Algebra: Converting From One Unit to Another

In the above video, astronaut Doug Wheelock explains that traveling in orbit requires a speed of 5 miles per second.  In my previous post, we determined that an object needs to travel at approximately 8 km/s, or kilometers per second.  How do we convert kilometers to miles to make sure that our calculations agree with those of an astronaut?

This is something that you can type into google to get the answer, but a teacher (or an exam) may require that you show your work.  So let's get down!


Converting from one distance to another


When you convert from one unit to another, you may have to start with a basic conversion and work your way outward. For kilometers to miles, let's start with a basic conversion that everyone was taught in high school: 1 inch is equal to 2.54 centimeters.  To properly use this, we need to convert out kilometers into meters and our inches into miles. We can then proceed as follows:

  • There are 1,000 meters in a kilometer (the prefix "kilo-" means one thousand)
  • There are 100 centimeters in a meter (the prefix "centi" means one hundred, as in century)
  • There are 2.54 centimeter in an inch
  • There are 12 inches in a foot
  • There are 5,280 feet in a mile

We can use these facts to convert our value of 8 km/s by multiplying by each of the ratios given above. Note that we must take care that corresponding units are written diagonally from each other.  

I color coded the units to emphasize that they must line up diagonally next to each other. This means that some conversion numbers will be in a numerator while others are in the denominator.  For example, I need the km in the denominator of the second ratio to cancel out the km in the numerator in the first ratio.

Converting from one time to another


We can use the same idea to prove Wheelock's assertion that astronauts see approximately 16 sunrises and sunsets each day. This would mean that the ISS makes a complete revolution around the Earth approximately 16 times a day. Additionally, we need to use the following ratios:

  • There are 60 seconds in an minute
  • There are 60 minutes in an hour
  • There are 24 hours in a day

Finally, we need to use a defined value for the distance around the Earth, also known as the circumference.  This value is defined as 24,901 miles.  So our final ratio is listed below.

  • There is one revolution around the Earth every 24,091 miles

We will apply the same math as earlier, making sure that the numerator and denominators which are diagonal to each other have the same units so that they can cancel out.

According to our calculations, we can conclude that orbiting at 5 mi does cause you circle the planet approximately 16 times in one day.