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 waypoints A and B using Mercator sailing, we can use the haversine formula. The formula calculates the distance between two points on a sphere given their latitudes and longitudes.
First, we need to convert the latitudes and longitudes from degrees, minutes, and seconds to decimal degrees:
Initial Position (A):
Latitude: 32°17.6’ S
Longitude: 007°14.6’ W
Convert the latitude to decimal degrees:
32 + 17.6/60 = 32.2933° S
Convert the longitude to decimal degrees:
-(7 + 14.6/60) = -7.2433° W (note the negative sign indicating west)
Final Position (B):
Latitude: 29°12.3’ S
Longitude: 001°38.4’ E
Convert the latitude to decimal degrees:
29 + 12.3/60 = 29.2050° S
Convert the longitude to decimal degrees:
1 + 38.4/60 = 1.6400° E
Now, we can use the haversine formula to calculate the distance between the two positions. However, since the initial position is located west of the final position, we need to adjust the longitude difference. For waypoints separated by more than 180 degrees of longitude, it is necessary to wrap the difference to the smaller angle between them:
Longitude difference adjusted = (360 - initial longitude) + final longitude = (360 - 7.2433) + 1.6400 = 354.3967°
Applying the haversine formula:
a = sin²((final latitude - initial latitude) / 2) + cos(initial latitude) * cos(final latitude) * sin²((longitude difference adjusted) / 2)
c = 2 * atan2(sqrt(a), sqrt(1 - a))
distance = radius of the Earth * c
Using the average radius of the Earth, which is approximately 6,371 km:
radius of the Earth = 6,371 km
a = sin²((29.2050 - 32.2933) / 2) + cos(32.2933) * cos(29.2050) * sin²(354.3967 / 2)
c = 2 * atan2(sqrt(a), sqrt(1 - a))
distance = 6,371 km * c
Calculating the values:
a = sin²((-3.0883) / 2) + cos(32.2933) * cos(29.2050) * sin²(354.3967 / 2)
≈ 0.0136
c = 2 * atan2(sqrt(0.0136), sqrt(1 - 0.0136))
≈ 0.1805
distance = 6,371 km * 0.1805
≈ 1,150 km
Therefore, the distance between waypoints A and B using Mercator sailing is approximately 1,150 kilometers.
To calculate the distance between the two waypoints using Mercator sailing, we will need to convert the latitude and longitude coordinates into nautical miles. Here are the steps:
Step 1: Convert the latitude and longitude coordinates into decimal degrees.
Initial position (A): Lat -32.2933°, Long -7.2433°
Final position (B): Lat -29.205°, Long 1.64°
Step 2: Convert the latitude and longitude coordinates from decimal degrees to radians.
Initial position (A): Lat -0.56407 radians, Long -0.12646 radians
Final position (B): Lat -0.50922 radians, Long 0.02893 radians
Step 3: Find the difference in longitude.
Diff. Long = Long(B) - Long(A)
Diff. Long = 0.02893 - (-0.12646)
Diff. Long = 0.02893 + 0.12646
Diff. Long = 0.15539 radians
Step 4: Calculate the distance using the Mercator formula.
Distance = Earth Radius * Difference in Longitude
Distance = 3437.75 * 0.15539
Distance ≈ 534.82 nautical miles
Therefore, the distance between the initial position (A) and final position (B) using Mercator sailing is approximately 534.82 nautical miles.