Question

Solve the problem below using Great Circle Sailing

Calculate the Great Circle Distance (nautical miles)
Initial Position, A: (11° 14’ N, 125° 03’ E)
Final Position, B: (08° 01’ S, 079° 34’ W)

Answer 1

To calculate the great circle distance between two positions using the Great Circle Sailing method, we can use the Haversine formula.

First, we need to convert the latitude and longitude from degrees, minutes, and seconds to decimal degrees.

Initial Position, A:
Latitude: 11° 14’ N
Longitude: 125° 03’ E

Latitude A in decimal degrees:
11 + (14/60) = 11.2333° N

Longitude A in decimal degrees:
125 + (3/60) = 125.05° E

Final Position, B:
Latitude: 08° 01’ S
Longitude: 079° 34’ W

Latitude B in decimal degrees:
-(8 + (1/60)) = -8.0167° S

Longitude B in decimal degrees:
-(79 + (34/60)) = -79.5667° W

Now, we can use the Haversine formula to calculate the great circle distance.

The Haversine formula is:
haversin(d) = sin²(dlat/2) + cos(lat1) * cos(lat2) * sin²(dlon/2)

Where:
dlat = lat2 - lat1
dlon = lon2 - lon1

Let's plug in the values and calculate the great circle distance.

dlat = (-8.0167 - 11.2333) * (π/180) = -19.25° * (π/180)
dlon = (-79.5667 - 125.05) * (π/180) = -204.6167° * (π/180)

haversin(d) = sin²(-19.25/2) + cos(-8.0167) * cos(11.2333) * sin²(-204.6167/2)
haversin(d) = sin²(-9.625) + cos(-8.0167) * cos(11.2333) * sin²(-102.30835)

Now, we can calculate the great circle distance (d) using the formula:
d = 2 * R * atan2(sqrt(haversin(d)), sqrt(1 - haversin(d)))

Where:
R is the radius of the Earth (mean radius = 6,371 km)

Let's plug in the values and calculate the great circle distance.

haversin(d) = sin²(-9.625) + cos(-8.0167) * cos(11.2333) * sin²(-102.30835)

haversin(d) ≈ 0.003188

d = 2 * 6,371 * atan2(sqrt(0.003188), sqrt(1 - 0.003188))
d ≈ 2 * 6,371 * atan2(0.056505, 0.998408)
d ≈ 2 * 6,371 * 3.137077
d ≈ 39,931.7126 km

Finally, we convert the distance from kilometers to nautical miles by multiplying by 0.5399568.

Great Circle Distance ≈ 39,931.7126 km * 0.5399568
Great Circle Distance ≈ 21,564.1471 nautical miles

Therefore, the Great Circle Distance between initial position A and final position B is approximately 21,564.1471 nautical miles.