Q082 - Physics For Engineers And Scientists - Solutions

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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.

Can the plane land?

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*} $

In [1]:
# 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:

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

In [2]:
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

سُوۡرَةُ حٰمٓ السجدة / فُصّلَت
ثُمَّ اسْتَوَىٰ إِلَى السَّمَاءِ وَهِيَ دُخَانٌ فَقَالَ لَهَا وَلِلْأَرْضِ ائْتِيَا طَوْعًا أَوْ كَرْهًا قَالَتَا أَتَيْنَا طَائِعِينَ ﴿١١
فَقَضَاهُنَّ سَبْعَ سَمَاوَاتٍ فِي يَوْمَيْنِ وَأَوْحَىٰ فِي كُلِّ سَمَاءٍ أَمْرَهَا ۚ وَزَيَّنَّا السَّمَاءَ الدُّنْيَا بِمَصَابِيحَ وَحِفْظًا ۚ ذَٰلِكَ تَقْدِيرُ الْعَزِيزِ الْعَلِيمِ ﴿١٢


Surah Fussilat: Thereafter turned He to the heaven and it was as smoke, and said Unto it and Unto the earth: come ye twain, willingly or loth. They said: we come willingly. (11) Then He decreed them as seven heavens in two days, and revealed Unto each heaven the command thereof; and We bedecked the nether heaven with lamps and placed therein a guard. That is the ordinance of the Mighty, the Knower. (12)

De hemelen en aarde (waren) een samenhangende massa. Wij hebben ze toen van elkaar gescheiden… Wij hebben de hemel tot een beschermend dak gemaakt … en de dag en de nacht, de zon en de maan geschapen. (Koran 41 11:12)


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