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
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).
To calculate the distance between the waypoints using Mercator sailing, you can follow these steps:
Step 1: Convert the latitude and longitude values from degrees, minutes, and seconds (DMS) to decimal degrees (DD). To convert DMS to DD, use the following formulas:
- Latitude in decimal degrees (DD) = degrees + (minutes/60) + (seconds/3600)
- Longitude in decimal degrees (DD) = degrees + (minutes/60) + (seconds/3600)
For the initial position (A):
Latitude in DD = 32 + (17.6/60) + (0/3600) = -32.2933° S
Longitude in DD = -7 + (14.6/60) + (0/3600) = -7.243° W
For the final position (B):
Latitude in DD = 29 + (12.3/60) + (0/3600) = -29.205° S
Longitude in DD = 1 + (38.4/60) + (0/3600) = 1.640° E
Step 2: Calculate the difference in longitude between the initial and final positions (ΔLO). The difference in longitude is the absolute value of the two longitudes (in decimal degrees) subtracted from each other:
ΔLO = |(-7.243) - 1.640| = 8.883°
Step 3: The distance in Mercator sailing can be calculated using the formula:
Tan Co = ΔLO / DMP
To find DMP, rearrange the equation:
DMP = ΔLO / Tan Co
Step 4: Calculate the tangent of the course (Co) which is equal to the difference in latitude (ΔLP) divided by the difference in longitude (ΔLO). The difference in latitude (ΔLP) is the absolute value of the two latitudes (in decimal degrees) subtracted from each other:
ΔLP = |-32.2933 - (-29.205)| = 3.0883°
Co = ΔLP / ΔLO
Step 5: Calculate the distance (DMP) in Mercator sailing:
DMP = ΔLO / Tan Co
To complete the calculation, substitute the values into the formula:
DMP = 8.883 / (3.0883 / 8.883)
Simplify:
DMP ≈ 28.739 nautical miles
Therefore, the distance between the waypoints using Mercator sailing is approximately 28.739 nautical miles.