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Re: Galileo's Balls (fwd)
------- start of forwarded message -------
From: [email protected]_spamfuscated (Steve Caskey)
Newsgroups: alt.folklore.urban
Subject: Re: Galileo's Balls ( Re: Gould, "The Paradox of the Visibly Irrelevant")
Date: Wed, 17 Dec 97 14:09:54 +1200
In Article <[email protected]>
[email protected] (Perry Friedman) writes:
>In article <[email protected]>,
>Ben Walsh <[email protected]'reirish.com> wrote:
>>The earth is round, but it's also very big. The curvature of the earth
>>over a couple of hundred metres isn't really significant.
>
>After 1/4 mile the earth has dropped away about 1/2 inch in my quick
>backof the napkin estimations, not an insignificant amount.
As a Junior Acolyte of the Church of the First Approximation, I will now
invoke the Doctrine of Facetious Accuracy to take a look at the
implications of this train of thought.
Assumptions: - four significant digits will suffice. (Actually, the
Church of the First Approximation considers this two and
a half digits too many, unless proselytising.)
- the bullet is fired from a height of 1.700 metres by a
person of typical height with a supernatural ability to
hold a rifle level.
- the bullet travels with a velocity of exactly 400.0
metres per second, or about 1,312 feet per second.
That's slightly supersonic, which is irrelevant
because...
- the Earth is a featureless perfect sphere with no
atmosphere, no rotation, and no massive bodies such as
the sun or moon in its vicinity. Its radius is 6,371,000
metres exactly. Any resemblance between the planet
depicted in this posting and any planetary body inhabited
by readers of this posting is unintentional.
Conversions: - 1 inch = 25.40mm, therefore...
- 1/4 mile = 402.3 metres, and...
- 1/2 inch = 0.01270 metres.
Constants: - g = 9.807 metres per second squared
*****
First, Perry's back of the napkin estimate is coincidentally spot on
given these assumptions, but after all this is the Doctrine of Facetious
Accuracy we are working with, and this sort of thing tends to happen.
Abusing Pythagoras and taking away the radius of the earth afterwards,
we get
_____________________
| 2 2
/\ | 6,371,000 + 402.3 - 6,371,000 = 0.001270 metres above the earth's
\| surface
*****
However, we're pretending a real person has fired a real gun. So how
long does the dropped bullet take to hit the ground?
_____ _______
1 2 | 2 d | 3.400
d = - g t => t = /\ | --- = /\ | ----- = 0.5888 seconds
2 \| g \| 9.807
*****
In that 0.5888 seconds the fired bullet has travelled 235.5 metres. A
tangent to the earth's surface for 235.5 metres will end up at
_____________________
| 2 2
/\ | 6,371,000 + 235.5 - 6,371,000 = 0.004353 metres above the earth's
\| surface, or just over 1/6 inch.
*****
Once again,
_____ _________________
| 2 d | 3.400 + 0.008706
t = /\ | --- = /\ | ---------------- = 0.5896 seconds
\| g \| 9.807
The time taken to fall that extra distance is about 0.0008 seconds, or
1/1250th sec.
*****
In conclusion, the fired bullet falls less than 0.26% further, and takes
just over 0.13% longer to hit the ground, than the dropped bullet.
Whether you consider this "significant" or not depends on your sense of
proportion.
It's a shame having an atmosphere, tides, and all that shit screws
things up so much.
*****
Oh yes, I forgot one more assumption: the earth's gravitational field is
linear, not inverse square. If anyone wants me to factor in the value
of g for a point 1.7 metres above the earth's surface, they can fuck
off.
Steve "another triumph of Bastard Science" Caskey
P.S. "Bastard Science" and "the Doctrine of Facetious Accuracy" are
registered trade marks of the Church of the First Approximation. Any
illegal use of these terms will result in people laughing at you.
P.P.S. All abuses of scientific method and sound mathematical practice
in the above posting are intentional, and made with tongue in cheek, but
the basic conclusions should be fundamentally correct. That's what
Bastard Science is all about.
--
Just another mindless public servant at the Ministry of Education
"Did you know that a uniform attracts women like flies?"
"I wondered why all your women looked like flies."
* See the alt.folklore.urban FAQ and archive at http://www.urbanlegends.com
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