Lighthouses Work
lighthouses work
How tall does a lighthouse need to be to be visible to a vessel (25km out to sea)?
(earth radius = 6370 km)
my mind is not working right now (a tad bit hungover)
i know it has something to do with trig. otherwise the radius would not have been given..
any help would be fab...
cheers
The curve distance to the vessel is s = r θ = 25 km with r = 6370 km
=> θ = s/r = 25/6370 (radians)
Since θ is so small, we can use the approximations sin θ ~ θ , cos θ ~ √(1-θ²) for θ in radians
The apparent height of the sea in between the lighthouse and vessel is thus
r (1 - cos (θ/2))
and (if we assume the observer in vessel is at sea level, which is a little pessimistic) drawing a straight line from observer through the sea, the height x of the lighthouse would need to be:
x = 2r (1 - cos (θ/2))
≈ 2r (1 - cos (θ/2))
≈ 2r (1 - √(1-(θ/2)²) )
by binomial approximation
x ≈ 2r (1 - (1- 2θ/2) )
x ≈ 2r θ
x ≈ 2r s/r = 2s (50km tall, seems hungover to me too...)
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Like a Lighthouse, True Love Shines for Everyone
True love--that is, unconditional love--is such a puzzle for so many of us. We give it names like agape or grace, and think it is reserved only for a certain few or that it is so rare that we have little hope of ever finding it.
Yet true love displays many of the same characteristics as the lighthouse, which is neither rare nor all that hard to locate.
Indeed, the lighthouse nearly always stands on an elevated promontory, deliberately exposed to all on sea and land. Everyone is supposed to see (and hear) the lighthouse; otherwise it could not fulfill its purpose.
The mission of the lighthouse is to guide us through perils. Night and day, in any kind of weather, its steady light rotates in every direction--360 degrees--illuminating the safe harbor to travelers on sea and land. When fog shrouds its beacon, the lighthouse still shines and adds a loud horn to make sure all those within its range get the message.
Notice a few things about the lighthouse. First, its beam falls on everyone alike. Pilgrim or pirate, saint or sinner, it does not concern the lighthouse. Its rays are available equally to all, without any kind of condition or limitation. The only step they need take is to keep open their eyes (and sometimes ears). It is always their choice.
Second, the lighthouse does not demand that travelers pay heed to its warning. It does not uproot itself, wade out into the ocean, take a ship by the prow, and insist that the vessel steer toward safety. The lighthouse, by its very nature, sets the example of the "tough love" that respects others' free will to follow their own spiritual paths.
Even if a million ships pass by without attending to the lighthouse, it does not become discouraged or start to think that there is no value to its purpose. It keeps shining its light because eventually, someone will notice and will avoid danger thanks to its illumination. That is true faith.
Third, the most brilliant part of its own light falls on the lighthouse itself, demonstrating that light--and love--are a two-way street, to the benefit of both the recipient and the giver. This is perhaps the hardest lesson of the lighthouse—that giving unconditional love (light) to others is not meant to be a painful sacrifice. We deserve to receive as much love as we lavish on others.
So what does the lighthouse teach us about true love? That we all deserve it, no matter who we are or what we have done. That we cannot force our love/light on others because doing so violates their free will. As puzzling as it may seem, others have the right to reject our light/love. And, last but by no means least, that we merit our light/love as much as anyone else to whom we give it.
These are deceptively simple lessons. Simple in this case should not be equated with easy. Most of us have a tough time understanding these lighthouse teachings at even the most basic level. We hoard our love and refuse to give it to certain people or groups. We foolishly rush in where angels (lighthouses) fear to tread because we lack faith in others' free will and thus are certain we know best for them. Or we try desperately to give love while insisting that we are not worthy of it, and giving then hurts because our own cup of love is bone dry.
Yet true love--lighthouse love, unconditional love--is the simplest of all love. Perhaps that is precisely why we don't get it. We're so accustomed to complexity that we discount the validity of anything so utterly simple.
Since complexity doesn't seem to be working for us, why not give simplicity a go? Let's simplify our love and look to the lighthouse as the example of how true love really works for others--and for us.
About the Author
Candace (C.L.) Talmadge is the author of the epic fantasy Green Stone of Healing(R) series and a political columnist syndicated by North Star Writers Group. As StoneScribe, she blogs about the intersection of politics and spirituality.
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