First we have to determine what you think you to see and what you see in fact.
You think you see the ship on a straight line. whereas it actually moves on a curve starting from the tip of the foot and extending to the skyline.
The eye has a detection sensitivity. we cannot see after a certain sensitivity. Let's do a test to measure this sensitivity.
Get remember this page:
we are unable to read the most down words on this paper. however, this is not due to the sensitivity of the eye, but the sensitivity of the computer screen. To measure it yourself, move away from an A4 sheet with a lettering and try to read it. then measure the distance from which you can no longer read the text. you can measure the sensitivity of your eye using the relation between this distance and the font size.
I did this experiment for you. I have opened a book. The words on it was 2 milimetres. I have began to move away from the book. When I came to a distance of 3 meters, I realized that I could not read the articles anymore.
Lets calculate sensitivity of my, an average human eye:
Distance to paper: 3 metres.
Highness of word: 2 mm.
Sensitivity of eye: 2mm. / (3m. x 1000) = 0,0007.
This value is also the distance we can no longer see on the skyline of an object approaching the skyline. What we need to do for this is to correct the skyline angle which should be 90 degrees with this value.
90 degrees - 90*0,0007 =
89,94 ° (for wave)
90 degrees + 90*0,0007 =
90,06 ° (for ship)
We understand that we cannot see the sensitivity of 0.06 degrees. with it, we can compute a common angle where we cannot distinguish between wave and ship. this will increase both values. in other words, we will assume that we can now see a distance from which we cannot notice two objects.
Corrected values. We can not distinguish wave and skyline whenever they are in angularly:
89,97 ° (for wave)
90,03 ° (for ship)
in other words, when an object reaches a horizontal angle of 89.94°, we now see it contiguous with the skyline.
Lets calculate distances for wave and ship seperately where we see them adjacent to skyline:
Calculating distance we see 1 metre wave adjacent to skylineα + β= arctan (L/2) + arctan (1/L) = 89,97°
>> L ~= 2000 metres.
After 2000 meters, we can no longer distinguish waves from other objects in the skyline. This value can be calculated differently according to your eye sensitivity.
Calculating distance we see 5 metres ship adjacent to skylineα + β= arctan (L/2) + arctan (1/L) = 90,03°
>> L ~= 4000 metres.
If we can not see this ship has 5 meter highness after
4 kms anymore because of we can not to distinguish it with a 1 meter wave in 2 kms distance.
Since our horizontal vision sensitivity has disappeared earlier, we now see both objects horizontally, so the ship begins to disappear behind the wave.
If you want to see the ship, even so you can use a camera has zoom property. but this is limited by the sensitivity of the camera. no matter how powerful a camera you have, the sensitivity will decrease at some point and you will see the ship disappearing behind the waves. this is not because the world is spherical, but because your angle of vision is limited to the tangent function and your sensitivity to observation is limited. As shown in the example, a tool can help you increase your visual sight. in this case, you can see that the object that just disappeared behind the waves is still in place.
in short, seeing a distant object is about property of seeing, it has nothing to do with the fact that the object is really visible.