DISTANCE between the beach and HORIZON// how do you do it?

Question

please do not give me any answer i can search for myself on the internet. clever one

Answer 1

first answer was close to right.

there is one qualification that seems need.
(do this all in meters for easyness of calculation)

R = radius of earth = 6400 km = 6.4 * 10^6 meters
d = distance to horizon in meters
h = height in meters

Similar triangles:

R / d = d / 2h

d^2 = 2Rh

d = sqrt (2Rh)

.... works pretty well for small "h"

eg 2 meters height =>

d = sqrt (2 * 6.4 * 10^6 * 2 ) = sqrt (25.6 * 10^6)
=5.06 km

[[ online calculator gives 5.5 km ]

mine's a LOT easier to use ...

I use "2h" instead of "h" for similar triangles
because of discrepancy of right angle at the
radius from eearth-center to observer and
r-a from earth-center to distant object.
[intersection happens "h" below earth surface]

that sounds confusing but I can't draw a picture

my eqn is easiest possible tho ..

Answer 2

Horizon is the line where you see earth and sky meeting. Simple geometrical considerations will tell you that the distance you see between you and horizon actually depends on your height. Consider earth as circle with radius R=6400 km, your height about h=2m, you can show that the length of the arc between you and the point where the tangent from your head meets the earth is R xSec-1(1+h/R), where Sec-1 is the sec inverse function. You will see the horizon about 290 km away.

Answer 3

Imagine a circle with one radius drawn and slightly extended. From its end, draw a tangent to the circle. The length of the tangent is the distance to the horizon, and the amount of the radius extension is the height of your eyeball above sea level. You can show by geometry that the horizon distance is the mean proportional between the distance of your eyeball above sea level and the distance from it to the center of the earth.