Pat Naughton wrote in USMA 24669:
From the easily remembered 1357 we have established the basic rule forfinding the distance to the horizon and we can use it for any other heights. Let's (for ease of calculation) consider 4 metres.From a height of 4 m the horizon is about 7 kilometres (sqrt (4) x 3.57).But knowing this requires that we remember the 'constant' 3.57 that I think is readily remembered as part of the sequence 1357. Cheers, Pat Naughtin LCAMS
This intrigued me. The formyla that I came up with was
a = arccosine (r/h+r),
where a = angle subtended by horizon distance at earth center
r = radius of the earth
h = height of observation
This formula led to the following results:
observation height horizon distance 3.57 X sqrt (height)
1 m 3.570 km 3.570 km
2 m 5.048 km 5.049 km
3 m 6.182 km 6.183 km
4 m 7.139 km 7.140 km
5 m 7.982 km 7.983 km
.........................
10 m 11.288 km 11.289 km
..........................
100 m 35.696 km 35.700 km
...........................
1 000 m 112.875 km 112.893 km
.............................
10 000 m 356.732 km 357.000 km
..............................
100 000 m 1 121 km 1 129 km
............................
1 000 000 m 3 357 km 3 570 km
5 000 000 m 6 218 km 7 983 km
10 000 000 m 7 461 km 11 289 km
The above results assume that the earth is a smooth sphere without an atmosphere instead of a spheroid with mountains and an atmosphere.
--
Joseph B. Reid
17 Glebe Road West
Toronto M5P 1C8 Telephone 416-486-6071
