Solve the problem below using Great Circle Sailing
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 determine the shortest distance and the initial and final courses.
1. Convert the coordinates from degrees, minutes, and seconds to decimal degrees:
Initial Position, A: (11° 14’ N, 125° 03’ E)
Latitude: 11 + 14/60 = 11.2333° N
Longitude: 125 + 3/60 = 125.05° E
Final Position, B: (08° 01’ S, 079° 34’ W)
Latitude: -8 - 1/60 = -8.0167° S
Longitude: 79 + 34/60 = 79.5667° W
2. Determine the shortest distance:
The shortest distance between two points on the Earth's curved surface is given by the great-circle distance formula, which is represented by the arc of a circle with its center at Earth's center and radius equal to Earth's radius.
d = R * √(2 - 2 * sin(lat₁) * sin(lat₂) - cos(lat₁) * cos(lat₂) * cos(lon₁ - lon₂))
Where:
d = great-circle distance
R = Earth's radius (approximately 6,371 km)
lat₁, lat₂ = initial and final latitudes in radians
lon₁, lon₂ = initial and final longitudes in radians
Converting the decimal degrees to radians:
Initial Position, A:
lat₁ = 11.2333° N = 11.2333° * (π/180) = 0.1961 rad
lon₁ = 125.05° E = 125.05° * (π/180) = 2.1821 rad
Final Position, B:
lat₂ = -8.0167° S = -8.0167° * (π/180) = -0.1400 rad
lon₂ = 79.5667° W = 79.5667° * (π/180) = -1.3877 rad
Calculating the distance:
d = 6,371 * √(2 - 2 * sin(0.1961) * sin(-0.1400) - cos(0.1961) * cos(-0.1400) * cos(2.1821 - (-1.3877)))
= 6,371 * √(2 - 2 * sin(0.1961) * sin(-0.1400) - cos(0.1961) * cos(-0.1400) * cos(3.5698))
≈ 9815.7 km (rounded to one decimal place)
Therefore, the shortest distance between the initial position A and the final position B is approximately 9815.7 km.
3. Determine the initial and final courses:
The initial course (θ₁) and final course (θ₂) are the bearings from the starting point (A) to the destination (B). They can be calculated using the following formulas:
θ₁ = atan2(sin(Δlon) * cos(lat₂), cos(lat₁) * sin(lat₂) - sin(lat₁) * cos(lat₂) * cos(Δlon))
θ₂ = atan2(sin(Δlon) * cos(lat₁), -cos(lat₂) * sin(lat₁) + sin(lat₂) * cos(lat₁) * cos(Δlon))
Where:
θ₁, θ₂ = initial and final courses
lat₁, lat₂ = initial and final latitudes in radians
Δlon = difference in longitudes (lon₂ - lon₁) in radians
Calculating the initial and final courses:
Δlon = lon₂ - lon₁ = (-1.3877) - 2.1821 = -3.5698
θ₁ = atan2(sin(-3.5698) * cos(-0.1400), cos(0.1961) * sin(-0.1400) - sin(0.1961) * cos(-0.1400) * cos(-3.5698))
= atan2(-0.1539 * 0.9900, 0.9803 * -0.1404 - 0.1960 * -0.9900 * -0.4470)
≈ 3.6132 rad (rounded to four decimal places)
≈ 207.33° (rounded to two decimal places)
θ₂ = atan2(sin(-3.5698) * cos(0.1961), -cos(-0.1400) * sin(0.1961) + sin(-0.1400) * cos(0.1961) * cos(-3.5698))
= atan2(-0.1539 * 0.9803, 0.1404 * 0.1960 + 0.9900 * -0.4470 * -0.1539)
≈ 5.7979 rad (rounded to four decimal places)
≈ 332.48° (rounded to two decimal places)
Therefore, the initial course (θ₁) is approximately 207.33°, and the final course (θ₂) is approximately 332.48°.
To solve the problem using Great Circle Sailing, we need to find the shortest distance and course (direction) between the initial position (A) and the final position (B).
Step 1: Convert the coordinates to decimal form:
Initial Position, A: (11° 14’ N, 125° 03’ E)
Final Position, B: (08° 01’ S, 079° 34’ W)
A: (11.233° N, 125.050° E)
B: (8.017° S, 79.567° W)
Step 2: Determine the difference in longitude:
∆Lon = LonB - LonA = 79.567° W - 125.050° E
Since the longitude values have opposite signs, we need to choose the smaller absolute value and the corresponding direction. In this case, we will use the absolute value of -79.567° W and travel westward.
∆Lon = 79.567° W
Step 3: Determine the average latitude:
Avg Lat = (LatA + LatB) / 2 = (11.233° N + (-8.017° S)) / 2
Since the latitude values are in different hemispheres, we need to convert one of them to the same hemisphere as the other by changing the sign.
Avg Lat = (11.233° N + 8.017° S) / 2 = 3.216 / 2 = 1.618° N
Step 4: Convert the average latitude to radians:
Avg Lat in Radians = Avg Lat * (π / 180) = 1.618° N * (π / 180)
Avg Lat in Radians = 0.0282 radians N
Step 5: Calculate the distance:
Distance = ∆Lon * cos(Avg Lat in Radians) = 79.567° * cos(0.0282 radians)
Distance = 79.567° * 0.999604 = 79.551 nautical miles
Step 6: Determine the course:
Course = arctan(sin(∆Lon) / (cos(LatA) * tan(LatB)) = arctan(sin(79.567° W) / (cos(11.233° N) * tan(8.017° S)))
Course = arctan(-0.986) = 45.032°
Since we are traveling westward, the course is 180° + 45.032° = 225.032°.
Therefore, the shortest distance from A to B using Great Circle Sailing is approximately 79.551 nautical miles, with a course of 225.032°.