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 calculate the Great Circle Distance between two positions, we can use the Haversine formula. The formula is as follows:
a = sin^2(Δφ/2) + cos(φ1) * cos(φ2) * sin^2(Δλ/2)
c = 2 * atan2(√a, √(1-a))
d = R * c
Where:
Δφ = φ2 - φ1
Δλ = λ2 - λ1
φ1, φ2 = latitude of point 1 and point 2 in radians
λ1, λ2 = longitude of point 1 and point 2 in radians
R = radius of the Earth (approximately 3440 nautical miles)
First, we need to convert the initial and final positions from degrees, minutes, and seconds to decimal degrees:
Initial Position, A:
Latitude: 11° 14' N = 11 + 14/60 = 11.2333° N
Longitude: 125° 03' E = 125 + 3/60 = 125.05° E
Final Position, B:
Latitude: 08° 01' S = -8 - 1/60 = -8.0167° S
Longitude: 079° 34' W = -(79 + 34/60) = -79.5667° W
Converting decimal degrees to radians:
φ1 = 11.2333° N * (π/180) = 0.195824 radians
λ1 = 125.05° E * (π/180) = 2.182502 radians
φ2 = -8.0167° S * (π/180) = -0.139996 radians
λ2 = -79.5667° W * (π/180) = -1.387781 radians
Now, we can substitute these values into the Haversine formula:
Δφ = (-0.139996) - (0.195824) = -0.33582 radians
Δλ = (-1.387781) - (2.182502) = -3.570283 radians
a = sin^2((-0.33582)/2) + cos(0.195824) * cos(-0.139996) * sin^2((-3.570283)/2)
= 0.596053863
c = 2 * atan2(√0.596053863, √(1-0.596053863))
= 1.81675787
d = 3440 * 1.81675787
= 6248.23 nautical miles (rounded to two decimal places)
Therefore, the Great Circle Distance between the initial position A and the final position B is approximately 6248.23 nautical miles.
To calculate the Great Circle Distance between two points on the Earth's surface, you can use the haversine formula. Here are the step-by-step instructions:
Step 1: Convert the coordinates from degrees, minutes, seconds (DMS) to decimal degrees (DD).
Initial position A:
Latitude: 11° 14' N = 11 + (14/60) = 11.2333° N
Longitude: 125° 03' E = 125 + (3/60) = 125.05° E
Final position B:
Latitude: 08° 01' S = -8 - (1/60) = -8.0167° S
Longitude: 079° 34' W = -79 - (34/60) = -79.5667° W
Step 2: Convert the decimal degrees to radians.
Initial position A:
Latitude in radians: 11.2333° * (π/180) = 0.1959 radians
Longitude in radians: 125.05° * (π/180) = 2.1834 radians
Final position B:
Latitude in radians: -8.0167° * (π/180) = -0.1398 radians
Longitude in radians: -79.5667° * (π/180) = -1.3873 radians
Step 3: Calculate the central angle between the two points.
Central angle (θ) = arccos(sin(latA) * sin(latB) + cos(latA) * cos(latB) * cos(lonA - lonB))
where latA, latB are the latitudes in radians, and lonA, lonB are the longitudes in radians.
θ = arccos(sin(0.1959) * sin(-0.1398) + cos(0.1959) * cos(-0.1398) * cos(2.1834 - -1.3873))
Step 4: Calculate the Great Circle Distance (d) using the Earth's radius.
Earth's radius = 3440 nautical miles
d = θ * Earth's radius
Now you can plug in the values and calculate the distance:
θ = arccos(sin(0.1959) * sin(-0.1398) + cos(0.1959) * cos(-0.1398) * cos(2.1834 - -1.3873))
d = θ * 3440
Let's carry out the calculations:
θ = arccos(sin(0.1959) * sin(-0.1398) + cos(0.1959) * cos(-0.1398) * cos(2.1834 - -1.3873))
θ ≈ 2.705 radians (rounded to 3 decimal places)
d = 2.705 * 3440
d ≈ 9309.52 nautical miles (rounded to 2 decimal places)
Therefore, the Great Circle Distance between the initial position A and final position B is approximately 9309.52 nautical miles.