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 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.
To calculate the great circle distance between two positions, we can use the haversine formula. The haversine formula is an equation that calculates the distance between two points on a sphere (in this case, the Earth) given their longitudes and latitudes.
Step 1: Convert the given 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
Step 2: Convert the decimal degrees to radians.
Initial Position, A:
Latitude: 11.2333° N => 11.2333° * (π/180) = 0.195732 radians N
Longitude: 125.05° E => 125.05° * (π/180) = 2.18279 radians E
Final Position, B:
Latitude: -8.0167° S => -8.0167° * (π/180) = -0.139876 radians S
Longitude: -79.5667° W => -79.5667° * (π/180) = -1.3887 radians W
Step 3: Apply the haversine formula to calculate the great circle distance.
The haversine formula is as follows:
distance = 2 * radius * arcsin(sqrt(sin²((lat₂ - lat₁) / 2) + cos(lat₁) * cos(lat₂) * sin²((lon₂ - lon₁) / 2)))
Assuming the radius of the Earth is approximately 3440 nautical miles, we can calculate the distance as follows:
distance = 2 * 3440 * arcsin(sqrt(sin²((-0.139876 - 0.195732) / 2) + cos(0.195732) * cos(-0.139876) * sin²((-1.3887 - 2.18279) / 2)))
Simplifying the equation:
distance = 2 * 3440 * arcsin(sqrt(sin²(-0.167304) + cos(0.195732) * cos(-0.139876) * sin²(-1.405245)))
Calculating the values inside the square root:
distance = 2 * 3440 * arcsin(sqrt((0.027973)^2 + 0.825250 * 0.012794))
distance = 2 * 3440 * arcsin(sqrt(0.000781 + 0.010407))
distance = 2 * 3440 * arcsin(sqrt(0.011188))
distance = 2 * 3440 * arcsin(0.105717)
Using a calculator:
distance = 2 * 3440 * 6.0643
distance = 7483.29 nautical miles
Therefore, the great circle distance between the initial position (A) and the final position (B) is approximately 7483.29 nautical miles.