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Q: Distance to horizon on ocean? ( Answered 5 out of 5 stars,   5 Comments )
Question  
Subject: Distance to horizon on ocean?
Category: Reference, Education and News > Education
Asked by: jbt152001-ga
List Price: $5.00
Posted: 26 Jan 2004 11:57 PST
Expires: 25 Feb 2004 11:57 PST
Question ID: 300417
When standing on a beach approximately 10 feet above sea level how far
in the distance is the horizon looking out at the ocean (assuming a
flat ocean or no waves)?  JT
Answer  
Subject: Re: Distance to horizon on ocean?
Answered By: omnivorous-ga on 26 Jan 2004 12:25 PST
Rated:5 out of 5 stars
 
JBT --

The simple answer is about 5.5 km or 3.4 miles.  The derived formula
is indicated on this web page:
Goddard Space Flight Center
"Distance to the Horizon" (Stern, Dec. 13, 2001)
http://www-istp.gsfc.nasa.gov/stargaze/Shorizon.htm

The formula is:
D = 112.88 km * h

Where:
D = distance to the horizon in kilometers
112.88 is actually the square root of the earth's diameter in km
h = height in meters

Since 10' is about 3 meters, D = 5.5 km.

Now some caveats:
*  why can you see a ship that's further offshore?  Because it's
taller, rising above the horizon -- a fact known to seafarers even in
Columbus' time (debunking the stories that Columbus' men believed in a
"flat earth").
*  atmospheric conditions can refract light over the horizon. 
Nathaniel Philbrick refers to this phenomenon called "looming" in his
book "Sea of Glory."  I've seen it on Lake Michigan and it's
apparently quite common in Antarctica, as Philbrick says maps created
by the Wilkes Expedition in 1838-1842 made some errors in judging
distance when it made the first maps of the Antarctic coast.  Here's
an explanation of looming:
SDSU
"Distance to the Horizon" (Young, 2003)
http://mintaka.sdsu.edu/GF/explain/atmos_refr/horizon.html

Best regards,

Omnivorous-GA
jbt152001-ga rated this answer:5 out of 5 stars and gave an additional tip of: $5.00
Wow!! Can it get any better than this!

Comments  
Subject: Re: Distance to horizon on ocean?
From: redhoss-ga on 26 Jan 2004 19:21 PST
 
Be sure that you add the distance from your feet to your eye to
calculate "h" if the beach elevation is actually 10 feet above sea
level.
Subject: Re: Distance to horizon on ocean?
From: jbt152001-ga on 26 Jan 2004 19:59 PST
 
Thanks for the extra comment about adding the h of our height.
Subject: Re: Distance to horizon on ocean?
From: jameswh-ga on 27 Jan 2004 11:47 PST
 
yes i think it can get any better than this...

looking at the site referred to by omnivorous, you can see that the
actual equation is:

D = 112.88 * sqrt(h)

where sqrt means 'square root'

further, h is the height in kilometers, not metres.

even further, the answer is actually 6.18km, i can't see where 5.5km
came from, whatever way you organise the units or the formula.

even further than that, 'perhaps' a more useful formula for working
out the distance is this:

D = acos(r/(r+h)) * ((2 * pi * r)/360)

where: 
acos() is the inverse cosine function
r is the radius of the earth in km (6370.9472km)
h is the height above the earth in km (0.003km in this case)
pi is about 3.142....something

this gives the actual distance from the point where you stand to the
horizon along the 'surface' of the planet, rather than in a straight
line through the air.

For this example, of 10' above the surface, the difference is
negligible, it is still 6.18km, but scales up better for example if
you were to look at the earth from the moon. the distance using the
first simple method would be about 400,000km to the horizon, but using
the second method, the distance along the surface of the earth from
the point directly below you to the horizon would be more like
10,007km which is one quarter of the earths circumference (approx).

i guess you wouldnt really gain much from using the second method, and
i cant refer you to a site that explains it better, because i just
worked it out myself.

anyway, just thought that if you were planning on throwing something
5.5km in the hope that it landed on the horizon, you might be
disappointed if it fell short.

it's 6.18km away.

and i did it for free... something wrong with me i think :)

james
Subject: Re: Distance to horizon on ocean?
From: jbt152001-ga on 27 Jan 2004 16:55 PST
 
James: Nothing wrong with a curious mind and a generous spirit...and
besides I am beginning to understand the super complexity of the
question!!  JBT
Subject: Re: Distance to horizon on ocean?
From: jameswh-ga on 28 Jan 2004 07:34 PST
 
well no, if you put it like that :)

james

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