Horizon distance and observers altitude problem

[trina1990]

Here's a problem from an astronomy book
"the top of the mountain 1000 m in height can just be visible seen from a ship approaching the land where the mountain is situated. . If the observer's eye is 30 m above sea level. Then how far the ship is from the mountain? "

The main problem is i don't know to measure the horizon distance from a mountain top when the observer itself has some altitude. . Plz suggest me what correction should i need here in the formula of horizon distance "root over 2RH " to determine this distance?

 

[BobG]

You essentially have two mountaintops - one 1000 meters high and one 30 meters high.

Find the distance to the horizon for each and add them together.

The concept is that your local horizontal is perpendicular to the Earth's radius. If you look along that local horizontal, you can see the top of the mountain because the sum of it's radius and elevation are high enough to reach that local horizontal. You have a right triangle consisting of the (Earth radius + mountain elevation) as the hypotenuse and the Earth's radius as the adjacent side. Just using the Pythagorean theorem you could get a pretty close answer for the distance to the horizon. A better answer is taking the arccos of the adjacent/hyp and multiplying by the Earth's radius (the angle from the arcos has to be in radians for this to work).

Since your elevation is 30 m high, your elevation + the Earth's radius forms a hypotenuse of a second triangle, etc. Both you and the mountain top lie on the local horizon, allowing you to just barely see the mountaintop "over" the Earth's curvature that lies between you.

I don't know what the letters in your formula stand for, since I don't know what book you have, but it's probably a shortcut for getting the approximate distance and the same principle almost certainly applies.

 

[trina1990]

Yes, i tried with this. . Thanks a lot for helping

 

[Redbelly98]

Yes, i tried with this. . Thanks a lot for helping

Can you show your calculation? Maybe there's just a arithmetic error somewhere, but we can't help with what went wrong unless we see the details of what you did.

 

[rwisz]

I would suggest using the formula for calculating the horizon distance, which takes into account the observer's altitude. This formula is given as d = √(2Rh + h^2), where d is the horizon distance, R is the radius of the Earth, and h is the observer's altitude. In this case, the mountain's height of 1000 m can be substituted for h, and the observer's altitude of 30 m can be added to the formula. This will give a more accurate calculation of the distance between the ship and the mountain. Additionally, it is important to note that atmospheric conditions and visibility can also affect the accuracy of this calculation. Therefore, it is important to consider these factors when using this formula to determine the horizon distance.