I was trying to calculate the maximum ground distance you can see on mountains, with your elvation given.
After some simple geometry, I was able to come up with the following formula:
Let $h$ be your elevation, $d(h)$ be the maximal distance you can see, then
$d(h)=2\pi R\arccos\frac{R}{R+h}$
where R is the radius of earth. We take $R=6378100m$ as its value.
But when I plot it in excel, here's what I got: The unit for the vertical axis is km while the unit for the horizontal axis is m.
Amazingly, for $d\in(0,20000m)$ (essentially the maximum elevation you can achieve without paying millions to board a spaceship) , $d(h)$ can be approximated by
$d(h)\approx22345\sqrt{h}$
here
with a $r^{2}$ value of 1!.
Only when $h>5\cdot10^{5}m$ dose the the approximation begin to deviate away significantly.
Does any one have a explanation of this from a numerical prespective?