So far we have dealt with the distance which you can see while observing from various heights. This has a few more applications.
In general, a sensing device at various heights would also be able to pick up electromagnetic radiation up to the horizon. Even if you cannot fly over enemy territory, you can fly near the border and pick up what you can on the other side. Infra-red cameras can also tell you what lies below the clouds.
As you have seen, the famous U-2 could fly at 70,000 ft which enabled it to observe a radius of over 500 Km. Initially the Soviet forces had nothing to catch them with but by 1960 they had succeeded in shooting down one of them and capturing the pilot. Meanwhile improvements in satellite technology gradually made it easier for the big powers to spy on each other’s vast territories.
For some years India used retired Boeing 707s to spy on our not-so-friendly neighbours. Later there were Mig-25Rs which had a similar ceiling as the U-2s. Thus, if you were flying one of these near Amritsar you could see 500 Km into Pakistan, enough to cover anything of interest (though not their nuclear test sites which are north-west of Quetta).
Spying on Tibet is a bit more challenging as the plateau is at an average height of 15,000 feet. So your effective height is actually 70,000 minus 15,000 or 55,000 feet. That gives you a horizon of 460 Km, which doesn’t get you into the Chinese heartland but enough to see what they are up to in Tibet. As it is a very sparsely populated area, it is relatively easy to spot new constructions. Sooner or later they will link up with Nepal with superhighways and probably railway lines, which would be great news for railfans but not good news for our military.
One last mathematical derivation concerns the visibility of tall objects from a distance.
Objects above the horizon
To compute the greatest distance at which an observer can see the top of an object above the horizon, compute the distance to the horizon for a hypothetical observer on top of that object, and add it to the real observer’s distance to the horizon. For example, for an observer with a height of 1.70 m standing on the ground, the horizon is 4.65 km away. For a tower with a height of 100 m, the horizon distance is 35.7 km. Thus an observer on a beach can see the top of the tower as long as it is not more than 40.35 km away. Conversely, if an observer on a boat (h = 1.7 m) can just see the tops of trees on a nearby shore (h = 10 m), the trees are probably about 16 km away.
Referring to the figure at the right, the top of the lighthouse will be visible to a lookout in a crow’s nest at the top of a mast of the boat if
where DBL is in kilometres and hB and hL are in metres.
Application-say you are on a ship off the coast of Dubai and you are at a height of 10 M above sea level. The Burj Khalifa is 828 M. So you can see it at 3.57 ( sqrt (10) + sqrt (828)) which is 114 Km away.
If you are looking for Mukesh Ambani’s Antilia ( a mere 170 M high) and you were at 10 M above sea level, you could see it at 3.57 (sqrt (10) + sqrt (170)) which is 58 km away.
To take something more topical, let us say a burning fishing boat in the Arabian sea has flames which are 10 M high. And your own small boat enables you to observe from 5 M high. Then it is easy to see that you will be able to see it only if you are less than 20 Km away (and less if the weather is bad, but then you should be able to see bright flames through rainy conditions). The point is, if you were more than 20 Km away you would not be able to see it. So it is not as if everyone in the Arabian sea could observe it.
Think of this when you see the news, particularly when military matters are being discussed. Also note that Wikipedia and other websites will give you a quick guide to even super-secret organizations such as RAW and the even less known ARC (Aviation Research Centre). Of course, you cannot expect as much transparency as in the US where the CIA comes for campus recruitment.
All the mathematics you need here is given in http://en.wikipedia.org/wiki/Horizon