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 $\varphi$ is:

   $$R=R_{earth}\cdot \cos (\varphi )$$

   Where:

   • $R$ is the radius of the circle at latitude $\varphi$
   • $R_{earth}$ is the Earth's radius (6370 km)
   • $\varphi$ is the latitude in radians.

   For 45° N:

   $$\varphi =45°=\frac{\pi }{4},\text{(in radians)}$$

   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.