Your line of sight is a tangent line from 500 m above the circle of the Earth. This tangent line is perpendicular to a radius of the Earth. The line from the center of the circle to a position 500 m above the circle is the hypotenuse of a right triangle with the radius and the tangent line as its sides.
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The Earth's radius is about 6400. The best way to understand this problem is to draw the problem!!! I used a 6 inch protractor to draw a semi-circle with its diameter horizontal. I drew a vertical line from the center of the semi-circle to a point ½ inch above the top edge of the semi-circle.
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How far out can you see-that is, how far is your horizon? The best way to understand this problem is to draw the problem!!! I used a 6 inch protractor to draw a semi-circle with its diameter horizontal. I drew a vertical line from the center of the semi-circle to a point ½ inch above the top edge of the semi-circle.
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Note that for E=0 (eyes at sea level) the observer is at the horizon, and for someone who is 6 feet tall the horizon is 3 miles away. That same six-footer will continue to see a 40-foot tall boat until it’s about 11 miles away. The foregoing assumes a perfectly calm ocean.
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Theoretically, we consider the Earth as a sphere, which is not, of radius Re. The observer O is located at a height Y above the ground. The distance D₁ between the observer and the point H on the horizon is: ……………D₁ = Y * √[1 + 2*Re/Y]
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For example, the phrase beyond the horizon is often used in the context of expressing a vast distance. For a person standing at sea level, however, the horizon is only about 3 miles away. Let’s do the calculations.
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Now that we’re thinking about horizons, here’s another cool thing to think about. The horizon is three miles away on earth. But elsewhere in the solar system it’s closer or farther, depending on the size of the body you’re standing on! Take the moon. The moon is two thousand, one hundred and sixty miles in diameter, only about one quarter the size of the earth. So the curvature at its surface is much more noticeable, and the horizon is much closer. It’s only about 1.5 miles away.
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A straight line drawn from the observer’s eyes tangent to the sea’s surface touches it at the distance d1, which is the distance to the horizon. If R is the radius of the earth, then, to first order in E/R, it is easy to show that.
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D₁ = Y * √[1 + 2*Re/Y] = 50 km * √[1 + 2*6356.8 km/50 km] = 799 km. D₂ = Re * acos (Re/(Re+Y)) =. = 6356.8 km * acos (6356.8 km /(6356.8 km +50 km)) = 795 km. Point “H” is the junction point between the tangent line of sight OH and the terrestrial surface.
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Those sailors should have known better. After all, they must have been aware that when a ship sails away it gradually sinks below the horizon, and when it returns it gradually rises out of the ocean. A strong clue that the earth’s surface is curved.
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how far is earth horizon
The Earth's horizon is three miles far. Earth's horizon is three miles far.