Units

What are units? Why are they important? What is unit analysis?

There are two main systems of measurement in the world—metric and English. The metric system is most widely used and is the standard of math, physics, engineering and medicine. The English system (also known as the United States customary system) is only used by three countries: the United States, Liberia and Myanmar.

The Importance of Units

Units give meaning to numbers. If your mom asks you to buy apples at the store, you might wonder, "How many apples? Five apples, or five pounds of apples?" Here, your unit of measure could be either apples or pounds. Without knowing the units, you might be bringing home way more apples than your mother intended.

Three of the most common units are those of length, mass and time. Mass and weight often are used interchangeably in the metric system, but they describe two different properties. Mass is the amount of matter (or "stuff") in an object and is measured in kilograms (kg), while weight is the acceleration of an object under the force of gravity and is measured in Newtons (N). We often say an object weighs 10 kg, but what we mean is that it has a mass of 10 kg. Its weight would be roughly 1 N. In the English system, I might weigh 150 pounds (lbs), but my mass is roughly 4.7 slugs. I would weigh less on the Moon because the Moon's gravitational pull is less than Earth's, but my mass would be the same.

As long as the units are given, you'll know if you're talking about mass (kg or slugs) or weight (N or pounds).

Unit Analysis

One of my favorite techniques of problem-solving is unit analysis. It's a great way to see if you're on the right track. If you're solving an equation and your units come out as expected, you've probably solved the problem correctly. If the numerical answer is wrong, it's likely just a clerical error.

Here's an example. Speed, or velocity, is in units of miles per hour, kilometers per hour, meters per second, feet per minute, etc. See the pattern? The units are all of the form "length per time." Instead of length, we'll say distance. So velocity (v) is distance (d) per time (t) or:

(1)

For example, suppose you want to know how long it takes to get to Grandma's house, which is 50 miles away, and you're going by sled pulled by a horse (who knows the way) at 5 miles per hour. If you think you're going to divide 5 by 50, consider the units:

Oops... is not the correct unit of time.

What about multiplying 5 by 50? Let's see:

We know the equation is wrong just because the units are not correct. No point in even doing any calculations.

To get units of time we need to divide distance by speed.

Unit analysis shows: .

Or, . We see that if we solve equation (1) for t, this is the correct answer.

Always check your units!

For The Interested Reader

The meter was first defined in 1791 as one ten-millionth of the distance from the Earth’s equator to the North Pole. In 1983 it was redefined to be the distance traveled by light in a vacuum in of a second.

For The Interested Reader

In 1999 NASA lost the Mars Climate Orbiter due to a mismatch between units. One engineering team was using the English system, while another was using metric. Because of the discrepancy the orbiter got too close to Mars before its engines fired and could not land properly. The mishap cost $125 million.

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