A ship sails for 5 hours from position A: (27° 38' N, 112° 45' W) at a speed of 16 knots on a course of 295°T.
Determine the course angle.
How far does the ship travel at the end of 5 hours?
Find the departure, DEP, for the final position.
Find the difference in latitude , DLAT ( to the nearest degree and minute) for the final position.
Determine the latitude of the final position B.
To determine the course angle, we subtract 180° from the course of 295°:
Course angle = 295° - 180° = 115°
To find the distance traveled by the ship, we multiply the speed of 16 knots by the duration of 5 hours:
Distance = 16 knots * 5 hours = 80 nautical miles
To find the departure (DEP) for the final position, we use the formula:
DEP = Distance * sin(course angle)
DEP = 80 nautical miles * sin(115°)
Using a calculator, we find that sin(115°) ≈ 0.9063, so
DEP ≈ 80 nautical miles * 0.9063 ≈ 72.504 nautical miles
To find the difference in latitude (DLAT) for the final position, we use the formula:
DLAT = DEP * sin(course)
DLAT = 72.504 nautical miles * sin(115°)
Again, using a calculator, we find that sin(115°) ≈ 0.9063, so
DLAT ≈ 72.504 nautical miles * 0.9063 ≈ 65.8116 nautical miles
To find the latitude of the final position B, we subtract the DLAT from the latitude of position A:
Latitude of final position = 27° 38' N - 65.8116 nautical miles
To convert nautical miles to degrees and minutes, we use the approximation that 1 nautical mile is roughly equal to 1 minute of latitude. Therefore:
65.8116 nautical miles ≈ 65.8116 minutes
So, the latitude of the final position B is approximately:
27° 38' N - 65.8116 minutes ≈ 27° 38' - 65' 49" = 27° 38' - 65' = 26° 33' N.
To determine the course angle, we need to subtract the variation or deviation from the true bearing. Let's assume the deviation is 10°E in this case.
1. Course angle = True bearing - Variation/Deviation
Course angle = 295°T - 10°E
= 285° (+E)
Therefore, the course angle is 285°E.
To find how far the ship travels at the end of 5 hours, we can use the formula:
2. Distance = Speed x Time
Given that the speed is 16 knots and the time is 5 hours, we can calculate the distance:
Distance = 16 knots x 5 hours
= 80 nautical miles
Therefore, the ship travels 80 nautical miles at the end of 5 hours.
Next, we need to find the departure (DEP) for the final position, which represents the east-west distance traveled from the starting position.
3. DEP = Distance x Sin(Course angle)
Given that the distance is 80 nautical miles and the course angle is 285°, we can calculate the departure:
DEP = 80 nautical miles x Sin(285°)
≈ 80 nautical miles x (-0.087)
≈ -6.96 nautical miles
The departure (DEP) for the final position is approximately -6.96 nautical miles.
To find the difference in latitude (DLAT) for the final position, we need to calculate the north-south distance traveled.
4. DLAT = Distance x Cos(Course angle)
Using the same values for distance and course angle:
DLAT = 80 nautical miles x Cos(285°)
≈ 80 nautical miles x (0.996)
≈ 79.68 nautical miles
The difference in latitude (DLAT) for the final position is approximately 79.68 nautical miles.
Finally, to determine the latitude of the final position B, we need to add the difference in latitude (DLAT) to the initial latitude of position A.
Latitude of position B = Latitude of position A + DLAT
= 27° 38' N + 79.68 nautical miles
Since 1 nautical mile is approximately equal to 1 minute of latitude, we can convert the nautical miles to minutes:
79.68 nautical miles ≈ 79.68 minutes
Latitude of position B ≈ 27° 38' N + 79.68'
≈ 27° 38' N + 80'
≈ 27° 118' N
Therefore, the latitude of the final position B is approximately 27° 118' N.