At $7:14$A.M. on June $30,1908$, a huge explosion 1 occurred above remote central Siberia, at latitude $61°N$and longitude $102°E$; the fireball thus created was the brightest flash seen by anyone before nuclear weapons. The Tunguska Event, which according to one chance witness “covered an enormous part of the sky,” was probably the explosion of a stony asteroid about$140m$ wide. (a) Considering only Earth’s rotation, determine how much later the asteroid would have had to arrive to put the explosion above Helsinki at longitude $25°E$. This would have obliterated the city. (b) If the asteroid had, instead, been a metallic asteroid, it could have reached Earth’s surface. How much later would such an asteroid have had to arrive to put the impact in the Atlantic Ocean at longitude$20°W$ ? (The resulting tsunamis would have wiped out coastal civilization on both sides of the Atlantic.)

Step by step solution

01

Given

Coordinates of explosion -$(61°\text{\hspace{0.17em}N},102°\text{\hspace{0.17em}E})$

02

Understanding the concept

Use the concept of angular velocity and angular displacement. First, find angular displacement and then, using the speed of earth, find the time taken by the asteroid to arrive to put in the explosion.

Formulae:

$\omega =\frac{\theta }{t}$

03

(a) Calculate how much later the asteroid would have had to arrive

Time taken by the asteroid to arrive to put explosion above Helsinki at longitude ${25}^0$E:

First we find angular separation:

$\begin{array}{c}\omega =\frac{1\text{\hspace{0.17em}rev}}{1\text{\hspace{0.17em}day}}\\ =\frac{{360}^{\text{o}}}{24\text{\hspace{0.17em}h}}\end{array}$

Angular velocity of earth in one day:

$\begin{array}{c}\omega =\frac{1\text{\hspace{0.17em}rev}}{1\text{\hspace{0.17em}day}}\\ =\frac{{360}^{\text{o}}}{24\text{\hspace{0.17em}h}}\end{array}$

Time taken for 1 day:

$\omega =\frac{\theta }{t}$

$\begin{array}{c}t=\frac{\theta }{\omega }\\ =\frac{{77}^{\text{o}}}{\frac{{360}^{\text{o}}}{24}}\\ =5.13\text{\hspace{0.17em}h}\end{array}$

$\begin{array}{c}t=5.13\text{\hspace{0.17em}h}\\ \approx 5.1\text{\hspace{0.17em}h}\end{array}$

04

(b) Calculate how much later would such an asteroid have had to arrive if the asteroid had been metallic

Time taken by the asteroid to arrive to put impact in the Atlantic Ocean at longitude${20}^0$ W:

Similarly, angular separation for this is

$\begin{array}{c}\theta ={\theta }_2-{\theta }_1\\ ={102}^{\text{o}}-(-{20}^{\text{o}})\\ ={122}^{\text{o}}\end{array}$

Time taken for 1 day:

$\omega =\frac{\theta }{t}$

$\begin{array}{c}t=\frac{\theta }{\omega }\\ =\frac{{122}^0}{\frac{{360}^0}{24}}\\ =8.13\text{\hspace{0.17em}h}\end{array}$

$\begin{array}{l}t=8.13\\ t\approx 8.1\text{\hspace{0.17em}h}\end{array}$

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