your vessel departs from yokohama from position A(35°27'N, 139°39' E) bound for San Francisco, CA at position B (37° 48.5' N, 122° 24' W). by great circle sailing, determine the great circle distance, the initial and final courses.
To determine the great circle distance between the two positions, we can use the haversine formula:
haversine(d) = sin²(Δφ/2) + cos(φ₁) * cos(φ₂) * sin²(Δλ/2)
Where:
- d is the great circle distance
- Δφ is the difference in latitude between the two positions
- Δλ is the difference in longitude between the two positions
- φ₁ and φ₂ are the latitudes of positions A and B, respectively
First, we need to convert the given positions from degrees, minutes, and seconds to decimal degrees:
Position A:
Latitude = 35°27'N = 35 + 27/60 = 35.45°N
Longitude = 139°39'E = 139 + 39/60 = 139.65°E
Position B:
Latitude = 37°48.5'N = 37 + 48.5/60 = 37.8083°N
Longitude = 122°24'W = -122 - 24/60 = -122.4°W
Next, we can calculate the differences in latitude and longitude:
Δφ = φ₂ - φ₁ = 37.8083° - 35.45° = 2.3583°
Δλ = λ₂ - λ₁ = -122.4° - 139.65° = -262.05°
Since the longitude difference is negative, we adjust it to positive:
Δλ = 360° + (-262.05°) = 97.95°
Now, we can plug these values into the haversine formula:
haversine(d) = sin²(Δφ/2) + cos(φ₁) * cos(φ₂) * sin²(Δλ/2)
haversine(d) = sin²(2.3583°/2) + cos(35.45°) * cos(37.8083°) * sin²(97.95°/2)
Using a scientific calculator, we find that:
sin(2.3583°/2) ≈ 0.03571
cos(35.45°) ≈ 0.81736
cos(37.8083°) ≈ 0.78895
sin(97.95°/2) ≈ 0.82315
Plugging in these values, we get:
haversine(d) = (0.03571)² + (0.81736)(0.78895)(0.82315)²
haversine(d) ≈ 0.0012809 + (0.81736)(0.78895)(0.6796)
haversine(d) ≈ 0.0012809 + 0.5340328
Adding the results, we have:
haversine(d) ≈ 0.5353137
Now, we need to solve for d. Rearranging the haversine formula, we get:
d = 2 * atan2(sqrt(haversine(d)), sqrt(1 - haversine(d)))
Using a scientific calculator, we find:
sqrt(haversine(d)) ≈ 0.73201
sqrt(1 - haversine(d)) ≈ 0.68168
atan2(0.73201, 0.68168) ≈ 0.8539
Multiplying by 2, we get:
d ≈ 2 * 0.8539 ≈ 1.7078
Therefore, the great circle sailing distance between Yokohama and San Francisco is approximately 1.7078 units (units could be in nautical miles, kilometers, etc.).
Next, we can determine the initial and final courses.
The initial course (also known as the initial heading or azimuth) is the angle between True North and the great circle path at position A.
Using the Vincenty formula, the initial course can be calculated as:
θ = atan2(sin(Δλ) * cos(φ₂), cos(φ₁) * sin(φ₂) - sin(φ₁) * cos(φ₂) * cos(Δλ))
Plugging in the values:
θ = atan2(sin(97.95°) * cos(37.8083°), cos(35.45°) * sin(37.8083°) - sin(35.45°) * cos(37.8083°) * cos(97.95°))
Using a scientific calculator, we find:
sin(97.95°) ≈ 0.80404
cos(37.8083°) ≈ 0.68819
sin(35.45°) ≈ 0.57720
cos(37.8083°) ≈ 0.68819
cos(97.95°) ≈ 0.59444
Plugging in these values, we get:
θ ≈ atan2(0.80404 * 0.68819, 0.57720 * 0.68819 - 0.80404 * 0.59444)
θ ≈ atan2(0.55334, 0.39668) ≈ 1.3400
Therefore, the initial course is approximately 1.3400 units.
The final course (also known as the final heading) is the angle between True North and the great circle path at position B.
The final course can be calculated as the complement of the initial course:
Final Course = θ + 180° = 1.3400 + 180° = 181.3400°
Therefore, the final course is approximately 181.3400 units.
To summarize:
- The great circle distance between Yokohama (position A) and San Francisco (position B) is approximately 1.7078 units.
- The initial course is approximately 1.3400 units.
- The final course is approximately 181.3400 units.