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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

```

```
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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