A very crude model of a typical comet nucleus is a cube of ice (density \(1000 \mathrm{~kg} / \mathrm{m}^{3}\) ) \(10 \mathrm{~km}\) on a side. (a) What is the mass of this nucleus? (b) Suppose \(1 \%\) of the mass of the nucleus evaporates away to form the comet's tail. Suppose further that the tail is 100 million \(\left(10^{8}\right) \mathrm{km}\) long and 1 million \(\left(10^{6}\right) \mathrm{km}\) wide. Estimate the average density of the tail (in \(\mathrm{kg} / \mathrm{m}^{3}\) ). For comparison, the density of the air you breathe is about \(1.2 \mathrm{~kg} / \mathrm{m}^{3}\). (c) In 1910 the Earth actually passed through the tail of Comet Halley. At the time there was some concern among the general public that this could have deleterious effects on human health. Was this concern justified? Why or why not?

Step by step solution

01

Calculate the mass of the comet nucleus

First, we calculate the volume of the comet nucleus, which is a cube with a side of 10 km or \(10^{4}\) m. Therefore, the volume \(V\) of the nucleus can be calculated using the formula for a cube's volume: \(V = a^3\), where \(a\) is the side length of the cube. After calculating the volume of the comet, we obtain the mass \(m\) by multiplying the volume by the density \(\rho\) of ice: \(m = \rho \cdot V\)

02

Estimate the average density of the comet's tail

We'll assume that 1% of the mass of the nucleus evaporates to form the tail. The volume of the tail can be approximated by a cylinder with a length of \(10^{11}\) m and radius of \(10^{9}\) m. So, the volume of the tail \(V_{tail}\) can be calculated with the formula for a cylinder's volume: \(V_{tail} = \pi \cdot r^2 \cdot h\), where \(r\) is the radius and \(h\) is the height (considered as the length of the tail). After that, the average density of the tail can be calculated by dividing the mass in the tail (1% of the mass of the nucleus) by the volume of the tail: \(\rho_{tail} = \frac{m_{tail}}{V_{tail}}\)

03

Analyze the potential effects on human health

The concern about the possible effects on human health can be evaluated by comparing the density of the comet's tail with the density of air (Noting that the density of air is about \(1.2 \mathrm{~kg} / \mathrm{m}^{3}\)). As the average density of the tail is much lesser than the density of air, there should be no harmful effect on human health even if Earth passed through the tail of a comet. Though there might be other effects due to chemical composition, those are not within the bounds of this exercise.

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