Question

Find the distance between the pair of points on the earth surface A(43°N,50°E) B(20°N,20°E)

Answer 1

To find the distance between two points on the Earth’s surface given in latitude and longitude, we can use the haversine formula, which calculates the great-circle distance between two points on a sphere given their longitudes and latitudes.

The formula is:
\[
d = 2r \cdot \text{asin}\left(\sqrt{\text{haversin}(\Delta \varphi) + \cos(\varphi_1) \cdot \cos(\varphi_2) \cdot \text{haversin}(\Delta \lambda)}\right)
\]
where:
- \(\varphi_1\) and \(\varphi_2\) are the latitudes of point A and B in radians,
- \(\Delta \varphi = \varphi_2 - \varphi_1\) (in radians),
- \(\Delta \lambda = \lambda_2 - \lambda_1\) (in radians),
- haversin(x) = \(\sin^2(x/2)\),
- \(r\) is the radius of the Earth (approximately 6371 km).

### Step 1: Convert degrees to radians
- Point A:
- Latitude (A): \(43°\) N = \(\frac{43 \cdot \pi}{180}\) radians
- Longitude (A): \(50°\) E = \(\frac{50 \cdot \pi}{180}\) radians

- Point B:
- Latitude (B): \(20°\) N = \(\frac{20 \cdot \pi}{180}\) radians
- Longitude (B): \(20°\) E = \(\frac{20 \cdot \pi}{180}\) radians

Calculating these values:
- \( \varphi_1 = \frac{43 \cdot \pi}{180} \approx 0.7505 \) radians
- \( \varphi_2 = \frac{20 \cdot \pi}{180} \approx 0.3491 \) radians
- \( \lambda_1 = \frac{50 \cdot \pi}{180} \approx 0.8727 \) radians
- \( \lambda_2 = \frac{20 \cdot \pi}{180} \approx 0.3491 \) radians

### Step 2: Calculate \(\Delta \varphi\) and \(\Delta \lambda\)
\[
\Delta \varphi = \varphi_2 - \varphi_1 = 0.3491 - 0.7505 = -0.4014 \text{ radians}
\]
\[
\Delta \lambda = \lambda_2 - \lambda_1 = 0.3491 - 0.8727 = -0.5236 \text{ radians}
\]

### Step 3: Calculate haversine of the angles
\[
\text{haversin}(\Delta \varphi) = \sin^2\left(\frac{\Delta \varphi}{2}\right) = \sin^2\left(\frac{-0.4014}{2}\right) \approx 0.1288
\]
\[
\text{haversin}(\Delta \lambda) = \sin^2\left(\frac{\Delta \lambda}{2}\right) = \sin^2\left(\frac{-0.5236}{2}\right) \approx 0.1346
\]

### Step 4: Substitute these into the formula
Now apply the haversine formula:
\[
d = 2 \cdot r \cdot \text{asin}\left(\sqrt{0.1288 + \cos(0.7505) \cdot \cos(0.3491) \cdot 0.1346}\right)
\]
Calculating:
- \(\cos(0.7505) \approx 0.7314\)
- \(\cos(0.3491) \approx 0.9438\)

Now compute:
\[
0.7314 \cdot 0.9438 \cdot 0.1346 \approx 0.0977
\]
Thus,
\[
\sqrt{0.1288 + 0.0977} = \sqrt{0.2265} \approx 0.476
\]

Finally:
\[
d \approx 2 \cdot 6371 \cdot \arcsin(0.476) \approx 2 \cdot 6371 \cdot 0.493 \approx 6288 \text{ km}
\]

The approximate distance between points A and B is **6288 km**.