Perform Unit Conversions for Speeds

Introduction

In this lesson, you will perform unit conversions with speeds. When you do unit conversions for speeds, you have units in the numerator and denominator that you need to change to what you want.

This video illustrates the lesson material below. Watching the video is optional.

Unit Conversions for Speeds (03:56 mins) | Transcript

Unit Conversions for Speeds

Steps for Speed Unit Conversions:

Start with what unit you know.
Determine what unit you want to get in the end.
Determine what conversion factors you should use. You will need more than one.
Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

Example 1
The average horse can run about 25 miles per hour. (Remember per can mean divided by. Therefore, this can also be written as $25 \space miles\div 1 \space hour$.) What is this speed in meters per second?

Start with what you know. The horse runs at 25 miles per hour, or $\frac{25miles}{hour}$ , and you want to convert this to meters per second, or $\frac{meters}{second}$. In other words, meters in the numerator and seconds in the denominator.

The first conversion factor to use is: $1\space mile = 1609.34 \space meters$, This particular unit conversion is one that is easily looked up online; you don't necessarily have to have it memorized.

The expression can now be written as:

\begin{align*} &\frac{25 \;\color{red}\cancel{miles}}{1\; hour}\times\frac{1609.34 \;meters}{1 \;\color{red}\cancel{miles}} \end{align*}

Notice how the miles units cancel out. Now you need to change the hour into seconds. To do this, multiply by another conversion factor that can change hours into seconds: $1\space hour = 3600\space seconds$.

\begin{align*} &\frac{25 \;\color{red}\cancel{miles}}{1\; \color{blue}\cancel{hour}}\times\frac{1609.34 \;meters}{1 \;\color{red}\cancel{miles}} \times \frac{1\;\color{blue}\cancel{hour}}{3600\;sec} \end{align*}

Now, hours can cancel each other out as well, and you are left with meters and seconds (meters in the numerator and seconds in the denominator), which is exactly what you wanted.

Now calculate to find the answer by using Method 1:

\begin{align*} 25\div 1\times 1609.34\div 1\times 1\div 3600\end{align*}

This equals 11.175 meters per second, or since you want to round up, 11.18 meters per second.

In method 2, you multiply the numerators across, then multiply the denominators across, and then simplify by dividing the numerator by the denominator to get the answer. Notice the answers using both methods are the same.

\begin{align*} \frac{25\times1609.34\times 1}{1\times 1\times 3600} = \frac{40233.5meters}{3600seconds} = \frac{11.18m}{1sec}\end{align*}

Therefore, if this horse is running at 25 miles per hour, it is also running at 11.18 meters per second.

Things to Remember

Per can also mean divided by.
Steps for speed unit conversions:
1. Start with what units you know.
2. Determine what units you want to get in the end.
3. Determine what conversion factors you should use. You will need more than one.
4. Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

Practice Problems

3. Kirk takes a ride on a train traveling 80 miles per hour (mph). Use the fact that 1 mile is approximately equal to 1609.344 meters and 1 hour is equal to 60 minutes to convert the speed of the train to meters per minute (m/min). Round to the nearest tenth.
1 mi = 1609.344 m
1 h = 60 min
(

Solution

Solution:
$\displaystyle \frac {80\text{mi}}{1\text{h}} \times \frac{1\text{h}}{60\text{min}} \times\frac {1609.344\text{m}}{1\text{mi}} = 2145.8 \frac {\text{m}}{\text{min}}$

Written Solution:

Step 1: Start with what you know on top and bottom of the fraction (numerator and denominator, respectively).

The train is traveling at a speed of 80 miles per hour, which can be written as: $ \dfrac {80\:\text {mi}}{1\:\text {hour}}$

Step 2: Determine what you want to get in the end (on top and bottom).

You want to know how fast the train is going in $ \dfrac {\text{m}} {\text{min}}$

Step 3: Determine what conversion factor(s) to use. You will need more than one (at least one for the top and one for the bottom).

You know the following:

$1\; mi = 1609.344\; m$, so either use $\dfrac{1\text{mi}}{1609.344\text{m}} $ or $\dfrac{1609.344\text{m}}{1\text{mi}}$

$1\; hour = 60\; min$, so either use $\dfrac{1\text{hour}}{60\text{min}}$ $\dfrac{60\text{min}}{1\text{hour}}$

Step 4: Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

You know that you need to change hours to minutes and miles to meters. You need to decide which conversion factors will allow you to do that.

$\displaystyle \frac{80\text{mi}}{1\text{hour}}\times\:?\:\times\:?\:=\frac{?\:\text{min}}{?\:\text{m}}$

Note that you chose the conversion factors that will enable you to cancel out the existing units and change them to the desired units. (For example, hour is in the bottom of the original fraction, so you chose the conversion factor that has hour in the top so they will cancel out, etc.)

$\displaystyle \frac{80\text{mi}}{1\text{hour}}\times\frac{1\text{hour}} {60\text{min}}\:\times\frac{1609.344\text{m}}{1\text{mi}}$

Next, cancel out miles:

$3 fractions multiplied together. 80 miles over 1 hour times 1 hour over 60 minutes times 1609.344 meters over 1 mile. The miles in the numerator of the first fraction and the denominator of the last fraction have been crossed out.$

Then cancel out hours:

$This is the same as the previous image of three fractions being multiplied together. This time the hours in the denominator of the first fraction and the hours in the numerator of the second fraction have been crossed out.$

Using the zig-zag method you make the calculations in a zig-zag pattern. Remember: Any time you move to the denominator you divide. And any time you move to the numerator, you multiply the following:

$This is the same image as the previous two images except now there are arrows indicating the zig-zag method. Going from left to right, 80 in the numerator of the first fraction is divided by the 1 in the denominator. Then multiply by the 1 in the numerator of the second fraction and divide by the 60 min in the denominator. Finally, multiply by the 1609.344 meters in the numerator of the third fraction and divide by 1 in the denominator.$

$80 \div 1 \times 1 \div 60\; min \times 1609.344\; m \div 1 = 2145.792$

So the train is traveling at a rate of $\displaystyle \frac{2145.8\:\text{m}}{1\:\text{min}} = 2145.8\frac{\text{meters}}{\text{minute}}$ when rounded to the nearest tenth.

)

6. A horse was observed galloping at a speed of 11 meters per second (m/sec). Use the following facts to convert this speed to kilometers per hour (km/h). Round to the nearest tenth.
1 km = 1000 m
1 min = 60 sec
1 hour = 60 min
(

Solution

Solution:
$ \displaystyle \frac{11\text{m}}{1\text{sec}}\times\frac{1\text{km}}{1000\text{m}}\times\frac{60\text{sec}}{1\text{min}}\times\frac{60\text{min}}{1\text{h}} = 39.6 \frac{\text{km}}{\text{h}}$

Written Solution:

Step 1: Start with what you know (on top and bottom).

The horse is galloping at a speed of 11 meters per second, which can be written as: $ \dfrac {11\:\text{m}}{1\:\text{sec}}$

Step 2: Determine what you want to get in the end (on top and bottom).

You want to know how fast the horse is going in $ \dfrac {\text{km}}{\text{h}}$

Step 3: Determine what conversion factor(s) to use. You will need more than one: at least one for the top (numerator) and one for the bottom (denominator).

You know the following:

1 km = 1000 m, so either use $\dfrac{1\text{km}}{1000\text{m}}$ or $\dfrac{1000\text{m}}{1\text{km}}$

1 min = 60 sec, so either use $\dfrac{1\text{min}}{60\text{sec}}$ or $\dfrac{60\text{sec}}{1\text{min}}$

1 hour = 60 min, so either use $\dfrac{1\text{hour}}{60\text{min}}$ or $\dfrac{60\text{min}}{1\text{hour}}$

Step 4: Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

You know that you need to change seconds to minutes to hours and meters to kilometers. You need to decide which conversion factors will allow you to do that.

$\displaystyle \frac{11\:\text{m}}{1\:\text{sec}}\:\times\:?\:\times\:?\:\times\:?\:=\frac{?\:\text{km}}{?\:\text{hour}}$

Note that you chose the conversion factors that will enable you to cancel out the existing units and change them to the desired units. (For example, seconds is in the bottom of the original fraction, so you chose the conversion factor with seconds in the top so they will cancel out, etc.)

$\displaystyle \frac{11\:\text{m}}{1\:\text{sec}}\times\frac{60\:\text{sec}}{1\:\text{min}}\times\frac{60\:\text{min}}{1\:\text{hour}}\times\frac{1\:\text{km}}{1000\:\text{m}}$

Next, cancel out seconds:

$\displaystyle \frac{11\:\text{m}}{1\:\cancel{{\color{Red} \text{sec}}}}\times\frac{60\:\cancel{{\color{Red} \text{sec}}}}{1\:\text{min}}\times\frac{60\:\text{min}}{1\:\text{hour}}\times\frac{1\:\text{km}}{1000\:\text{m}}$

Then cancel out meters:

$\displaystyle \frac{11\:\cancel{{\color{Purple} \text{m}}}}{1\:\cancel{{\color{Red} \text{sec}}}}\times\frac{60\:\cancel{{\color{Red} \text{sec}}}}{1\:\text{min}}\times\frac{60\:\text{min}}{1\:\text{hour}}\times\frac{1\:\text{km}}{1000\:\cancel{{\color{Purple} \text{m}}}}$

Then cancel minutes:

$\displaystyle \frac{11\:\cancel{{\color{Blue} \text{m}}}}{1\:\cancel{{\color{Red} \text{sec}}}}\times\frac{60\:\cancel{{\color{Red} \text{sec}}}}{1\:\cancel{{\color{Teal} \text{min}}}}\times\frac{60\:\cancel{{\color{Teal} \text{min}}}}{1\:\text{hour}}\times\frac{1\:\text{km}}{1000\:\cancel{{\color{Blue} \text{m}}}}$

Using the zig-zag method you make the calculations in a zig-zag pattern. Remember: Any time you move to the denominator you divide. And any time you move to the numerator, you multiply the following:

$Four fractions multiplied together with arrows indicating the zig-zag method. 11 in the numerator of the first fraction divided by the 1 in its denominator. Next multiply by 60 in the numerator of the second fraction and divide by the 1 in its denominator. Next, multiply by 60 in the numerator of the third fraction and divide by the 1 hour in its denominator. Finally, multiply by the 1 kilometer in the numerator of the last fraction and divide by the 1000 in its denominator.$

$11 \div 1 \times 60 \div 1 \times 60 \div 1 hour \times 1 km \div 1000 = 39.6 kilometers\; per\; hour$

So the horse is traveling at a rate of $\dfrac{39.6\:\text{km}}{\text{hr}}$ or 39.6 km/h.

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