A and B are points on the parallel of latitude 58.4N their longitudes being 148W and 32E respectively.What is their distance apart (a)along the parallel of latitude (b)along a great circle
Well, I will do this in nautical miles since that is my thing. You have to correct if you are using feet or land miles or meters or kilometers.
Now around the earth at the equator is360 degrees which is 360 * 60 = 21600 nautical miles but due to a brilliant decree by Napoleon Bonaparte a minute of arc is a nautical mile around a great circle so it is 21600 nautical miles around the world at the equator.
now what is a minute of longitude at 58.4 North ?
draw a horizontal circle around at 58.4 north. What is its radius?
r = Rearth cos 58.4 = .524 Rearrth
so
a minute of arc on that horizontal circle is .524 Nautical miles
and 148W and 32E = 180 degrees or halfway around
so
.524 * 180 * 60 = 5659 nautical miles
Now
ON a Great Circle we still go 180 degrees in longitude but what angle do we subtend at the center?
to do this 180 on a great circle, constant radius, we go right over the North pole. Our angle will be 2 (90 - 58.4) = 63.2
so our distance will be great circle circumference *(63.2/360) = 21600*.1756
= 3792 nautical miles (ALWAYS take the great circle route if possible, trip over North Pole becoming more feasible for your ship)
Now in case your class uses other units instead of navigation units:
1 nautical mile = 1.1507794480235 statute (American Land) miles
or 1.852 Kilometers
By the way this is why a ship between England and New York is likely to go way up near Newfoundland and Nova Scotia (in some cases braving icebergs).
https://www.movable-type.co.uk/scripts/latlong.html is a great circle calculator, it uses the Haversine formula in spherical trig, which is seldom used anymore (I remember it from High School). Due the radius of Earth is not a perfect sphere, professor Damon's answer is off about .5 percent, as is the Haversine fromula, which for navigation is not much, until we discovered ICBMs which needed much more precise calculations...but that is another topic. I noticed the difference between Prof Damon's calculations, and the Haversine calculation on the great circle was approximately 40 nm, again, not much for sea or air naviagation, both within the error on the Earth Radius at the poles.
I am sure they made that longitude difference 180 degrees on purpose so we would not really have to use spherical trig.
I know the earth is squished a little but we who plod around on the surface of the ocean do not worry about it much :)
Yeah, I miss those old functions like haversine and Gudermannian
To find the distance between two points on Earth's surface, you can use the Haversine formula. The Haversine formula calculates the distance between two points on a sphere given their longitudes and latitudes.
(a) To find the distance along the parallel of latitude, we can use the formula:
Distance = Δlongitude * cos(latitude) * Earth's radius
Given:
- Point A: Latitude = 58.4N, Longitude = 148W (or -148)
- Point B: Latitude = 58.4N, Longitude = 32E (or +32)
Calculations:
1. Convert the longitudes from degrees to radians:
- Longitude A: -148 * (π/180) = -2.584 radians
- Longitude B: 32 * (π/180) = 0.558 radians
2. Calculate the difference in longitude:
- Δlongitude = |Longitude A - Longitude B| = |-2.584 - 0.558| = 3.142 radians
3. Calculate the distance along the latitude parallel:
- Distance = Δlongitude * cos(latitude) * Earth's radius
- Latitude = 58.4N, so cos(latitude) = cos(58.4 * π/180)
- Earth's radius is approximately 6,371 km
- Distance = 3.142 * cos(58.4 * π/180) * 6,371 km
(b) To find the distance along a great circle (shortest distance between two points on a sphere), we can use the haversine formula:
Distance = 2 * Earth's radius * arcsin(√(sin²(Δlatitude/2) + cos(latitude1) * cos(latitude2) * sin²(Δlongitude/2)))
Given:
- Point A: Latitude = 58.4N, Longitude = 148W (or -148)
- Point B: Latitude = 58.4N, Longitude = 32E (or +32)
Calculations:
1. Convert the longitudes and latitudes from degrees to radians:
- Latitude A: 58.4 * (π/180) = 1.019 radians
- Latitude B: 58.4 * (π/180) = 1.019 radians
- Longitude A: -148 * (π/180) = -2.584 radians
- Longitude B: 32 * (π/180) = 0.558 radians
2. Calculate the differences in latitude and longitude:
- Δlatitude = |Latitude A - Latitude B| = |1.019 - 1.019| = 0 radians
- Δlongitude= |Longitude A - Longitude B| = |-2.584 - 0.558| = 3.142 radians
3. Calculate the distance along the great circle:
- Distance = 2 * Earth's radius * arcsin(√(sin²(Δlatitude/2) + cos(latitude1) * cos(latitude2) * sin²(Δlongitude/2)))
- Earth's radius is approximately 6,371 km
- Distance = 2 * 6,371 km * arcsin(√(sin²(0/2) + cos(1.019) * cos(1.019) * sin²(3.142/2)))
These calculations will give you the distance in kilometers.