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:
-
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.
-
Earth's Radius: You're given the radius of the Earth as 6370 km.
-
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 \( \cos(45°) = \frac{1}{\sqrt{2}} \approx 0.7071 \):
\[ R = 6370 \cdot 0.7071 \approx 4500 , \text{km} \]
-
Finding the Circumference: The circumference of the circle at latitude 45° N is:
\[ C = 2 \pi R = 2 \pi \cdot 4500 \approx 28274 , \text{km} \]
-
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.
0 answers