Horizon

Hagar

I live on a hill over looking the sea. I would love to know how far away the horizon is on a clear day. My viewpoint according to Google Earth is 683m above sea level. For the really pedantic I'm 178cm tall.

Could somebody work out for me how far away the horizon is? Or even tell me how to go about doing it myself. It's a long time since I left school so my math is fairly rusty. Thanks.

Pherekydes

Draw a circle. Draw a line from the centre through the circle and extend it a small distance (the 684.78m). Draw a tangent from the mountain peak to the circle. Join the point of tangency to the centre. You now have a right-angled triangle. Apply Pythagoras' theorem...

smiles

I'm not sure... these all seem to have diff approaches... too lazy to figure it out!

http://www.nctm.org/high/asolutions.asp?ID=346
http://www.boatsafe.com/kids/distance.htm
http://science.howstuffworks.com/question198.htm

Or maybe they're all in diff. measurements... soz!


Cant seem to follow your explanation Slow Coach, explain it a bit more? I'm on a stupid day.

Hagar

I just came back for another black marker.
Does Slow Coach have any idea how big this circle is...

breadmonkey

Earth's radius (R) = 6,378,135m (from Wikipedia)
http://en.wikipedia.org/wiki/Earth_radius

Using the method Slow Coach proposed:

Height above see level = 683m
Height of Hagar = 1.78m;)

Distance from centre of the Earth to Hagar's gaf
T = 6,378,819.78m (hypotheneuse)

R = 6,378,135m (adjacent)
Distance to horizon = h

Pythagoras:

T^2 = h^2 + R^2
h^2 = (6,378,819.78)^2 - (6,378,135)^2
h^2 = 8.735*10^9m
h = 93,465m

So distance to the horizon is approximately 93km

I hope I didn't mess that up, that would be embarrassing!

Hagar

Thanks for the help guys.

Now just one last thing I don't think I can actually see a 93kms horizon. The ships on the horizon are too big to be that far away.Have you taken the curvature of the Earth into account? I think I can only see about 8 to kms away.

Before anyone says anything, yes I can see the Sun if I look up during the day.:p

Capt'n Midnight

From Fix It Friday (Ray D'Arcy etc.) FIXED page 177
The distance you can see to the horizon is the square root of the product of the elevation of they eye and the diameter of the earth. Thus if the elevation of the eye is 1.75 meters and the diameter of the earth is 12,714,000 meters, you can see for a distance of approximately 4,717 meters

Sqrt( 6,378,135 m * 2 * 684.78 m = 8,735,238,570.6 m2 ) = 93,462

breadmonkey's equations overstate the distance you can see by a massive 3m !
This I trust will explain the apparant size of the ships.

Hagar

I see

Squirrel

Well, also depends on where in the earth, with the world not being perfectly spherical, and also the air temperature which can cause refraction which would change the measurement. This would only change the distance by a few metres, maybe up to a kilometre.

Just to be really pedantic