How to Convert Units (Dimensional Analysis)

Pop Quiz! No internet or calculators allowed!

How fast is 60 miles per hour in feet per second?

Take a few minutes to work this out on your own. Here's a hint: use fractions. 

We can write 60 MPH as (60 miles)/(1 hour). A mile = 5,280 feet, which we can write as 

(5,280 feet)/(1 mile)

This is true because 1 mile = 5,280 feet. If we divide both sides by 1 mile, we get:

 1 = (5,280 feet)/(1 mile)

which is the same as

 (5,280 feet)/(1 mile) = 1

When we make fractions like this, they will always equal 1, so we can multiply as many such fractions together as we like. Remember that multiplying anything by 1 gives whatever the other number is. 

If we multiply these fractions together, the mile units will cancel out because one is on top and the other is on the bottom. We need two more fractions to get the right time units: 

(1 hour)/(60 minutes) and (1 minute)/(60 seconds). 

Now we multiply all four fractions together like so:

(60 miles)/(1 hour)*(5,280 feet)/(1 mile)*(1 hour)/(60 minutes)*(1 minute)/(60 seconds)

When we cancel units and two of the "60", we are left with:

(5,280 feet)/(60 seconds) = 88 feet per second

When doing such problems by hand, it is written like this:

This is the way scientists, engineers, and others who get paid to do math solve such problems. It's a good thing to know.