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
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.
To calculate the distance between the initial position (A) and the final position (B) using Mercator sailing, we can follow these steps:
Step 1: Convert the latitude and longitude coordinates from degrees, minutes, and seconds (DMS) format to decimal degrees (DD) format.
For the initial position (A):
Latitude (A) = 32°17.6’ S = -32.2933°
Longitude (A) = 007°14.6’ W = -7.2433°
For the final position (B):
Latitude (B) = 29°12.3’ S = -29.2050°
Longitude (B) = 001°38.4’ E = 1.6400°
Step 2: Convert the latitudes to radians.
Latitude (A) in radians = -32.2933° * π / 180 = -0.5638 rad
Latitude (B) in radians = -29.2050° * π / 180 = -0.5092 rad
Step 3: Calculate the difference in longitudes, converted to radians.
Δ Longitude = (Longitude (B) - Longitude (A)) * π / 180
= (1.6400° - (-7.2433°)) * π / 180
= 8.8833° * π / 180
= 0.1548 rad
Step 4: Calculate the distance using the Mercator formula:
Distance = Earth's mean radius * Δ Longitude * cos((Latitude (A) + Latitude (B)) / 2)
The Earth's mean radius used in this formula is approximately 6,371 km.
So, the distance between the waypoints A and B is:
Distance = 6371 km * 0.1548 rad * cos((-0.5638 rad + (-0.5092 rad)) / 2)
= 6371 km * 0.1548 rad * cos(-0.5365 rad)
≈ 6371 km * 0.1548 * 0.8587
≈ 6371 km * 0.1330
≈ 847.43 km
Therefore, the distance between waypoints A and B, calculated using Mercator sailing, is approximately 847.43 kilometers.