Calculate the distance between these waypoints using Mercator sailing:
Solve for DMP
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 A and B using Mercator sailing, you can follow the steps below:
1. Convert the latitude and longitude coordinates from degrees, minutes, and seconds (DMS) to decimal degrees (DD).
For position A (initial position):
Latitude: 32°17.6' S = -32.2933°
Longitude: 007°14.6' W = -7.2433°
For position B (final position):
Latitude: 29°12.3' S = -29.2050°
Longitude: 001°38.4' E = 1.6400°
2. Convert the decimal degrees to radians by multiplying them by π/180.
For position A:
Latitude: -32.2933° * (π/180) = -0.563828 rad
Longitude: -7.2433° * (π/180) = -0.126371 rad
For position B:
Latitude: -29.2050° * (π/180) = -0.509699 rad
Longitude: 1.6400° * (π/180) = 0.0286174 rad
3. Calculate the difference in longitude (Δλ) between the two positions.
Δλ = LongitudeB - LongitudeA
Δλ = 0.0286174 rad - (-0.126371 rad)
Δλ = 0.1559884 rad
4. Calculate the distance using the Mercator formula:
DMP = R * Δλ
Where R is the Earth's radius, which is approximately 6,371 km.
DMP = 6,371 km * 0.1559884 rad
DMP = 990.65 km
So, the distance between waypoints A and B using Mercator sailing is approximately 990.65 kilometers.
To calculate the distance between the waypoints A and B using Mercator sailing, we can follow these steps:
Step 1: Convert the latitude and longitude from degrees, minutes, and seconds (DMS) to decimal degrees (DD) format.
A: Latitude 32°17.6’ S, Longitude 007°14.6’ W
Latitude: -32.2933°, Longitude: -7.2433° (converted to DD)
B: Latitude 29°12.3’ S, Longitude 001°38.4’ E
Latitude: -29.2050°, Longitude: 1.6400° (converted to DD)
Step 2: Convert the longitudes to the same hemisphere.
Since A has a longitude in the western hemisphere (W) and B has a longitude in the eastern hemisphere (E), we need to convert the longitudes so that they are in the same hemisphere. By convention, when doing Mercator sailing, longitudes are considered positive in the western hemisphere and negative in the eastern hemisphere. Therefore, we need to convert B's longitude to -1.6400°.
Step 3: Calculate the difference in longitude (∆L).
∆L = Longitude(B) - Longitude(A)
= -1.6400° - (-7.2433°)
= 5.6033°
Step 4: Calculate the distance using the Mercator formula:
DMP = √(∆L^2 + (Latitude(B))^2)
= √((5.6033)^2 + (-29.2050)^2)
= √(31.4004 + 853.0140)
= √884.4144
≈ 29.7244 nautical miles
Therefore, the distance between waypoints A and B using Mercator sailing is approximately 29.7244 nautical miles.