Lesson Name: Unit Conversion
Test Drive Lesson

Instructional Block

Unit Conversion

Learning Objectives

By the end of this section, you will be able to:

Use conversion factors to express the value of a given quantity in different units.

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. 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:

$80\;\not{m} \times {1\;km \over 1000\;\not{m}}=0.080\;km$

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

$80\;m=8.0×10^1\;m=8.0×10^{−2}\;km=0.080\;km$

since “kilo-” means 10³ and 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.

EXAMPLE 1: Converting Nonmetric Units to Metric

The distance from the university to home is 10 mi and it usually takes 20 min to drive this distance. Calculate the average speed 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 factors and multiplying by them. The correct conversion factors are those that cancel the unwanted units and leave the desired units in their place. In this case, we want to convert miles to meters, so we need to know the fact that there are 1609 m in 1 mi. We also want to convert minutes to seconds, so we use the conversion of 60 s in 1 min.

Solution

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,

$\text{Average speed} = {\text{Distance} \over \text{Time}}$
Substitute the given values for distance and time:

$\text{Average speed} = {10\;mi \over 20\;min} = 0.50\;{mi \over min}$
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:

$0.50\;{\not{mile} \over \not{min}} \times {1609\;m \over 1\;\not{mile}} \times {1\;\not{min} \over 60\;s} = {(0.50)(1609) \over 60}\;m/s = 13\;m/s$

$0.50 \frac{mile}{min} \times \frac{1609 m}{1 mile} \times \frac{1 min}{60 s} = \frac{(0.50) (1609)}{60} m/s = 13 m/s .$

Significance

Check the answer in the following ways:

Be sure the units in the unit conversion cancel correctly. If the unit conversion factor was written upside down, the units do not cancel correctly in the equation. We see the “miles” in the numerator in 0.50 mi/min cancels the “mile” in the denominator in the first conversion factor. Also, the “min” in the denominator in 0.50 mi/min cancels the “min” in the numerator in the second conversion factor.
Check that the units of the final answer are the desired units. The problem asked us to solve for average speed in units of meters per second and, after the cancellations, the only units left are a meter (m) in the numerator and a second (s) in the denominator, so we have indeed obtained these units.

EXAMPLE 2: Converting between Metric Units

The density of iron is 7.86 g/cm³ under standard conditions. Convert this to kg/m³.

Strategy

We need to convert grams to kilograms and cubic centimeters to cubic meters. The conversion factors we need are 1 kg = 10³ $g$ and 1 cm = 10^-2 m. However, we are dealing with cubic centimeters (cm³ = cm x cm x cm) $,$ so we have to use the second conversion factor three times (that is, we need to cube it). The idea is still to multiply by the conversion factors in such a way that they cancel the units we want to get rid of and introduce the units we want to keep.

Solution

$7.86\;{\not{g} \over \not {cm^3}} × {kg \over 10^3\not {g}} × \bigg({\not {cm} \over 10^{−2}\;m}\bigg)^3={7.86 \over (10^3)(10^{−6})}kg/m3=7.86×10^3\;kg/m^3$

Significance

Remember, it’s always important to check the answer.

Be sure to cancel the units in the unit conversion correctly. We see that the gram (“g”) in the numerator in 7.86 g/cm3 cancels the “g” in the denominator in the first conversion factor. Also, the three factors of “cm” in the denominator in 7.86 g/cm3 cancel with the three factors of “cm” in the numerator that we get by cubing the second conversion factor.
Check that the units of the final answer are the desired units. The problem asked for us to convert to kilograms per cubic meter. After the cancellations just described, we see the only units we have left are “kg” in the numerator and three factors of “m” in the denominator (that is, one factor of “m” cubed, or “m3”). Therefore, the units on the final answer are correct.

Unit conversions may not seem very interesting, but not doing them can be costly. One famous example of this situation was seen with the Mars Climate Orbiter. This probe was launched by NASA on December 11, 1998. On September 23, 1999, while attempting to guide the probe into its planned orbit around Mars, NASA lost contact with it. Subsequent investigations showed a piece of software called SM_FORCES (or “small forces”) was recording thruster performance data in the English units of pound-seconds (lb-s). However, other pieces of software that used these values for course corrections expected them to be recorded in the SI units of newton-seconds (N-s), as dictated in the software interface protocols. This error caused the probe to follow a very different trajectory from what NASA thought it was following, which most likely caused the probe either to burn up in the Martian atmosphere or to shoot out into space. This failure to pay attention to unit conversions cost hundreds of millions of dollars, not to mention all the time invested by the scientists and engineers who worked on the project.

No hints found for this question

Multiple Choice

The volume of Earth is on the order of $10^{21}\;m^{3}$ . What is this in cubic kilometers ( $km^{3}$ )?

$15^{10}\;km^3$

$12^{10}\;km^3$

$10^{12}\;km^3$

$2^{15}\;km^3$

Answer	Rubric	% Of Choosen
$15^{10}\;km^3$	In-Correct	0 %
$12^{10}\;km^3$	In-Correct	0 %
$10^{12}\;km^3$	Correct	0 %
$2^{15}\;km^3$	In-Correct	0 %

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Multiple Choice

The volume of Earth is on the order of $10^{21}m^3$ . What is it in cubic miles ( $mi^3$ )?

$10^{21}\;mi^3$

$10^{11}\;mi^3$

$10^{14}\;mi^3$

$10^{25}\;mi^3$

Answer	Rubric	% Of Choosen
$10^{21}\;mi^3$	In-Correct	0 %
$10^{11}\;mi^3$	Correct	0 %
$10^{14}\;mi^3$	In-Correct	0 %
$10^{25}\;mi^3$	In-Correct	0 %

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Multiple Choice

The volume of Earth is on the order of $10^{21}cm^3$ . What is it in cubic centimeters ( $cm^3$ )?

$10^{5}cm^3$

$10^{27}cm^3$

$10^{32}cm^3$

$10^{45}cm^3$

Answer	Rubric	% Of Choosen
$10^{5}cm^3$	In-Correct	0 %
$10^{27}cm^3$	Correct	0 %
$10^{32}cm^3$	In-Correct	0 %
$10^{45}cm^3$	In-Correct	0 %

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Multiple Choice

The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second?

$15.9\;m/s$

$27.8\;m/s$

$22.85\;m/s$

$16.45\;m/s$

Answer	Rubric	% Of Choosen
$15.9\;m/s$	In-Correct	0 %
$27.8\;m/s$	Correct	0 %
$22.85\;m/s$	In-Correct	0 %
$16.45\;m/s$	In-Correct	0 %

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Multiple Choice

The speed limit on some interstate highways is roughly $100 \;km/h$ . How many miles per hour is this?

$45\;mi/h$

$85\;mi/h$

$62\;mi/h$

$38\;mi/h$

Answer	Rubric	% Of Choosen
$45\;mi/h$	In-Correct	0 %
$85\;mi/h$	In-Correct	0 %
$62\;mi/h$	Correct	0 %
$38\;mi/h$	In-Correct	0 %

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Multiple Choice

A car is traveling at a speed of $33 \;m/s$ . What is its speed in kilometers per hour?

$12.1\times10^5\;km/h$

$1.2\times10^2\;km/h$

$2.8\times10^4\;km/h$

$9.8\times10^3\;km/h$

Answer	Rubric	% Of Choosen
$12.1\times10^5\;km/h$	In-Correct	0 %
$1.2\times10^2\;km/h$	Correct	0 %
$2.8\times10^4\;km/h$	In-Correct	0 %
$9.8\times10^3\;km/h$	In-Correct	0 %

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Multiple Choice

A car is traveling at a speed of $33 \;m/s$ . Is it exceeding the $90\; km/h$ speed limit?

$100 \;km/h$ , the car is exceeding the speed limit.

$120\; km/h$ , the car is exceeding the speed limit.

Cannot tell.

$132\; km/h$ , the car isn't exceeding the speed limit.

Answer	Rubric	% Of Choosen
At $100 \;km/h$ , the car is exceeding the speed limit.	In-Correct	0 %
At $120\; km/h$ , the car is exceeding the speed limit.	Correct	0 %
Cannot tell.	In-Correct	0 %
At $132\; km/h$ , the car isn't exceeding the speed limit.	In-Correct	0 %

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Multiple Choice

American football is played on a $100\; yd$ long field, excluding the end zones. How long is the field in meters? (Assume that $1\; m = 3.281\; ft.$ )

$12.2\;m$

$36.4\;m$

$91.4\;m$

$9.8\;m$

Answer	Rubric	% Of Choosen
$12.2\;m$	In-Correct	0 %
$36.4\;m$	In-Correct	0 %
$91.4\;m$	Correct	0 %
$9.8\;m$	In-Correct	0 %

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Multiple Choice

Soccer fields vary in size. A large soccer field is $115 \;m$ long and $85.0\; m$ wide. What is its area in square feet? (Assume that $1\; m = 3.281\; ft$ .)

$4.12\times10^4\;ft$

$1.05\times10^5\;ft$

$1.26\times10^3\;ft$

$8.9\times10^8\;ft$

Answer	Rubric	% Of Choosen
$4.12\times10^4\;ft$	In-Correct	0 %
$1.05\times10^5\;ft$	Correct	0 %
$1.26\times10^3\;ft$	In-Correct	0 %
$8.9\times10^8\;ft$	In-Correct	0 %

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Multiple Choice

What is the height in meters of a person who is $6\;ft$ $1.0\; in$ . tall?

$1.85\;m$

$5.6\;m$

$2.7\;m$

$7.6\;m$

Answer	Rubric	% Of Choosen
$1.85\;m$	Correct	0 %
$5.6\;m$	In-Correct	0 %
$2.7\;m$	In-Correct	0 %
$7.6\;m$	In-Correct	0 %

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Multiple Choice

Mount Everest, at $29,028\; ft$ , is the tallest mountain on Earth. What is its height in kilometers? (Assume that $1 \;m = 3.281\; ft$ .)

$1.265\;km$

$4.568\;km$

$8.847\;km$

$5.26\;km$

Answer	Rubric	% Of Choosen
$1.265\;km$	In-Correct	0 %
$4.568\;km$	In-Correct	0 %
$8.847\;km$	Correct	0 %
$5.26\;km$	In-Correct	0 %

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Multiple Choice

The speed of sound is measured to be $342\; m/s$ on a certain day. What is this measurement in kilometers per hour?

$4.26\times10^6\;km/h$

$7.18\times10^4\;km/h$

$1.23\times10^3\;km/h$

$5.93\times10^3\;km/h$

Answer	Rubric	% Of Choosen
$4.26\times10^6\;km/h$	In-Correct	0 %
$7.18\times10^4\;km/h$	In-Correct	0 %
$1.23\times10^3\;km/h$	Correct	0 %
$5.93\times10^3\;km/h$	In-Correct	0 %

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Multiple Choice

Tectonic plates are large segments of Earth’s crust that move slowly. Suppose one such plate has an average speed of $4.0 \;cm/yr.$ What distance does it move in $1.0\; s$ at this speed?

$1.3\times10^{-9}m$

$9.8\times10^{-4}m$

$2.6\times10^6m$

$1.5\times10^9m$

Answer	Rubric	% Of Choosen
$1.3\times10^{-9}m$	Correct	0 %
$9.8\times10^{-4}m$	In-Correct	0 %
$2.6\times10^6m$	In-Correct	0 %
$1.5\times10^9m$	In-Correct	0 %

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Multiple Choice

Tectonic plates are large segments of Earth’s crust that move slowly. Suppose one such plate has an average speed of $4.0 \;cm/yr$ . What is its speed in kilometers per million years

$40\;km/My$

$0.04\;km/My$

$60\;km/My$

$8\;km/My$

Answer	Rubric	% Of Choosen
$40\;km/My$	Correct	0 %
$0.04\;km/My$	In-Correct	0 %
$60\;km/My$	In-Correct	0 %
$8\;km/My$	In-Correct	0 %

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Multiple Choice

The average distance between Earth and the Sun is $1.5\times10^{11}\;m$ . Calculate the average speed of Earth in its orbit (assumed to be circular) in meters per second.

$3.65\times10^{45}\;m/s$

$3.0\times10^4\;m/s$

$6.5\times10^{12}\;m/s$

$4.5\times10^8\;m/s$

Answer	Rubric	% Of Choosen
$3.65\times10^{45}\;m/s$	In-Correct	0 %
$3.0\times10^4\;m/s$	Correct	0 %
$6.5\times10^{12}\;m/s$	In-Correct	0 %
$4.5\times10^8\;m/s$	In-Correct	0 %

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Multiple Choice

The average distance between Earth and the Sun is $1.5\times10^{11}\;m$ . What is this speed in miles per hour?

$3.0\times10^4\;mi/h$

$7.6\times10^{11}\;mi/h$

$6.7\times10^{4}\;mi/h$

$3.21\times10^4\;mi/h$

Answer	Rubric	% Of Choosen
$3.0\times10^4\;mi/h$	In-Correct	0 %
$7.6\times10^{11}\;mi/h$	In-Correct	0 %
$6.7\times10^{4}\;mi/h$	Correct	0 %
$3.21\times10^4\;mi/h$	In-Correct	0 %

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Multiple Choice

The density of nuclear matter is about $10^{18}\;kg/m^3$ . Given that $1 \;mL$ is equal in volume to $cm^3$ , what is the density of nuclear matter in megagrams per microliter (that is, $Mg/\mu{L}$ )?

$10^6\;Mg/\mu{L}$

$10^3\;Mg/\mu{L}$

$12^5\;Mg/\mu{L}$

$20^9\;Mg/\mu{L}$

Answer	Rubric	% Of Choosen
$10^6\;Mg/\mu{L}$	Correct	0 %
$10^3\;Mg/\mu{L}$	In-Correct	0 %
$12^5\;Mg/\mu{L}$	In-Correct	0 %
$20^9\;Mg/\mu{L}$	In-Correct	0 %

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Multiple Choice

The density of aluminum is $2.7\;g/cm^{3}$ . What is the density in kilograms per cubic meter?

$10.7\times10^3\;kg/m^3$

$6.35\times10^6\;kg/m^3$

$20.6\times10^8\;kg/m^3$

$2.45\times10^3\;kg/m^3$

Answer	Rubric	% Of Choosen
$10.7\times10^3\;kg/m^3$	Correct	0 %
$6.35\times10^6\;kg/m^3$	In-Correct	0 %
$20.6\times10^8\;kg/m^3$	In-Correct	0 %
$2.45\times10^3\;kg/m^3$	In-Correct	0 %

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Multiple Choice

A commonly used unit of mass in the English system is the pound-mass, abbreviated lbm, where 1 lbm = 0.454 kg. What is the density of water in pound-mass per cubic foot?

$22.6\;lbm/ft^3$

$62.4 \;lbm/ft^{3}$

$10.64\;lbm/ft^3$

$60.45\;lbm/ft^3$

Answer	Rubric	% Of Choosen
$22.6\;lbm/ft^3$	In-Correct	0 %
$62.4 \;lbm/ft^{3}$	Correct	0 %
$10.64\;lbm/ft^3$	In-Correct	0 %
$60.45\;lbm/ft^3$	In-Correct	0 %

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Multiple Choice

A furlong is 220 yd. A fortnight is 2 weeks. Convert a speed of one furlong per fortnight to millimeters per second.

$0.166\;mm/n$

$0.100\;mm/n$

$0.200\;mm/n$

$0.350\;mm/n$

Answer	Rubric	% Of Choosen
$0.166\;mm/n$	Correct	0 %
$0.100\;mm/n$	In-Correct	0 %
$0.200\;mm/n$	In-Correct	0 %
$0.350\;mm/n$	In-Correct	0 %

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Multiple Choice

It takes $2\pi$ radians (rad) to get around a circle, which is the same as 360°. How many radians are in 1°?

$0.201\;rad$

$0.170\;rad$

$0.020\;rad$

$0.017\;rad$

Answer	Rubric	% Of Choosen
$0.201\;rad$	In-Correct	0 %
$0.170\;rad$	In-Correct	0 %
$0.020\;rad$	In-Correct	0 %
$0.017\;rad$	Correct	0 %

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Multiple Choice

Light travels a distance of about $3\times10^8\;m/s$ . A light-minute is the distance light travels in 1 min. If the Sun is $1.5\times10^{11}\;m$ from Earth, how far away is it in light-minutes?

8 light minutes

10 light minutes

6 light minutes

25 light minutes

Answer	Rubric	% Of Choosen
8 light minutes	Correct	0 %
10 light minutes	In-Correct	0 %
6 light minutes	In-Correct	0 %
25 light minutes	In-Correct	0 %

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Multiple Choice

A light-nanosecond is the distance light travels in 1 ns. Convert 1 ft to light-nanoseconds.

1 light-nanosecond

20 light-nanosecond

4 light-nanosecond

5 light-nanosecond

Answer	Rubric	% Of Choosen
1 light-nanosecond	Correct	0 %
20 light-nanosecond	In-Correct	0 %
4 light-nanosecond	In-Correct	0 %
5 light-nanosecond	In-Correct	0 %

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Numeric

In SI units, speeds are measured in meters per second (m/s). But, depending on where you live, you’re probably more comfortable of thinking of speeds in terms of either kilometers per hour (km/h) or miles per hour (mi/h). Convert 25 m/s to km/h.

Is correct?	Answer (midpoint)	Rounding Margin	Answer Range	Units	Wrong Answer Feedback
Correct	90	0	[90,90]	-	NULL

$25 \; \frac{m}{s} \left[ \frac{1\; km}{1000 \; m} \right] \left[ \frac{3600\; s}{1 \; h} \right]$

Answer: 90

Numeric

Convert 62 m/s to mi/h.

Is correct?	Answer (midpoint)	Rounding Margin	Answer Range	Units	Wrong Answer Feedback
Correct	150	50	[100,200]	-	NULL

$62 \; \frac{m}{s} \left[ \frac{1\; mi}{1609 \; m} \right] \left[ \frac{3600\; s}{1 \; h} \right]$

Answer: 138.7197

Numeric

Is correct?	Answer (midpoint)	Rounding Margin	Answer Range	Units	Wrong Answer Feedback
Correct	65	5	[60,70]	-	NULL

$18 \; \frac{m}{s} \left[ \frac{1\; km}{1000 \; m} \right] \left[ \frac{3600\; s}{1 \; h} \right]$

Answer: 64.8

Numeric

Convert 55 m/s to mi/h.

Is correct?	Answer (midpoint)	Rounding Margin	Answer Range	Units	Wrong Answer Feedback
Correct	150	50	[100,200]	-	NULL

$55 \; \frac{m}{s} \left[ \frac{1\; mi}{1609 \; m} \right] \left[ \frac{3600\; s}{1 \; h} \right]$

Answer: 123.0578

Numeric

Is correct?	Answer (midpoint)	Rounding Margin	Answer Range	Units	Wrong Answer Feedback
Correct	150	50	[100,200]	-	NULL

$40 \; \frac{m}{s} \left[ \frac{1\; km}{1000 \; m} \right] \left[ \frac{3600\; s}{1 \; h} \right]$

Answer: 144

Numeric

Convert 8 m/s to mi/h.

Is correct?	Answer (midpoint)	Rounding Margin	Answer Range	Units	Wrong Answer Feedback
Correct	15	5	[10,20]	-	NULL

$8 \; \frac{m}{s} \left[ \frac{1\; mi}{1609 \; m} \right] \left[ \frac{3600\; s}{1 \; h} \right]$

Answer: 17.8993

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Lesson Name: Unit Conversion
Test Drive Lesson

Unit Conversion

Learning Objectives

EXAMPLE 1: Converting Nonmetric Units to Metric

EXAMPLE 2: Converting between Metric Units

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Lesson Name: Unit Conversion Test Drive Lesson

EXAMPLE 1: Converting Nonmetric Units to Metric

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Start with Key2PHYSICS, our all-in-one Learning Platform for Physics Educators. It has the tools you need to teach physics—lessons, assessments, and learning reports—all in one place.

Lesson Name: Unit Conversion
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