Calculate the distance between these waypoints using Mercator sailing:
Solve for true course
Initial position (A) Lat 32°17.6’ S, Long 007°14.6’W
Final position (B)Lat 29°12.3’ S, Long 001°38.4’ E
To calculate the distance between the waypoints using Mercator sailing, you can follow these steps:
1. Convert the latitude and longitude values from degrees, minutes, and seconds to decimal degrees.
For initial position (A):
Latitude: 32°17.6’ S = -32.2933°
Longitude: 007°14.6’ W = -7.2433°
For final position (B):
Latitude: 29°12.3’ S = -29.205°
Longitude: 001°38.4’ E = 1.640°
2. Calculate the true course (TC) using the formula:
TC = atan((sin(longitudeB - longitudeA) * cos(latitudeB)) / (cos(latitudeA) * sin(latitudeB) - sin(latitudeA) * cos(latitudeB) * cos(longitudeB - longitudeA)))
Substituting the values:
TC = atan((sin(1.640 - (-7.2433)) * cos(-29.205)) / (cos(-32.2933) * sin(-29.205) - sin(-32.2933) * cos(-29.205) * cos(1.640 - (-7.2433))))
3. Convert the true course from radians to degrees.
4. The distance between the waypoints can be calculated using the formula:
Distance = 60 * 1.852 * d * cos(TC)
Where d is the difference in longitude between the waypoints.
Substituting the values:
Distance = 60 * 1.852 * abs(1.640 - (-7.2433)) * cos(TC)
Now, let's calculate the true course and distance:
Step 1:
Latitude A: -32.2933
Longitude A: -7.2433
Latitude B: -29.205
Longitude B: 1.640
Step 2:
TC = atan((sin(1.640 - (-7.2433)) * cos(-29.205)) / (cos(-32.2933) * sin(-29.205) - sin(-32.2933) * cos(-29.205) * cos(1.640 - (-7.2433))))
TC ≈ 65.513°
Step 3:
True Course: 65.513°
Step 4:
Distance = 60 * 1.852 * abs(1.640 - (-7.2433)) * cos(TC)
Distance ≈ 976.32 nautical miles
Therefore, the distance between the waypoints (A and B) is approximately 976.32 nautical miles.
To calculate the distance between the given waypoints using Mercator sailing, we need to follow these steps:
Step 1: Convert the latitude and longitude values from degrees, minutes, and seconds (DMS) format to decimal degrees (DD) format.
Convert the latitude coordinates of the initial position (A):
Latitude 32°17.6’ S = -32.2933°
Convert the longitude coordinates of the initial position (A):
Longitude 007°14.6’ W = -7.2433°
Convert the latitude coordinates of the final position (B):
Latitude 29°12.3’ S = -29.2050°
Convert the longitude coordinates of the final position (B):
Longitude 001°38.4’ E = 1.6400°
Step 2: Calculate the course using the formula:
Course = arctan(tan(lonB - lonA) / sin(latB - latA)) * 180 / π
Where:
lonA is the longitude of the initial position (A) in radians,
lonB is the longitude of the final position (B) in radians,
latA is the latitude of the initial position (A) in radians,
latB is the latitude of the final position (B) in radians,
π is the mathematical constant.
Step 3: Calculate the distance using the formula:
Distance = ArcCos(sin(latA) * sin(latB) + cos(latA) * cos(latB) * cos(lonB - lonA)) * R
Where:
latA is the latitude of the initial position (A) in radians,
latB is the latitude of the final position (B) in radians,
lonB is the longitude of the final position (B) in radians,
lonA is the longitude of the initial position (A) in radians,
R is the Earth's radius (mean radius = 6371 km).
Now, let's calculate the true course and the distance between the waypoints (A) and (B) step-by-step.
Step 1: Convert the latitude and longitude values from DMS to DD format.
Initial position (A):
Latitude: 32°17.6’ S = -32.2933°
Longitude: 007°14.6’ W = -7.2433°
Final position (B):
Latitude: 29°12.3’ S = -29.2050°
Longitude: 001°38.4’ E = 1.6400°
Step 2: Calculate the course.
Course = arctan(tan(lonB - lonA) / sin(latB - latA)) * 180 / π
lonA = -7.2433° * π / 180 ≈ -0.1266 radians
lonB = 1.6400° * π / 180 ≈ 0.0286 radians
latA = -32.2933° * π / 180 ≈ -0.5643 radians
latB = -29.2050° * π / 180 ≈ -0.5097 radians
π ≈ 3.14159
Course = arctan(tan(0.0286 - (-0.1266)) / sin(-0.5097 - (-0.5643))) * 180 / π
Course ≈ 159.646°
Step 3: Calculate the distance.
Distance = ArcCos(sin(latA) * sin(latB) + cos(latA) * cos(latB) * cos(lonB - lonA)) * R
R = 6371 km (mean Earth's radius)
Distance = ArcCos(sin(-0.5643) * sin(-0.5097) + cos(-0.5643) * cos(-0.5097) * cos(0.0286 - (-0.1266))) * 6371
Distance ≈ 463.01 km (rounded to two decimal places)
Therefore, the distance between waypoints (A) and (B) using Mercator sailing is approximately 463.01 km, and the true course is approximately 159.646°.