Calculate the distance between these waypoints using Mercator sailing:
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
Formulas:
Distance= DLAT/COS Course
Tan Co= DLO/DMP
MP= 7915.7logtan(lat2+45)-23.4sinlat
To calculate the distance between the waypoints using Mercator sailing, we will follow these steps:
1. Convert the latitude and longitude values from degrees, minutes, and seconds (DMS) to decimal degrees (DD).
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 = -29.2050°
Longitude: 001°38.4’ E = 1.6400°
2. Calculate the difference in latitude (DLAT) and the difference in longitude (DLO) between the positions.
DLAT = Latitude(B) - Latitude(A) = (-29.2050°) - (-32.2933°) = 3.0883°
DLO = Longitude(B) - Longitude(A) = 1.6400° - (-7.2433°) = 8.8833°
3. Calculate the course (CO) using the formula:
CO = arctan(DLO / (DMP * cos(latitude(A))))
DMP: Distance to the Mercator Projection (10000 nautical miles)
Latitude(A) = -32.2933°
CO = arctan(8.8833 / (10000 * cos(-32.2933°)))
4. Calculate the distance (Distance) using the formula:
Distance = DLAT / cos(CO)
Distance = 3.0883° / cos(CO)
5. Substitute the CO value obtained in step 3 to calculate the final distance.
Alternatively, we can simplify the formulas using the values provided.
MP = 7915.7 * log(tan(latitude(B) + 45°)) - 23.4 * sin(latitude(B))
Latitude(B) = -29.2050°
MP = 7915.7 * log(tan(-29.2050° + 45°)) - 23.4 * sin(-29.2050°)
Step 1: Convert latitude and longitude to decimal degrees:
Initial position (A):
- Latitude: 32°17.6’ S
Convert to decimal degrees: -32.2933°
- Longitude: 007°14.6’ W
Convert to decimal degrees: -7.2433°
Final position (B):
- Latitude: 29°12.3’ S
Convert to decimal degrees: -29.2050°
- Longitude: 001°38.4’ E
Convert to decimal degrees: 1.6400°
Step 2: Calculate the course (Co) between the two positions:
DLO = B.Longitude - A.Longitude = 1.6400° - (-7.2433°) = 8.8833°
DMP = ln(tan((B.Latitude+45)/2°) / tan((A.Latitude+45)/2°))
= ln(tan((-29.2050+45)/2°) / tan((-32.2933+45)/2°))
Step 3: Calculate the distance (DLAT) between the two positions:
DLAT = B.Latitude - A.Latitude = -29.2050° - (-32.2933°) = 3.0883°
Step 4: Calculate the course (Co):
Co = DLON / DMP = 8.8833° / DMP
Step 5: Calculate the distance:
Distance = DLAT / cos(Co) = 3.0883° / cos(Co)
Please note that the DMP value is missing in the given information. To proceed with the calculation, we need the correct DMP value.