Chapter 1: Introduction
1.4 Conversions
Authors: William Moebs, Samuel Ling, Jeff Sanny
Adapted by: Rob Pryce, Alix Blacklin
Learning Objectives
By the end of this section, you will be able to:
- Use conversions to express the value of a given quantity in different units.
- Derive conversion factors to quickly convert between commonly used units of measure.
It is often necessary to convert from one unit to another. For example, in the field of biomechanics we often refer to velocity in units of m/s, however most people are more familiar with units of km/hr or even miles per hour. 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, meters, or kilometers 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 just 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. Refer to the Appendix for a more complete list of conversion factors. In this case, we know that there are 1000 m in 1 km. Now we can set up our unit conversion. The last step is to write the units we have and then multiply them by the conversion factor so the units cancel out, as shown:
[latex]80m \times \frac{1km}{1000m} = 0.080 km[/latex]
Note that the unwanted meter unit cancels, leaving only the desired kilometer unit. Another way of looking at the conversion factor is that it is just a fraction equal to 1, and therefore can be used to change the units, but will keep the overall magnitude intact. That is, multiplying any number by 1 will not change the value, just the units.
This process works on any conversion, as long as you write the conversion factor correctly and ensure the correct units cancel out. For example, if you had written the conversion factor as 1000m/1km, the units would not cancel properly:
[latex]80 m \times \frac{1000m}{1km} = 80000m^2/km[/latex]
This still follows the correct math principles, and even represents the same value, but the units are nonsensical.
Using the same example, if you were to convert from 0.08 km back to metres, you would write the conversion as follows:
[latex] 0.08 km \times \frac{1000m}{1km} = 80m[/latex]
Notice in this case the conversion factor 1000m/1km was correct (because we were converting from km to m and needed to cancel the km).
Now, these conversions like these can also easily be done using the SI prefixes (adding and subtracting exponents and/or moving decimal places), however the conversion factor method works to to convert between any type of unit. For example when converting between units that are not not metric or when converting between derived units (e.g., m/s) as the following examples illustrate.
Example 1.2
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:
[latex]\text{Average speed = Distance / Time}[/latex]
- Substitute the given values for distance and time:
[latex]\text {Average speed} = \frac {10mi}{20min} = 0.50 mi/min[/latex]
- Convert miles per minute to meters per second by multiplying by the conversion factor that cancels miles and leave meters. This will leave the average speed in m/min:
[latex] \frac {0.50 mi}{min} \times \frac {1609 m}{1 mi} = \frac{0.50}{1609} = 804.5 m/min.[/latex]
- Then multiply by the conversion factor that cancels minutes and leave seconds:
[latex]\frac{804.5 m}{min} \times \frac{1 min}{60 sec} = 13 m/s[/latex]
Discussion
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.
note 1: you could also do step 3 and 4 all at once.
note 2: the order of converting doesn’t matter – you could multiply by 1min/60sec first and then 1609 m/1mile and get the same answer
Check your Understanding
Competitive swimming pools come in three different lengths: 25 yards, 25m and 50m.
(a)If a swimmer takes 15 seconds to swim a 25 yard pool, what is their average speed in m/s?
(b) How much time would it take them to swim a lap in a 25m or 50m pool? (assuming no change in their average speed)
Example 1.3
Converting between Metric Units
The Tour de France is one of the most famous bicycle races in the world, where riders compete in over 20 days of racing across France and nearby countries. In 2021 the race was won by Tadej Pogacar (SLO) with an average speed of 41.2 km/hr across a distance of 3,414 km. Convert this speed to m/s..
Strategy
We need to convert kilometers to meters and hours to seconds. The conversion factors we need are 1km = 1000m and 1 hour = 3600 seconds.
Discussion
[latex]\frac{41.2 km}{hr} x \frac {1000m}{1km} = \frac {41,200 m}{hr} x \frac{1hr}{3600 sec} = 11.4 m/s[/latex]
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.
Example 1.4
Deriving conversion factors
Sometimes it can also be useful to calculate a ‘simple’ conversion factor that can be used to quickly change between different frequently used units.
One such example is the conversion between m/s and km/hr above. While most speed data in biomechanics will be given in units of m/s we often want to convert to units of km/hr – either when working with the general public or for our own reference. For instance, if someone said they were running at a speed of 10.4 m/s does that seem fast or slow? If we know how to quickly convert between m/s to km/hr we could easily answer that question.
Instead of having to first convert the meters to kilometers and then the seconds to hours, it would be easier if we knew just one number that could multiply (or divide) the starting number (10.4 m/s) to convert to km/hr in one step.
Strategy
In order to derive such a ‘simple’ conversion factor we can perform the usual conversion factor process above, but start with a value of ‘one’ or 1 m/s.
This would look like:
[latex]\frac{1m}{s} \times \frac{3600s}{1hr} \times \frac{1km}{1000m} = 3.6 km/hr[/latex]
This means that there 1 m/s is equal to 3.6 km/hr, and we can now multiply any speed in m/s by the ‘simple’ conversion factor of 3.6 to quickly convert to km/hr.
Similarly, if we want to convert from km/hr to m/s then we can divide by 3.6. Or, we could calculate a conversion factor that would allow us to multiply km/hr to m/s instead of divide by starting with 1 km/hr and converting to m/s. That number is 0.28. Can you solve for it yourself?
Solution
10.4 m/s x 3.6 = 37.4 km/hr
That seems like a very fast running speed – in fact, 37.4 km/hr is the average speed of the world record for the 100m dash (9.58 seconds set by Usain Bolt at the 2009 World Championships in Berlin. So, if we happen to calculate an average running speed of 10.4 m/s we should make sure it makes sense for the circumstances – hopefully you are dealing with elite level track runners otherwise you might want to check for an error in your calculation and/or an error in the measurement technique.
Discussion
We can also use these ‘simple’ conversion factors to do quick ‘mental’ estimates. For instance, we could round down 10.4 m/s to 10 m/s (the 0.4 is a relatively small amount of the overall value) and do the simple estimate in our head of 10 x 3.6 = 36 km/hr. This is very close to the exact value and certainly enough for us to gain a rough idea of the speeds involved.