Students will learn how to convert units from metric to English system and vice versa using dimensional analysis.

### Key Equations

\begin{align*} 1\ \text{meter} & = 3.28\ \text{feet} && \\
1\ \text{mile} & = 1.61 \text{~kilometers} && \\
1\ \text{lb. (1\ pound)} & = 4.45\ \text{Newtons}\end{align*}

Guidance

- The key to converting units is to multiply by a clever factor of one. You can always multiply by 1, because it does not change the number. Since 1 in. is equal to 2.54 cm, then \begin{align*}1 = {\frac{2.54 \;\mathrm{cm}}{1 \;\mathrm{in}}} = {\frac{1 \;\mathrm{in}}{2.54 \;\mathrm{cm}}}\end{align*}
1=2.54cm1in=1in2.54cm . Thus, one can multiply by this form of 1 in order to cancel units (see video below). - Write out every step and show all your units cancelling as you go.
- When converting speeds from metric to American units, remember the following rule of thumb: a speed measured in mi/hr is about double the value measured in m/s (
*i.e.,*\begin{align*}10 \mathrm{m/s}\end{align*}10m/s is equal to about 20 MPH). Remember that the speed itself hasn’t changed, just our representation of the speed in a certain set of units. - When you’re not sure how to approach a problem, you can often get insight by considering how to obtain the units of the desired result by combining the units of the given variables. For instance, if you are given a distance (in meters) and a time (in hours), the only way to obtain units of speed (meters/hour) is to divide the distance by the time. This is a simple example of a method called
*dimensional analysis*, which can be used to find equations that govern various physical situations without any knowledge of the phenomena themselves.

#### Example 1

**Question**: 20 m/s = ? mi/hr

**Solution**:

20 m/s (1 mi/1600 m) = .0125 mi/s

.0125 mi/s (60 s/min) = .75 mi/min

.75 mi/min (60 min/hr) = 45 mi/hr

### Watch this Explanation

### Explore More

- Estimate or measure your height.
- Convert your height from feet and inches to meters.
- Convert your height from feet and inches to centimeters \begin{align*}(100 \;\mathrm{cm} = 1 \;\mathrm{m})\end{align*}
(100cm=1m)

- Estimate or measure the amount of time that passes between breaths when you are sitting at rest.
- Convert the time from seconds into hours
- Convert the time from seconds into milliseconds \begin{align*}\;\mathrm{(ms)}\end{align*}
(ms)

- Convert the French speed limit of \begin{align*}140 \;\mathrm{km/hr}\end{align*}
140km/hr into \begin{align*}\;\mathrm{mi/hr}\end{align*}mi/hr . - Estimate or measure your weight.
- Convert your weight in pounds into a mass in \begin{align*}kg\end{align*}
kg - Convert your mass from \begin{align*}kg\end{align*}
kg into \begin{align*}\mu g\end{align*}μg - Convert your weight into Newtons

- Convert your weight in pounds into a mass in \begin{align*}kg\end{align*}
- Find the \begin{align*}SI\end{align*}
SI unit for pressure. - An English lord says he weighs \begin{align*}12\end{align*}
12 stone.- Convert his weight into pounds (you may have to do some research online)
- Convert his weight in stones into a mass in kilograms

- If the speed of your car increases by \begin{align*}10 \;\mathrm{mi/hr}\end{align*}
10mi/hr every 2 seconds, how many \begin{align*}\;\mathrm{mi/hr}\end{align*}mi/hr is the speed increasing every second? State your answer with the units \begin{align*}\;\mathrm{mi/hr/s}\end{align*}mi/hr/s .

**Answers**

- a. A person of height \begin{align*}5 \;\mathrm{ft}\end{align*}
5ft . \begin{align*}11 \;\mathrm{in}\end{align*}. is \begin{align*}1.80 \;\mathrm{m}\end{align*} tall b. The same person is \begin{align*}180 \;\mathrm{cm}\end{align*} - a. \begin{align*}3 \;\mathrm{seconds} = 1/1200 \;\mathrm{hours}\end{align*} b. \begin{align*}3x10^3 \;\mathrm{ms}\end{align*}
- \begin{align*}87.5 \;\mathrm{mi/hr}\end{align*}
- If the person weighs \begin{align*}150 \;\mathrm{lb}\end{align*} then a. 67.9 kg (on Earth) b. 67.9 billion \begin{align*}\mu g\end{align*} c. this is equivalent to \begin{align*}668 \;\mathrm{N}\end{align*}
- Pascals (Pa), which equals \begin{align*}\;\mathrm{N/m}^2\end{align*}
- a. \begin{align*}168 \;\mathrm{lb}\end{align*}.,b. \begin{align*}76.2 \;\mathrm{kg}\end{align*}
- \begin{align*}5 \;\mathrm{mi/hr/s}\end{align*}