How far can you see?

This begins an occasional series on applied mathematics. Do not disdain your boring old maths classes, though remembering up to 10th grade maths will be useful.

Today we examine a formula which tells you how far you can see on an unobstructed flat surface (land or water).

The entire derivation and formulae can be seen here:

http://en.wikipedia.org/wiki/Horizon#Distance_to_the_horizon

Actually the derivation should be within the capacity of a smart 10th grader, but here we skip straight to the results:

I) Ignoring the effect of atmospheric refraction, distance to the horizon from an observer close to the Earth’s surface is about

d \approx 3.57\sqrt{h} \,,

where d is in kilometres and h is height above ground level in metres.

Examples:

  • For an observer standing on the ground with h = 1.70 metres (5 ft 7 in) (average eye-level height), the horizon is at a distance of 4.7 kilometres (2.9 mi).
  • For an observer standing on the ground with h = 2 metres (6 ft 7 in), the horizon is at a distance of 5 kilometres (3.1 mi).
  • For an observer standing on a hill or tower of 100 metres (330 ft) in height, the horizon is at a distance of 36 kilometres (22 mi).
  • For an observer standing at the top of the Burj Khalifa (828 metres (2,717 ft) in height), the horizon is at a distance of 103 kilometres (64 mi).
  • For an observer atop Mount Everest (8,848 metres (29,029 ft) in altitude), the horizon is at a distance of 336 kilometres (209 mi).

II)  With d in miles and h in feet,

d \approx 1.22\sqrt{h} \,.

Examples, assuming no refraction:

  • For an observer on the ground with eye level at h = 5 ft 7 in (1.70 m), the horizon is at a distance of 2.9 miles (4.7 km).
  • For an observer standing on a hill or tower 100 feet (30 m) in height, the horizon is at a distance of 12.2 miles (19.6 km).
  • For an observer on the summit of Aconcagua (22,841 feet (6,962 m) in height), the sea-level horizon to the west is at a distance of 184 miles (296 km).
  • For an U-2 pilot, whilst flying at its service ceiling 70,000 feet (21,000 m), the horizon is at a distance of 324 miles (521 km)

III)  Also, if d is in nautical miles, (i.e. 6080 feet or 1.15 statute miles or 1.85 km) and h in feet, the constant factor is about 1.06, which is close enough to 1 that it is often ignored, giving:

d \approx \sqrt h

  Now let us refine this a little further. Current estimates are that an average Indian male adult is 5’5″ or 1.65 M, and that his female counterpart is 5’0″ or 1.52 M. (This may seem low, but remember that it is an average from all over the country). However the height of their eyes above the ground would be about 0.12 M less. Neglecting the thickness of footwear, the average height to be considered is 1.65-0.12 = 1.53 M for a male and 1.52-0.12 = 1.40 M for a female.

Using formula I) above, assuming there is nothing to obstruct your view:

Mr X can see for a radius of 4.42 Km and Ms Y can see for 4.22 Km.

On land, this would generally be true only on a featureless plain such as a desert. More realistically, you would be interested in knowing how far you could see from a building or a vessel at sea. Some examples are given above. More examples are:

From the top of the Qutab Minar (if you were allowed to go there) and neglecting your own height, take h = 73 M which gives the horizon at 30.5 Km. From the top of the Eiffel Tower (301 M) it will be 61.9 Km.

A more practical example would be on a small fishing boat where your eyes may be 2.5 M above the water. Then you could see for 5.6 Km.

A typical jet airliner flies at 30,000 to 35,000 feet. Let us take a round figure of 10,000 M for this. This gives a horizon of 357 Km, though it is somewhat meaningless as there will usually be clouds below it.

So far we have dealt with objects on the surface of the horizon. For objects at a height a small refinement is needed. To be continued.

Some material has been taken from Wikipedia.

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