multiplying by it. The correct conversion factor is the one that cancels the unwanted unit and leaves the desired unit in its place.

### Solution for a

(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 will be covered in later lessons.) In equation form,

\(\mathrm{average \; speed} = \cfrac{\mathrm{distance}}{\mathrm{time}}.\)

(2) Substitute the given values for distance and time.

\(\mathrm{average \; speed} = \cfrac{10.0\;\mathrm{km}}{20.0\;\mathrm{min}}.\)\( = 0.500 \cfrac{\mathrm{km}}{\mathrm{min}}.\)

(3) Convert km/min to km/h: multiply by the conversion factor that will cancel minutes and leave hours. That conversion factor is \(60\;\mathrm{min/hr}.\) Thus,

\(\mathrm{average \; speed} = 0.500\;\cfrac{\mathrm{km}}{\mathrm{min}} × \cfrac{60\;\mathrm{min}}{1 \mathrm{h}}\)\( = 30.0\;\cfrac{\mathrm{km}}{\mathrm{h}}.\)

**Discussion for a**

To check your answer, consider the following:

(1) Be sure that you have properly cancelled the units in the unit conversion. If you have written the unit conversion factor upside down, the units will not cancel properly in the equation. If you accidentally get the ratio upside down, then the units will not cancel. Rather, they will give you the wrong units as follows:

\(\cfrac{\mathrm{km}}{\mathrm{min}} × \cfrac{1\;\mathrm{hr}}{60\;\mathrm{min}} \)\(= \cfrac{1}{60}\cfrac{\mathrm{km\;.hr}}{\mathrm{min}^2},\)

which are obviously not the desired units of km/h.

(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 km/h and we have indeed obtained these units.

(3) Check the significant figures. Because each of the values given in the problem has three significant figures, the answer should also have three significant figures. The answer 30.0 km/hr does indeed have three significant figures, so this is appropriate. Note that the significant figures in the conversion factor are not relevant because an hour is *defined* to be 60 minutes, so the precision of the conversion factor is perfect.

(4) Next, check whether the answer is reasonable. Let us consider some information from the problem—if you travel 10 km in a third of an hour (20 min), you would travel three times that far in an hour. The answer does seem reasonable.

**Solution for b**

There are several ways to convert the average speed into meters per second.

(1) Start with the answer to (a) and convert km/h to m/s. Two conversion factors are needed—one to convert hours to seconds, and another to convert kilometers to meters.

(2) Multiplying by these yields

\(\mathrm{Average\;speed} = \)\(30.0\cfrac{\mathrm{km}}{\mathrm{h}} × \cfrac{1\;\mathrm{h}}{3,600\;\mathrm{s}} × \cfrac{1,000\;\mathrm{m}}{1\;\mathrm{km}},\)

\(\mathrm{Average\;speed} = 8.33\;\cfrac{\mathrm{m}}{\mathrm{s}}\)

**Discussion for b**

If we had started with 0.500 km/min, we would have needed different conversion factors, but the answer would have been the same: 8.33 m/s.

You may have noted that the answers in the worked example just covered were given to three digits. Why? When do you need to be concerned about the number of digits in something you calculate? Why not write down all the digits your calculator produces? The next lessons on Accuracy, Precision, and Significant Figures will help you answer these questions.

## Nonstandard Units

While there are numerous types of units that we are all familiar with, there are others that are much more obscure. For example, a **firkin** is a unit of volume that was once used to measure beer. One firkin equals about 34 liters. To learn more about nonstandard units, use a dictionary or encyclopedia to research different “weights and measures.” Take note of any unusual units, such as a barleycorn, that are not listed in this tutorial. Think about how the unit is defined and state its relationship to SI units.

## More Examples

### Question 1

Some hummingbirds beat their wings more than 50 times per second. A scientist is measuring the time it takes for a hummingbird to beat its wings once. Which fundamental unit should the scientist use to describe the measurement? Which factor of 10 is the scientist likely to use to describe the motion precisely? Identify the metric prefix that corresponds to this factor of 10.

The scientist will measure the time between each movement using the fundamental unit of seconds. Because the wings beat so fast, the scientist will probably need to measure in milliseconds, or \(10^{-3}\) seconds. (50 beats per second corresponds to 20 milliseconds per beat.)

### Question 2

One cubic centimeter is equal to one milliliter. What does this tell you about the different units in the SI metric system?

The fundamental unit of length (meter) is probably used to create the derived unit of volume (liter). The measure of a milliliter is dependent on the measure of a centimeter.

## Check Your Understanding

- A car is traveling at a speed of 33 m/s. (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90 km/h speed limit?
- American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 3.281 feet.)
- The speed of sound is measured to be 342 m/s on a certain day. What is this in km/h?
- The average distance between the Earth and the Sun is 1.496 × 10
^{11}m. Calculate the average speed of the Earth in its orbit in kilometers per second. What is this in meters per second?