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Conversions in Physics

7 min readdecember 10, 2021

Briana Dokken

Briana Dokken

Briana Dokken

Briana Dokken

How to Convert Units in Physics Calculations

If you're taking high school physics, chemistry, and/or math, you're likely going to run into problems that require you to convert a measurement in one unit into a measurement of a different unit. For example, a question in physics might give you the speed of an object in miles per hour and then ask you to find the speed in kilometers per second. How do we go between units like this in physics? How can we start with a bunch of different measurements and find the one number we need? This is where KHDUDCM and dimensional analysis come into play!

Metric Units

One kind of conversion you will see all the time in physics is converting from one metric prefix to another (without changing what type of measurement we're looking at). For example, a problem might give us a length measured in millimeters (mm) and then want an answer in centimeters (cm). Or maybe we have the volume of a liquid in kiloliters (kL), but we need it in just regular liters (L). When the metric base unit stays the same, but we want to change its prefix, we use the acronym "King Henry Died Unusually Drinking Chocolate Milk" or KHDUDCM.

Here are the metric base units and what they measure:

Quantity

Base Unit

Abbreviation

length

meter

m

mass

gram

g

volume

Liter

L

time

second

s

force

Newton

N

energy

Joule

J

temperature

Kelvin

K

electric charge

Coulomb

C

Using KHDUDCM

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-dVDOZw8HnpEi.png?alt=media&token=b83406a2-3731-4ab0-904b-5ab82002d2db

We use the saying "King Henry Died Unusually Drinking Chocolate Milk" 👑 to get the acronym KHDUDCM. Each of these letters stands for a metric prefix.

To take a measurement with a specific prefix and find out what it is equal to with a different prefix, we move the decimal point in that number. Let's look at how that works with an example.

Let's say we measured the length of a couch 🛋️ to be 226 centimeters (cm), but we want to know the length of the couch in meters (m). To figure this out, we write KHDUDCM and put our pencil on the starting prefix, which is C (C stands for centi). Then we draw bumps over to our ending prefix, U (we want to get to meters which is a base unit (it doesn't have a prefix), and the U stands for base units). We count the number of bumps we drew (the number of letters we moved over), and we carry the decimal point from our original measurement over that many times to get our final measurement. The decimal place moves in the same direction as our pencil did.

There are more metric prefixes than just KHDUDCM, but KHDUDCM is most likely the common one you will encounter. You can see additional prefixes in the picture on the right, and how much bigger or smaller they are than a base unit.

Example

1) Put your pencil on the starting prefix.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-P0rIBkcrOgjk.png?alt=media&token=54939380-fb9d-4873-97e2-e1e40a567751

2) Draw bumps by having the end of a bump on a letter, the top of the bump in between two letters, and the other bottom of the bump at the 2nd letter. Keep drawing these bumps stopping at each letter until you get to the letter representing the prefix you want to end on.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Ptlv98cIR9gu.png?alt=media&token=f9810e73-a0ea-4b93-b2bb-1a7574946277

3) Count the number of bumps as you go. The number of bumps is how many times you move the decimal place over in your original measurement to get your measurement from the starting prefix to the ending prefix.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-tU5fzD2udT4l.png?alt=media&token=666cd486-0f8e-4429-bb1a-981d7eb2f77b

If your pencil moves to the right, the decimal moves to the right. If your pencil moves to the left, the decimal moves to the left.

There are more metric prefixes than just KHDUDCM, but KHDUDCM are most likely the common ones you will run into. You can see additional prefixes in the picture on the right and how much bigger or smaller they are than a base unit.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-XscAgYzsXt44.jpg?alt=media&token=15c33aa1-34df-4ccf-83fd-2c13f5e32a80

Photo courtesy of Reddit

Practice Problems

Converting units can seem like it takes many steps at first, but it becomes pretty easy once you do it a few times. After a while, you will probably be able to do this in your head. Try out these practice problems to get used to doing metric conversions. Also, you do not have to use the KHDUDCM method! There are many different ways to remember how to do metric conversions; this is just one of them. If you know another way to do these, use whichever way works best for you to find the answers to these questions!

  1. Write 10L in kL

  2. Write 12.3mm in Km

Answers

1) 10L -> ?kL

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-hoN8jpE1zHzi.png?alt=media&token=ea9f41b5-9a5c-4328-b4c0-0738f7fe84a6

Whenever we move the decimal over a place without a number, we fill in a 0 there.

0.10kL

It makes sense that we have less kL than L, since kL are much larger than L.

2) 12.3 mm -> ?km

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-T2XCfhDaeIiO.png?alt=media&token=8d23658a-27aa-4f6d-8a15-fb7ef80d8549

.0000123km

It makes sense that we have way less km than mm, because km are much bigger (106 times bigger) than mm.

Converting Between Different Kinds of Units

Another type of conversion you'll see a lot of in physics (and other areas of science) is a conversion from one kind of unit to another. For example, you might have to get meters/second (m/s) into miles/hours (mi/hr). Let's check out how that works with an example. 

Let's say a car is going 27m/s, and we want to know how many miles/hour (mi/hr) the car is going. First, we'll write what we start with (27m/s) as a fraction.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-0tJ1K4D2jINm.png?alt=media&token=4054d2a0-974b-42d1-9995-30100bc0b610

Now, we'll multiply this fraction by another fraction, changing the meters on the top of the fraction into miles and the seconds on the bottom into hours. The final number will be in mi/hr. 

To get meters into miles, we can multiply our original fraction by another fraction with meters in the denominator and miles in the numerator. That way, we are canceling out meters and leaving us with a measurement in miles. What numbers should be in this fraction we multiply by? 

Since we're trying to find an answer equivalent to the fraction we started with (just with different units), we multiply the original fraction by an equal one. That means whatever number in the multiplying fraction's numerator has to be an equal measurement to the number in the denominator in the fraction, even though they are in different units. We need to know how many meters are in one mile (or how many miles are in one meter, we could do this either way) and set that up as a fraction. 

A quick search tells us that there are about 1609 meters in 1 mile. 1mi/1609meters is a conversion factor; it's a fraction equal to one (because 1 mi is the same length as 1609m), and it can change our initial measurement into the units we want. If we set that up as a fraction and multiply to cancel out meters, we get

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1QVN0E2WQMvF.png?alt=media&token=4f977409-1bb7-4cff-b752-7a71f7a72fc1

Next, we need to get the seconds in the denominator into hours. To do this, we'll need to multiply 0.017mi/s by a different fraction with seconds in the numerator and hours in the denominator so that seconds cancel out and we're left with hours in the denominator. There are 3600s in 1 hr. We'll use this to get

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-KiYkXc0ubLIJ.png?alt=media&token=510aca34-c2c8-4dc9-a957-f793886b8164

Now that we changed the meters into miles and the seconds into hours, we have our final answer as 61.2 mi/hr! Typically we would write these two steps all in one line like this.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-XWMsQasgfMIo.png?alt=media&token=b9ace78d-6614-41b4-b656-4411cfda7327

Summary of Solving Unit Conversion (aka Dimensional Analysis) Problems

1️⃣Write your initial value as a fraction.

2️⃣Multiply this fraction by other "conversion factor fractions" (fractions with equal measurements in the numerator and denominator, but in different units). Units diagonal from each other cancel out, so use this to get your answer in the units you need.

We can also convert between metric prefixes (the KHDUDCM stuff we did earlier) using this technique. Here's an example of that.

Let's say we want to change 30m into km. We're going to have to multiply 30m by a fraction with km in the numerator and m in the denominator, so m cancels out, and we're left with km. To get a fraction that has m/km, we'll need to know how many meters are in a kilometer. We can use our trick from before (KHDUDCM) to figure this out.

1m -> ?km

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fproblem2-xoePOQNOhHwn.png?alt=media&token=eb1fa7fc-2fc6-4fe4-b810-db0e794cb318

Now we can set it up so that meters cancel out:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fproblem2-Xw2Ly4sgoDWv.png?alt=media&token=71432f09-aca6-42b0-a405-71e206c31b1a

This has the same effect as just using our trick of counting bumps and moving the decimal in the 30.m as we did before. Still, if you have a problem with multiplying a bunch of fractions by each other, sometimes it can be helpful to write out these prefix conversions as fractions. Either way, you'll get the same answer, so use whichever technique works for you.

Dimensional Analysis Practice Problems

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1VRYMU3N6KF1.png?alt=media&token=40725239-6735-457b-b0a4-a01aec85e346

What to Know

You can use the acronym "King Henry Died Unusually Drinking Chocolate Milk" 🍫 to convert metric prefixes. We use dimensional analysis and conversion factors to convert between units. To go from one unit to another, we multiply by a fraction where the units we want to cancel out are opposite each other (one is in the numerator of a fraction, the other is in the denominator of another fraction). The fraction must equal 1 (we put two measurements equal to each other in the numerator and denominator of the fraction). These conversions are a little tricky at first, but they are all over in physics and chemistry, and after some practice, they'll get super easy! Good luck studying!

Conversions in Physics

7 min readdecember 10, 2021

Briana Dokken

Briana Dokken

Briana Dokken

Briana Dokken

How to Convert Units in Physics Calculations

If you're taking high school physics, chemistry, and/or math, you're likely going to run into problems that require you to convert a measurement in one unit into a measurement of a different unit. For example, a question in physics might give you the speed of an object in miles per hour and then ask you to find the speed in kilometers per second. How do we go between units like this in physics? How can we start with a bunch of different measurements and find the one number we need? This is where KHDUDCM and dimensional analysis come into play!

Metric Units

One kind of conversion you will see all the time in physics is converting from one metric prefix to another (without changing what type of measurement we're looking at). For example, a problem might give us a length measured in millimeters (mm) and then want an answer in centimeters (cm). Or maybe we have the volume of a liquid in kiloliters (kL), but we need it in just regular liters (L). When the metric base unit stays the same, but we want to change its prefix, we use the acronym "King Henry Died Unusually Drinking Chocolate Milk" or KHDUDCM.

Here are the metric base units and what they measure:

Quantity

Base Unit

Abbreviation

length

meter

m

mass

gram

g

volume

Liter

L

time

second

s

force

Newton

N

energy

Joule

J

temperature

Kelvin

K

electric charge

Coulomb

C

Using KHDUDCM

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-dVDOZw8HnpEi.png?alt=media&token=b83406a2-3731-4ab0-904b-5ab82002d2db

We use the saying "King Henry Died Unusually Drinking Chocolate Milk" 👑 to get the acronym KHDUDCM. Each of these letters stands for a metric prefix.

To take a measurement with a specific prefix and find out what it is equal to with a different prefix, we move the decimal point in that number. Let's look at how that works with an example.

Let's say we measured the length of a couch 🛋️ to be 226 centimeters (cm), but we want to know the length of the couch in meters (m). To figure this out, we write KHDUDCM and put our pencil on the starting prefix, which is C (C stands for centi). Then we draw bumps over to our ending prefix, U (we want to get to meters which is a base unit (it doesn't have a prefix), and the U stands for base units). We count the number of bumps we drew (the number of letters we moved over), and we carry the decimal point from our original measurement over that many times to get our final measurement. The decimal place moves in the same direction as our pencil did.

There are more metric prefixes than just KHDUDCM, but KHDUDCM is most likely the common one you will encounter. You can see additional prefixes in the picture on the right, and how much bigger or smaller they are than a base unit.

Example

1) Put your pencil on the starting prefix.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-P0rIBkcrOgjk.png?alt=media&token=54939380-fb9d-4873-97e2-e1e40a567751

2) Draw bumps by having the end of a bump on a letter, the top of the bump in between two letters, and the other bottom of the bump at the 2nd letter. Keep drawing these bumps stopping at each letter until you get to the letter representing the prefix you want to end on.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Ptlv98cIR9gu.png?alt=media&token=f9810e73-a0ea-4b93-b2bb-1a7574946277

3) Count the number of bumps as you go. The number of bumps is how many times you move the decimal place over in your original measurement to get your measurement from the starting prefix to the ending prefix.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-tU5fzD2udT4l.png?alt=media&token=666cd486-0f8e-4429-bb1a-981d7eb2f77b

If your pencil moves to the right, the decimal moves to the right. If your pencil moves to the left, the decimal moves to the left.

There are more metric prefixes than just KHDUDCM, but KHDUDCM are most likely the common ones you will run into. You can see additional prefixes in the picture on the right and how much bigger or smaller they are than a base unit.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-XscAgYzsXt44.jpg?alt=media&token=15c33aa1-34df-4ccf-83fd-2c13f5e32a80

Photo courtesy of Reddit

Practice Problems

Converting units can seem like it takes many steps at first, but it becomes pretty easy once you do it a few times. After a while, you will probably be able to do this in your head. Try out these practice problems to get used to doing metric conversions. Also, you do not have to use the KHDUDCM method! There are many different ways to remember how to do metric conversions; this is just one of them. If you know another way to do these, use whichever way works best for you to find the answers to these questions!

  1. Write 10L in kL

  2. Write 12.3mm in Km

Answers

1) 10L -> ?kL

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-hoN8jpE1zHzi.png?alt=media&token=ea9f41b5-9a5c-4328-b4c0-0738f7fe84a6

Whenever we move the decimal over a place without a number, we fill in a 0 there.

0.10kL

It makes sense that we have less kL than L, since kL are much larger than L.

2) 12.3 mm -> ?km

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-T2XCfhDaeIiO.png?alt=media&token=8d23658a-27aa-4f6d-8a15-fb7ef80d8549

.0000123km

It makes sense that we have way less km than mm, because km are much bigger (106 times bigger) than mm.

Converting Between Different Kinds of Units

Another type of conversion you'll see a lot of in physics (and other areas of science) is a conversion from one kind of unit to another. For example, you might have to get meters/second (m/s) into miles/hours (mi/hr). Let's check out how that works with an example. 

Let's say a car is going 27m/s, and we want to know how many miles/hour (mi/hr) the car is going. First, we'll write what we start with (27m/s) as a fraction.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-0tJ1K4D2jINm.png?alt=media&token=4054d2a0-974b-42d1-9995-30100bc0b610

Now, we'll multiply this fraction by another fraction, changing the meters on the top of the fraction into miles and the seconds on the bottom into hours. The final number will be in mi/hr. 

To get meters into miles, we can multiply our original fraction by another fraction with meters in the denominator and miles in the numerator. That way, we are canceling out meters and leaving us with a measurement in miles. What numbers should be in this fraction we multiply by? 

Since we're trying to find an answer equivalent to the fraction we started with (just with different units), we multiply the original fraction by an equal one. That means whatever number in the multiplying fraction's numerator has to be an equal measurement to the number in the denominator in the fraction, even though they are in different units. We need to know how many meters are in one mile (or how many miles are in one meter, we could do this either way) and set that up as a fraction. 

A quick search tells us that there are about 1609 meters in 1 mile. 1mi/1609meters is a conversion factor; it's a fraction equal to one (because 1 mi is the same length as 1609m), and it can change our initial measurement into the units we want. If we set that up as a fraction and multiply to cancel out meters, we get

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1QVN0E2WQMvF.png?alt=media&token=4f977409-1bb7-4cff-b752-7a71f7a72fc1

Next, we need to get the seconds in the denominator into hours. To do this, we'll need to multiply 0.017mi/s by a different fraction with seconds in the numerator and hours in the denominator so that seconds cancel out and we're left with hours in the denominator. There are 3600s in 1 hr. We'll use this to get

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-KiYkXc0ubLIJ.png?alt=media&token=510aca34-c2c8-4dc9-a957-f793886b8164

Now that we changed the meters into miles and the seconds into hours, we have our final answer as 61.2 mi/hr! Typically we would write these two steps all in one line like this.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-XWMsQasgfMIo.png?alt=media&token=b9ace78d-6614-41b4-b656-4411cfda7327

Summary of Solving Unit Conversion (aka Dimensional Analysis) Problems

1️⃣Write your initial value as a fraction.

2️⃣Multiply this fraction by other "conversion factor fractions" (fractions with equal measurements in the numerator and denominator, but in different units). Units diagonal from each other cancel out, so use this to get your answer in the units you need.

We can also convert between metric prefixes (the KHDUDCM stuff we did earlier) using this technique. Here's an example of that.

Let's say we want to change 30m into km. We're going to have to multiply 30m by a fraction with km in the numerator and m in the denominator, so m cancels out, and we're left with km. To get a fraction that has m/km, we'll need to know how many meters are in a kilometer. We can use our trick from before (KHDUDCM) to figure this out.

1m -> ?km

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fproblem2-xoePOQNOhHwn.png?alt=media&token=eb1fa7fc-2fc6-4fe4-b810-db0e794cb318

Now we can set it up so that meters cancel out:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fproblem2-Xw2Ly4sgoDWv.png?alt=media&token=71432f09-aca6-42b0-a405-71e206c31b1a

This has the same effect as just using our trick of counting bumps and moving the decimal in the 30.m as we did before. Still, if you have a problem with multiplying a bunch of fractions by each other, sometimes it can be helpful to write out these prefix conversions as fractions. Either way, you'll get the same answer, so use whichever technique works for you.

Dimensional Analysis Practice Problems

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1VRYMU3N6KF1.png?alt=media&token=40725239-6735-457b-b0a4-a01aec85e346

What to Know

You can use the acronym "King Henry Died Unusually Drinking Chocolate Milk" 🍫 to convert metric prefixes. We use dimensional analysis and conversion factors to convert between units. To go from one unit to another, we multiply by a fraction where the units we want to cancel out are opposite each other (one is in the numerator of a fraction, the other is in the denominator of another fraction). The fraction must equal 1 (we put two measurements equal to each other in the numerator and denominator of the fraction). These conversions are a little tricky at first, but they are all over in physics and chemistry, and after some practice, they'll get super easy! Good luck studying!



© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.