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How far is that?
This past weekend we were in San Diego for a soccer tournament and had some time to kill so we went to the beach. As we looked out on the water we noticed a large black smoke plume on the horizon, but could see nothing else. A man was standing in the bed of his pickup with binoculars looking at it and said he could see flames, we could not. One of the group asked how far away that was and I said more than 3 miles away. They asked how did I know and I explained a thumbrule that I learned in the Navy. The rule states that the distance to the horizon (in Nautical miles) is 1.1 times the Height of Eye (in feet)
I’m 6’ tall so in round numbers the square root is 2.4 * 1.1 = 2.6NM or about 3 statute miles.
My friend just laughed and said something like, “who knows that?” I explained it briefly and then someone else walking along said he knew that the horizon was 21miles away. I didn’t want to get into an argument with the guy so I left it alone. But that made me think what is the geometry involved?
The applicable rule is the “Secant Tangent Theorem”
It says: the length of a tangent squared is equal to the length of the two secants that go to the same point.
We’ll choose a secant that is the diameter of the earth plus the height of eye (~6ft) and the second secant is just the height of eye. The formula becomes.
d2 = h * (D + h)
Where:
d = distance from the point to the tangent
= distance to the horizon
D = Earth’s diameter (7900 – 7926miles)
H = height of eye
We can simplify the formula by virtue of the fact that the diameter of the earth is much, much greater than the height of eye. (At a 100:1 ratio the error is <1%) the equation becomes:
d2 = h * (D )
take the square root:
d = √(h*D)
this is commutative, so:
d = √(h) * √(D)
D = 41.7e6 – 41.85e6 ft
Square Root = 6.47e3 = 6470
Nm = 6076ft => 6470 = 1.065 * 6076
d = √(h) [ft] * 1.065 * 6076 [ft]
d / 6076 = √(h) * 1.065
d [NM] = 1.065 * √(h) [ft] now simplify for ease of using your noggin only, no calculators
d [NM] = 1.1 * √(h) [ft]
Remember that if you’re looking at something that has some height to it, like a boat, lighthouse whatever that you have to add the two distances to the horizon. Your eyeball to the horizon and then the distance of that object to the same point on the horizon from the other direction.
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