Calculating Horizon as a Function of Altitude
If I am in an airplane flying at 37,000 ft. and I look out
my window, how far in the distance (land) can I see?
Roughly 240 miles.
To get this answer, draw a circle, representing the Earth, and define the
following three points:
A - where the plane is, at height h=37000 ft. above the Earth's surface,
and at distance R+h from the center of the Earth. (R is the Earth's
radius. R ~ 4000 mi. ~ 21,000,000 ft.)
B - the horizon as seen from point A
C - the center of the Earth
The line AB is tangent to the circle, so it must make a right angle with
the line BC. This makes ABC a right triangle, so (AB)^2 + (BC)^2 = (AC)^2
(AB) is the distance we want; let's call it 'd'. (BC) is just the Earth's
radius, R. (AC) is R+h.
In these terms, the equation is
d^2 + R^2 = (R+h)^2
d^2 = (R+h)^2 - R^2 = h^2 + 2hR, which we can approximate as 2hR, since h
is small compared to R.
So d ~ sqrt(2hR)
Assuming the earth is a perfect sphere with radius about 3950 miles
and that you can see to the horizon, you should be able to see about
16.6 miles. The idea is that the line segments joining the center of
the earth to you in the plane and to the horizon along with the
segment from you to the horizon form a right triangle with the
segment from you to the center being the hypotenuse. Let h be your
altitude, r the radius of the earth and x the distance from you to
the horizon, x^2 + r^2 = (h + r)^2. Multiplying out, yields x =
sqrt(h^2 + 2hr).
Substituting (3950*5280ft for r and 37000ft for h ) yields x = 1242860
feet or about 235 miles, approximately.
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Update: June 2012