P and Q are two points on latitude 55 degrees North and their longitudes are 33 degrees West and 23 degrees East respectively. calculate the distance between P and Q measured along (a)along the parallel of latitude (b)along a great circle
Let R be the radius of the earth.
The radius of a parallel at latitude θ is r = R cosθ
arc length s = rØ where Ø is the difference in longitude.
getting the great circle distance is a bit trickier, but as usual, google is your friend. Here's a calculator, and also the formulas used to do the math.
https://keisan.casio.com/exec/system/1224587128
To calculate the distance between P and Q measured along the parallel of latitude (a) and along a great circle (b), we can use the Haversine formula, which is commonly used to calculate distances between points on a sphere.
To calculate the distance measured along the parallel of latitude (a), we can assume that the Earth is a perfect sphere and use the average radius of the Earth, which is approximately 6,371 kilometers.
(a) Along the parallel of latitude:
To calculate the distance between P and Q measured along the parallel of latitude, we need to determine the difference in longitudes between the two points and then calculate the corresponding fraction of the Earth's circumference.
The difference in longitudes between P and Q is:
33 degrees West - (-23 degrees East) = 56 degrees
To calculate the distance along the latitude, we can use the formula:
Distance = (Circumference of the Earth) × (cos(latitude)) × (difference in longitude / 360 degrees)
Using the circumference of the Earth (approximately 2π × radius) and substituting the values:
Distance = (2π × 6,371 km) × (cos(55 degrees)) × (56 degrees / 360 degrees)
Calculating this expression will give you the distance between P and Q measured along the parallel of latitude.
(b) Along a great circle:
To calculate the distance between P and Q along a great circle, we need to determine the shortest path on the Earth's surface connecting the two points using the Haversine formula.
The Haversine formula is given by:
Distance = 2 × (radius) × arcsin(√(sin²(Δlatitude/2) + cos(latitude1) × cos(latitude2) × sin²(Δlongitude/2)))
Where:
- radius is the radius of the Earth
- Δlatitude is the difference in latitudes between the two points
- Δlongitude is the difference in longitudes between the two points
Using the radius of the Earth and substituting the values of latitudes and longitudes, we can calculate the distance between P and Q along a great circle.
Note that the distance calculated along the great circle will typically be shorter than the distance measured along the parallel of latitude.