A plane flies from A (60°N, 40°W) to town B (60°N, 140°E)
What is your question?
since the towns are at the same latitude, the radius of the circle flown is Rcos60 = 1/2 earth radius
so the distance flown is s = rθ = R/2 * (140+60)/360 * 2π
unless the plane flies along a great circle ...
yes I response to maths
To find the distance and direction between point A (60°N, 40°W) and point B (60°N, 140°E), you can use the Haversine formula.
Here's how you can use this formula to calculate the distance between two points on the Earth's surface:
1. Convert the latitude and longitude of the two points from degrees to radians. Since the Haversine formula uses radians, we need to convert the given latitude and longitude values to radians. To convert from degrees to radians, multiply the degree value by π/180.
For point A:
Latitude of A in radians = 60° * π/180 ≈ 1.0472 radians (rounded to four decimal places)
Longitude of A in radians = -40° * π/180 ≈ -0.6981 radians (rounded to four decimal places)
For point B:
Latitude of B in radians = 60° * π/180 ≈ 1.0472 radians (rounded to four decimal places)
Longitude of B in radians = 140° * π/180 ≈ 2.4435 radians (rounded to four decimal places)
2. Use the Haversine formula to calculate the central angle between the two points:
Δσ = Haversin(Δφ) + cos(φ1) * cos(φ2) * Haversin(Δλ)
Δφ = Latitude of B - Latitude of A
Δλ = Longitude of B - Longitude of A
Haversin(x) = sin²(x/2)
Substitute the values into the formula:
Δφ = 1.0472 - 1.0472 = 0 radians
Δλ = 2.4435 - (-0.6981) ≈ 3.1416 radians (rounded to four decimal places)
Haversin(Δφ) = sin²(0/2) = 0
Haversin(Δλ) = sin²(3.1416/2) ≈ 1
Δσ = 0 + cos(1.0472) * cos(1.0472) * 1 ≈ 0.3773 (rounded to four decimal places)
3. Calculate the arc length using the Earth's radius:
Arc Length = 2 * radius * arcsin(√Δσ)
The radius of the Earth is approximately 6,371 kilometers.
Arc Length = 2 * 6,371 km * arcsin(√0.3773) ≈ 5,738.12 km (rounded to two decimal places)
So, the distance between point A and point B is approximately 5,738.12 kilometers.
To determine the direction or bearing between point A and point B, you can use the formula for the spherical law of cosines.
The formula for calculating the initial bearing (or forward azimuth) from point A to point B is:
θ = atan2(sin(Δλ) * cos(φ2), cos(φ1) * sin(φ2) - sin(φ1) * cos(φ2) * cos(Δλ))
Substitute the values into the formula:
θ = atan2(sin(3.1416) * cos(1.0472), cos(1.0472) * sin(1.0472) - sin(1.0472) * cos(1.0472) * cos(3.1416))
θ ≈ 49.68° (rounded to two decimal places)
Therefore, the plane is flying at an initial bearing of approximately 49.68° from A to B.