I recently had a chance to go to a sea-shore and to also ponder the question:
"What is the farthest point on the horizon that I can theoretically see, ignoring the effects of fog, clouds, waves, tide, and non-uniform curvature of Earth?"
So, I got to work, and derived the formula thus:
Earth to be a perfect sphere of radius,
My height (at eye-level) to be,
The linear distance to the horizon to be,
$(r+h)^2 = d^2 + r^2$ (via Pythagoras theorem)
$d = (h^2 + 2rh) ^.5$
Also, the curved distance,
c, to the horizon, would be:
$c = r . arccos (r / (r+h) )$
However, here begins my difficulty. When I substitute the values...
r = 6370 km h = 180 cm (or, 6 ft)
... I get,
d = 4.58 km (reasonable, appears correct) c = 274 km (!!!)
... which doesn't make sense to me!
Why so much difference between
d? Intuitively, I am expecting
c to be very close to
d. Why? Because, the surface of Earth is more or less flat, and when dealing with a linear distance,
d, in the vicinity of merely
5 km, the curved distance,
c, should also be very close to
d. Isn't it?
Where am I wrong in my intuitive thinking (as, afaik, my formula and computation are correct)?