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##### Flat Earth Debate / Clouds vs Mountains
« on: May 07, 2013, 01:55:11 PM »
WARNING: The following post contains basic trigonometry!

Let's start from two basic assumptions:
1) Distant mountains are not visible beyond ~100km because the air is not transparent enough, i.e. it becomes hazy.
2) When there's a full cloud cover, it appears to touch the horizon because of perspective, but in reality it's two parallel planes.

Let's calculate how far away the line of clouds is, which appears to touch the horizon due to perspective. According to Wikipedia, the angular resolution of the naked eye is about one arcminute [http://en.wikipedia.org/wiki/Naked_eye]. Objects that fall within this angle on your visual field cannot be told apart and appear as one. It can vary among individuals, but let's take this as the value (it cannot vary too dramatically anyway). Let's also assume a cloud cover at a height of 10km, for simplicity. NOTE that you can do the following simple calculation using your own values for angular resolution and cloud cover height, just to compare.

Cue epic MS Paint skills:

To get the distance d between the observer and the cloud cover at the point it 'touches' the horizon due to perspective, we must divide the cloud cover height by the sine of the resolution angle.

d = 10km/(sin(1')) = 10km/0.00029 = 34483 km

We get that the point where the cloud cover meets the horizon is almost 34500 km from the viewer. It's basically outside the 'habitable world'. Yet your eyes can clearly see it through all that air. We can deduce that assumptions 1 and 2 can not be true at the same time.

Addendum: assumption 1 is, by itself, false, proven by simple observation. You can see the moon rising on the horizon, yet its light travels much further through the same layers of air that would stand between you and a mountain.

Just for funzies, here is the same calculation for cloud cover distance at the point it touches the visible horizon, on a globe:

Taking Ro (Earth radius) as 6371 km, we can apply the Pythagorean theorem:

d2 = 63812 - 63712 (in km of course)
d = 357 km

This is the distance that'd be between you and the cloud cover on the horizon. Unless there's a mountain in the way

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