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Dimensional Analysis and Unit Conversions

Date: 09/15/2002 at 19:47:25
From: Jenn
Subject: Dimensional analysis

Using dimensional analysis, convert 8 feet/second to kilometers/hour.

I'm not sure how to work from hours to seconds. Here's what I tried, 
but the answer doesn't make sense to me.

8ft/sec (60s/1min)(60min/1hour)(1 yard/3ft)(1mile/1760yards)
(1.609/1mile)=46339.2km/5280hours------8.78km/hours

If you could please help, that would be wonderful. Thank you.


Date: 09/16/2002 at 14:22:32
From: Doctor Ian
Subject: Re: Dimensional analysis

Hi Jenn,

The only metric length conversion I have memorized is that 2.54 cm is 
_exactly_ one inch (it's how the inch is defined), so I like to use 
that:

  8 ft    60 sec   60 min   12 in   2.54 cm     1 m       1 km
  ----- * ------ * ------ * ----- * ------- * ------ * -------
  1 sec    1 min    1 hr    1 ft      1 in    100 cm   1000 m

    8 * 60 * 60 * 12 * 2.54
  = -----------------------
         100 * 1000

  = 8.78 km/hr

which is the same thing you got.  

Actually, there are two other conversions that I've memorized, both of 
which come in quite handy: 60 mph is the same as 88 ft/sec. And there 
are about 8 kilometers in 5 miles. (The first is exact, the second is 
approximate.) 

So these can shorten the calculation a little:

             60 mph      8 kilometers
  8 ft/sec * --------- * ------------
             88 ft/sec   5 miles


    8 * 60 * 8
  = ----------
      88 * 5

    60 * 8
  = ------
    11 * 5

    12 * 8
  = ------
      11

  = 8.72 km/hr

This is less accurate, but it's easier, and in many situations it will 
be close enough. 

Note that we can use the 60/88 ratio to make a quick check of the 
result: 8 ft/sec is 1/11 of 88 ft/sec, so it's 1/11 of 60 mph, or a 
little less than 6 mph. So 8-point-something km/hr is at least in the 
right ballpark.   

The 8/5 ratio is handy when you see road signs with distances in 
kilometers instead of miles. (Some of these were put up during the 
1980's, when there was some thought that the United States might 
switch to the metric system.) 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Measurement

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