Question 82 - Review - Chapter 1

Problem:

You are crossing the Atlantic in a small sailboot and hoping to make landfall in the Azores. The highest peak of the Azores is 2300 m in altitude.

Question:

At what distance you can see the peak just emerging over the horizon?

(Assume that your eye is (almost) at the level of the water.

Solution:

The tangent of a circle is perpendicular to the radius of the circle in the tangent point, so the triangle abc forms a right triangle. The line on which we see the mountain top the earliest is just the tangent of the circle at the point where our eye is. The side A of the triangle is known, this is the radius of the earth. The side C is also known, this is the radius + the height of the mountain.

Using the Pythagorean theorem, we can therefore calculate B side

$ \begin{align*} & C^2 = A^2 + B^2 \iff B = \sqrt{C^2 - A^2} \end{align*} $

# Calculations
import numpy as np
from astropy.constants import R_earth
Height = 2300
A = R_earth.value
C = R_earth.value + Height
B = np.sqrt(C**2 - A**2)

print(f'The maximum distance at which we see the mountain top {B:.0f} m -or- {B/1000:.1f} km')

The maximum distance at which we see the mountain top 171303 m -or- 171.3 km

We can also determine the angle a

$ a_{rad} = \arccos{\frac{A}{B}} $

We then can calculate the distance over the surface:

The circumference of the earth is $Radius * 2 * \pi$

The arc length is:

$$\frac{circumference * a_{rad}}{2 * \pi}$$

a = np.arccos(A/C)
circ = 2* np.pi * A
distance = (circ * a )/(2*np.pi)
print(f'The angle in rad is: {a}')
print(f'The maximum distance over surface at which we see the mountain top {distance:.0f} m -or- {distance/1000:.1f} km')

The angle in rad is: 0.026851466493741847
The maximum distance over surface at which we see the mountain top 171261 m -or- 171.3 km