Calculating the Drop Due to Earth's Curvature: A Comprehensive Guide

Introduction

Understanding the drop due to the curvature of the Earth is crucial for various applications, from laying out infrastructure to scientific research. This guide provides a detailed explanation of how to calculate this drop using both a simple formula and a more precise Pythagorean approach.

Leveraging the Simple Formula

The curvature of the Earth can be approximated using a straightforward formula derived from spherical geometry. For practical purposes, the Earth is considered a sphere with a radius of approximately 6,371 kilometers (3,959 miles).

The formula for the drop over a distance (d) in kilometers is:

Drop ( approx frac{d^2}{2R} )

Where:

R is the radius of the Earth (approximately 6,371 km). d is the distance from the observer to the object in kilometers.

Steps to Calculate the Drop

Determine the distance (d): Measure the distance to the object in kilometers. Use the formula: Plug the distance into the formula to find the drop.

Example Calculation

Let's illustrate with an example. Suppose you want to calculate the drop over a distance of 10 kilometers:

Set (d 10) km. Use the formula:

Drop ( approx frac{10^2}{2 times 6371} frac{100}{12742} approx 0.00785 ) km (approx 7.85) m.

Therefore, the drop due to the curvature of the Earth over a distance of 10 kilometers is approximately 7.85 meters.

Refining the Calculation Using Pythagorus

For more precision, especially at longer distances or with specific applications, a more accurate method is needed. This involves the Pythagorean theorem.

Consider a spherical Earth with radius (r 6371) km. The horizontal distance over which you want to calculate drop is (d), and the required drop is (h). The relationship is:

( r - h sqrt{r^2 - d^2} )

Rearranging for (h), we get:

( h r - sqrt{r^2 - d^2} )

Example Calculation

For (r 6371) km and (d 10) km:

( sqrt{6371^2 - 10^2} sqrt{40589641 - 100} sqrt{40589741} 6371.007848056695... ) km. ( h 6371 - 6371.007848056695 approx 0.007848056695 ) km (approx 7.848) m.

Thus, the drop over 10 kilometers using the Pythagorean method is very close to 7.85 meters, aligning with the simple formula.

Handling Precision Issues

When dealing with very small distances, precision issues arise due to the subtraction of two large numbers. A solution to mitigate this is to use the binomial theorem:

( h r - r sqrt{1 - left(frac{d}{r}right)^2} approx r left(1 - left(frac{d}{r}right)^2 frac{1}{2} left(frac{d}{r}right)^4 right) )

For (d 10) km and (r 6371) km, this simplifies to:

( frac{d^2}{2r} frac{10^2}{2 times 6371} frac{100}{12742} approx 0.007848 ) km (approx 7.848) m.

This match confirms the accuracy of both methods, especially for small distances.

Further Considerations

The use of the Earth's curvature in practical applications is not just theoretical. For example, when laying out long stretches of railway or pipeline, the drop must be accurately calculated using the square of the distance. If you need to lay out a 2-mile stretch, the drop is (4 times 8) inches, which is 32 inches, and for a 3-mile stretch, it is (9 times 8) inches, or 72 inches (6 feet).

Conclusion

Understanding and calculating the drop due to the curvature of the Earth is essential for accurate measurements and applications. Whether using a simple formula or a more precise method, these techniques provide reliable results for various scenarios.

References

[1] Robert Paxon's diagram.

[2] Microsoft Calculator for high precision results.