Question

Calculate the distance between these waypoints using Mercator sailing:

Tan Co= DLO/DMP
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

Answer 1

To calculate the distance between these waypoints using Mercator sailing, we can use the formula:

DMP = Tan Co * DLO

where:
- DMP is the desired distance
- Tan Co is the tangent of the course angle
- DLO is the difference in longitude between the initial and final positions

First, we need to calculate the course angle (Co) using the formula:

Co = ArcTan(Tan(Co') * Cos L1 / Cos L2)

where:
- Co' is the initial course angle in degrees (measured clockwise from true north)
- L1 is the initial latitude in radians
- L2 is the final latitude in radians

Co' = (180° + 90°) - (180° + 7°14.6') = 90° - 7°14.6' = 82°45.4'

L1 = -32°17.6' converted to radians = -32.2933° * (π/180°) = -0.563
L2 = -29°12.3' converted to radians = -29.2050° * (π/180°) = -0.509

Co = ArcTan(Tan(82°45.4') * Cos(-0.563) / Cos(-0.509))
= ArcTan(1.2498 * 0.8539 / 0.8628)
= ArcTan(0.8294)
= 39.43°

Next, we need to calculate the difference in longitude (DLO):

DLO = LongB - LongA

LongA = 007°14.6'W, which can be treated as -7°14.6' converted to radians = -7.2433° * (π/180°) = -0.126
LongB = 001°38.4'E, which can be treated as 001°38.4' converted to radians = 1.64° * (π/180°) = 0.029

DLO = 0.029 - (-0.126) = 0.155

Finally, we can substitute these values into the main formula:

DMP = Tan(Co) * DLO
= Tan(39.43°) * 0.155

Using a calculator, we find:

DMP = 0.810 * 0.155
= 0.12555

Therefore, the distance between the waypoints using Mercator sailing is approximately 0.126 units (the actual unit is not provided).