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)
To solve this problem using Great Circle Sailing, we need to find the shortest distance between the two given positions A and B.
Step 1: Convert the given coordinates from degrees, minutes, and seconds to decimal degrees.
Initial Position, A:
Latitude: 11° 14' N = 11.233° N (approximately) [converted: 11 + 14/60]
Longitude: 125° 03' E = 125.05° E (approximately) [converted: 125 + 3/60]
Final Position, B:
Latitude: 08° 01' S = -8.017° S (approximately) [converted: -8 - 1/60]
Longitude: 079° 34' W = -79.567° W (approximately) [converted: -79 - 34/60]
Step 2: Use the haversine formula to calculate the great circle distance.
The haversine formula is given by:
d = 2r * arcsin(sqrt(sin²((Lat2 - Lat1) / 2) + cos(Lat1) * cos(Lat2) * sin²((Lon2 - Lon1) / 2)))
Where:
- d is the great circle distance
- r is the radius of the Earth (approximately 3440.07 nautical miles)
- Lat1 and Lat2 are the initial and final latitudes in radians
- Lon1 and Lon2 are the initial and final longitudes in radians
Let's calculate the great circle distance:
r = 3440.07 nautical miles
Lat1 = 11.233° N = 11.233 * (pi / 180) ≈ 0.1960 radians
Lon1 = 125.05° E = 125.05 * (pi / 180) ≈ 2.1822 radians
Lat2 = -8.017° S = -8.017 * (pi / 180) ≈ -0.1398 radians
Lon2 = -79.567° W = -79.567 * (pi / 180) ≈ -1.3872 radians
Using the haversine formula:
d = 2 * 3440.07 * arcsin(sqrt(sin²((0.1960 - (-0.1398))/2) + cos(0.1960) * cos(-0.1398) * sin²((-1.3872 - 2.1822)/2)))
Calculating the value inside the square root:
sin²((0.1960 - (-0.1398))/2) + cos(0.1960) * cos(-0.1398) * sin²((-1.3872 - 2.1822)/2) ≈ 0.000576
Using the value in the haversine formula:
d = 2 * 3440.07 * arcsin(sqrt(0.000576))
≈ 2 * 3440.07 * arcsin(0.024)
Using the arcsin value ≈ 0.024:
d ≈ 2 * 3440.07 * 0.024
≈ 165.61 nautical miles
Therefore, the Great Circle Distance between position A and B is approximately 165.61 nautical miles.
To calculate the Great Circle Distance between two points using Great Circle Sailing, we can use the haversine formula:
haversine(θ) = sin²(θ/2)
where θ = central angle between two points on the surface of a sphere.
To calculate the Great Circle Distance (nautical miles), follow these steps:
Step 1: Convert the latitude and longitude coordinates from degrees, minutes, and seconds (DMS) to decimal degrees (DD).
Initial Position, A: (11° 14’ N, 125° 03’ E)
- Latitude in decimal degrees: 11 + (14/60) = 11.2333° N
- Longitude in decimal degrees: 125 + (3/60) = 125.05° E
Final Position, B: (08° 01’ S, 079° 34’ W)
- Latitude in decimal degrees: -(8 + (1/60)) = -8.0167° S (negative because it is in the southern hemisphere)
- Longitude in decimal degrees: -(79 + (34/60)) = -79.5667° W (negative because it is in the western hemisphere)
Step 2: Convert the decimal degrees to radians by multiplying by π/180.
Latitude A in radians: 11.2333° × (π/180) = 0.1955 radians N
Longitude A in radians: 125.05° × (π/180) = 2.1818 radians E
Latitude B in radians: -8.0167° × (π/180) = -0.1399 radians S
Longitude B in radians: -79.5667° × (π/180) = -1.3879 radians W
Step 3: Calculate the central angle (θ) using the haversine formula.
haversine(θ) = sin²[(LatitudeB - LatitudeA)/2] + cos(LatitudeA) × cos(LatitudeB) × sin²[(LongitudeB - LongitudeA)/2]
haversine(θ) = sin²[(0.1955 - (-0.1399))/2] + cos(0.1955) × cos(-0.1399) × sin²[(2.1818 - (-1.3879))/2]
haversine(θ) = sin²[0.1677/2] + cos(0.1955) × cos(-0.1399) × sin²[1.2849/2]
haversine(θ) = sin²[0.0839] + cos(0.1955) × cos(-0.1399) × sin²[0.6424]
haversine(θ) = 0.0069 + cos(0.1955) × cos(-0.1399) × 0.1228
haversine(θ) = 0.0069 + 0.9789 × 0.9902 × 0.1228
haversine(θ) = 0.0069 + 0.9483 × 0.9902 × 0.1228
haversine(θ) = 0.0069 + 0.9429 × 0.1228
haversine(θ) = 0.0069 + 0.1158
haversine(θ) = 0.1227
Step 4: Calculate the distance using the formula:
Distance = 2 × radius × arcsin(√haversine(θ))
Assuming the radius of the Earth is 3440 nautical miles:
Distance = 2 × 3440 × arcsin(√0.1227)
Distance = 2 × 3440 × arcsin(0.35)
Distance = 2 × 3440 × 0.3569
Distance = 2385.37 nautical miles
Therefore, the Great Circle Distance between the initial position A and final position B is approximately 2385.37 nautical miles.