Question

Place x and Y are both on latitude 45°N and are 180° of the longtitude apart . Calculate the great circle distance(via North) between them in km. Take the earths radius =6370km. Pls explain in a simple way that i can understand.

Answer 1

To calculate the great circle distance between two points (X and Y) on the same latitude (45° N) that are 180° apart in longitude, we can use the following steps:

1. **Understanding the Geometry**: Since both points are on the same line of latitude, we can visualize them on a circle parallel to the equator. This circle is smaller than the Earth's equator but retains the same relationship between degrees of longitude.

2. **Earth's Radius**: You're given the radius of the Earth as 6370 km.

3. **Calculating the Circumference at 45° N**: The distance along a circle (the circumference) is determined by the radius of the circle. However, the radius of the circle at a certain latitude (other than the equator) is not the same as the Earth's radius. The formula for the radius of the circle at a latitude \( \phi \) is:

\[
R = R_{earth} \cdot \cos(\phi)
\]

Where:
- \( R \) is the radius of the circle at latitude \( \phi \)
- \( R_{earth} \) is the Earth's radius (6370 km)
- \( \phi \) is the latitude in radians.

For 45° N:

\[
\phi = 45° = \frac{\pi}{4} \, \text{(in radians)}
\]

Now, we can calculate the radius at this latitude:

\[
R = 6370 \cdot \cos\left(45°\right)
\]

Since \( \cos(45°) = \frac{1}{\sqrt{2}} \approx 0.7071 \):

\[
R = 6370 \cdot 0.7071 \approx 4500 \, \text{km}
\]

4. **Finding the Circumference**: The circumference of the circle at latitude 45° N is:

\[
C = 2 \pi R = 2 \pi \cdot 4500 \approx 28274 \, \text{km}
\]

5. **Great Circle Distance**: Since the two points are 180° apart in longitude, they are half the circumference apart.

To find the distance:

\[
\text{Distance} = \frac{C}{2} = \frac{28274 \, \text{km}}{2} \approx 14137 \, \text{km}
\]

So the great circle distance via North between points X and Y is approximately **14137 kilometers**.