Calculate the distance between these waypoints using Mercator sailing:

Question

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

Answer 1

To calculate the distance between the initial position (A) and the final position (B) using Mercator sailing, we can use the formula for the approximate length of a rhumb line:

d = R * Δφ * cos(ϕavg)

where:
- d is the distance between the two waypoints
- R is the radius of the Earth (assumed to be 6,371 km)
- Δφ is the difference in latitude between the waypoints in radians
- ϕavg is the average latitude between the waypoints in radians.

First, we need to convert the given latitudes and longitudes from degrees, minutes, and seconds (DMS) to decimal degrees (DD).

For position A:
Latitude: 32°17.6' S
Longitude: 7°14.6' W

To convert to DD:
Latitude (A) = -32 - (17.6/60) = -32.2933°
Longitude (A) = -7 - (14.6/60) = -7.2433°

For position B:
Latitude: 29°12.3' S
Longitude: 1°38.4' E

To convert to DD:
Latitude (B) = -29 - (12.3/60) = -29.205°
Longitude (B) = 1 + (38.4/60) = 1.64°

Next, we need to convert the decimal degrees to radians:

Latitude (A) = (-32.2933 * π) / 180 = -0.5635 radians
Longitude (A) = (-7.2433 * π) / 180 = -0.1264 radians

Latitude (B) = (-29.205 * π) / 180 = -0.5094 radians
Longitude (B) = (1.64 * π) / 180 = 0.0286 radians

Now, we can calculate the difference in latitude and the average latitude:

Δφ = Latitude (B) - Latitude (A) = -0.5094 - (-0.5635) = 0.0541 radians

ϕavg = (Latitude (A) + Latitude (B)) / 2 = (-0.5635 + (-0.5094)) / 2 = -0.5364 radians

Substituting these values into the formula, with R = 6,371 km:

d = 6,371 * 0.0541 * cos(-0.5364) ≈ 368.35 km

Therefore, the distance between the waypoints A and B using Mercator sailing is approximately 368.35 km.