Dear Pat:
I have been brooding over the difference between our results. I think one difference is that you calculated the length of the sight line from the top of the observation tower to the horizon, whereas I calculated the distance on the ground from the foot of the observation tower to the horizon. Another difference is due to your analysis being only valid for small distances or short towers, whereas my analysis is valid for towers of any height.
Joe
Dear Joe, Thanks for your views. You took a different path but you arrived at the same point. Let me explain my approach.1 I made the same assumptions as you did. 'The above results assume that the earth is a smooth sphere without an atmosphere instead of a spheroid with mountains and an atmosphere'. 2 I drew a circle showing a radius to the point where the horizon could be seen and I labelled this 'r' for 'radius of the earth'. I drew another radius to the location of the observer and extended it by an additional height labelled 'h' for 'height of observation'. I then joined the ends of these two lines and labelled the third line of the triangle 'd' for distance to the horizon. 3 Knowing that the line of sight to the horizon is a tangent, I knew that the triangle that had been formed was a right angle triangle with the right angle at the point of the horizon. 4 I then used Pythagoras' theorem to calculate the following results: h sqrt (h) horizon distance sqrt (h) x 3.57 1 m 1 3.570 km 2 m 1.4142 5.049 km 3 m 1.7321 6.184 km 4 m 2 7.140 km 5 m 2.2361 7.983 km ......................... 10 m 3.1622 11.289 km .......................... 100 m 10 35.700 km ........................... 1 000 m 31.6228 112.893 km ............................. 10 000 m 100 357.000 km .............................. 100 000 m 316.228 1 129 km ............................ 1 000 000 m 1000 3 570 km 5 000 000 m 2236.1 7 983 km 10 000 000 m 3162.3 11 289 km We are in fairly close agreement on all of these except for your last value (for a height of 10 000 000) that I suspect is a slip of the calculator. Pat Naughtin LCAMS
-- Joseph B. Reid 17 Glebe Road West Toronto M5P 1C8 Telephone 416-486-6071
