Unit Conversions

Introduction As scientists and engineers, we need to measure all kinds of things, like distances, times, masses, speeds, and so on. Measuring things requires units: standard, agreed-upon quantities that are compared to the quantities to be measured. For example, if you are five feet tall, your height is equal to five of those things we all agree upon, called feet. Historically, the "foot" probably derived from the length of peoples' feet (perhaps with shoes).

There are many different units used to measure all kinds of quantities. For example, for length or distances, there are feet, inches, miles, leagues, fathoms, nautical miles, meters, kilometers, and many others. Visit http://www.unc.edu/~rowlett/units/index.html to get a flavor for the variety of units that get used.

In physics we often have to convert between units, to change a measurement from one set of units into another. Even in everyday life it is often necessary to do so as well.

In cooking, you often have to convert between cups, teaspoons, tablespoons, ounces, pints and quarts.

What is your height? Most of you probably know your height in feet and inches. But what if you needed to know your height in centimeters — what then? What is your height in nanometers? How about miles?

With simple conversions it is often easy enough to know to just multiply or divide by a conversion factor, but I want you to avoid the temptation to do that. Instead, I want you to learn the systematic way, called the Factor-Label Method.

The key to the Factor-Label Method is to remember that multiplying anything by the number one changes nothing. Here's the method, step-by-step.

  1. Write the given measurement, the measurement that you are trying to convert into different units.
  2. Multiply by a conversion factor, a fraction that relates one unit to another, so the starting unit "cancels" and is replaced by a new unit. Cross out the canceled units. The conversion factor equals one, so you are not changing the quantity, only the units it is expressed in.
  3. If this new unit is what you want, stop!
  4. If this new unit isn't what you want, multiply by another conversion factor to get rid of it. Repeat until you have the final units you need.

Watch this video, and it'll make more sense.

Activities & Practice
to do as you read

Example (easy) Convert the length of a marathon (26.2 miles) into kilometers. This can be done in a single step...
A mile equals 1.61 km — they are the same thing. Anything divided by itself equals one, so that conversion factor in parentheses equals one, and therefore the original quantity (26.2 miles) is not being changed. Since miles is in both the numerator and denominator of the fraction, though, the fraction equals one and can be ignored. People often say they “cancel out”. This works because units multiply and divide just like numbers.


You must use the factor-label method and show all your work.

1. Convert your weight (in pounds) to newtons. (1 pound equals 4.45 newtons.)

2. Convert your height (in inches) to centimeters.

3. Convert your height (in centimeters) to meters.

Example (harder) Convert the time 1 year into seconds. Since you probably don't know how many seconds there are in a year, convert to days instead...

...then multiply by another conversion factor, to turn days into hours...

...then turn the hours into minutes and the minutes into seconds with two more conversion factors. Lastly, get out your calculator and punch in the numbers.

4. Convert your height (in inches) to miles. (NOTE: there are 5280 feet in a mile.)

5. Convert your age into seconds.

6. Convert your weight to tons.

Example (even harder) Convert the common highway speed limit of 55 miles per hour into furlongs per fortnight. A furlong is one-eighth of a mile, and a fortnight is 2 weeks.

7. Convert the usual highway speed limit of 55 miles/hr into km/hr.

8. Convert the usual highway speed limit of 55 miles/hr into m/sec.

9. Convert the speed of light (3x108 m/sec) into miles/hr.

There are many good online unit-conversion resources. Two of them are http://www.unit-conversion.info/length.html and http://www.onlineconversion.com/. The Google search engine will also do unit conversions — for example, try typing "1 mile in meters" into the Google search field and see what it does.

If you need to convert areas or volumes, read Unit Conversions B.


Converting, or not converting units, can have big consequences in the real world.

NASA lost a Mars probe called Mars Climate Orbiter (MCO) because a data file used the English force unit pounds instead of the SI force unit newtons. This data file was used to calculate trajectory corrections using small rocket thrusters on the spacecraft. Because the units in the data were wrong, the trajectory corrections were done incorrectly, and MCO entered Mars's atmosphere and burned up instead of going into orbit around the planet.

In 1983 an Air Canada 767 airliner, flying from Montreal to Edmonton, ran out of fuel in mid-flight. The pilots and ground crew had incorrectly calculated how much fuel to take onboard, because they had confused kilograms, pounds, liters and gallons. The pilots were able to land the aircraft without engine power safely in Gimli, Manitoba. The airplane was forever known thereafter as the "Gimli Glider".



Additional Activities & Practice

10. The average peak blood flow velocity in the human aorta is about 92 cm/sec, according to an article in the American Heart Journal (Volume 107, Issue 2, Pages 310-319). What is that speed in miles per hour?

11. Human nerve impulses travel at about 270 miles per hour. What is that speed in cm/sec?


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