Chapter 18, Problem 12P

To determine

The number of impacts expected to strike a $100{\text{m}}^{\text{2}}$ region in one hour during Earth’s formation.

Answer to Problem 12P

The number of impacts expected to strike a $100{\text{m}}^{\text{2}}$ region in one hour during Earth’s formation is $133$.

Explanation of Solution

Write the expression for the total particles needed.

    $\text{total}\text{particle}\text{needed}=\frac{M_{\text{E}}}{m_{\text{p}}}$        (I)

Here, $M_{\text{E}}$ is the mass of Earth and $m_{\text{p}}$ is the average mass of each particle.

Write the expression for the particles captured by Earth.

    $\text{Particles}\text{captured}\text{by}\text{Earth}=\frac{\text{Total}\text{particles}\text{needed}}{t}$        (II)

Here, $t$ is the time required to capture the particles.

Write the expression for the surface area of the Earth.

    $A=4\pi R_{\text{E}}^2$        (III)

Here, $R_{\text{E}}$ is the radius of the Earth.

Write the expression for the fraction of the total surface area.

    $\text{fraction}\text{of}\text{total}\text{surface}\text{area}=\frac{S_{\text{r}}}{S_{\text{E}}}$        (IV)

Here, $S_{\text{r}}$ is the surface area of the region and $S_{\text{E}}$ is the surface area of Earth.

Write the expression for the number of impacts in the given region.

    $\text{number}\text{of}\text{impacts}\text{in}\text{100}{\text{m}}^2/\sec =\left[\begin{array}{l}\left(\text{fraction}\text{of}\text{total}\text{surface}\text{area}\right)\times \\ \left(\text{particles}\text{captured}\text{by}\text{Earth}/\sec \right)\end{array}\right]$        (V)

Conclusion:

Refer to Table A-10, “Properties of the planets” to obtain the value of $M_{\text{E}}$ as $5.97\times {10}^{24}\text{kg}$.

Substitute $5.97\times {10}^{24}\text{kg}$ for $M_{\text{E}}$ and $100\text{gm}$ for $m_{\text{p}}$ in equation (I) to find the total particles needed.

    $\begin{array}{c}\text{total}\text{particle}\text{needed}=\frac{5.97\times {10}^{24}\text{kg}}{100\text{gm}\times \left(\frac{1\text{kg}}{1000\text{gm}}\right)}\\ =5.97\times {10}^{25}\end{array}$

Substitute $5.97\times {10}^{25}$ for total particles needed and 10 million years for $t$ in equation (II) to find the particles captured by Earth.

    $\begin{array}{c}\text{Particles}\text{captured}\text{by}\text{Earth}=\frac{5.97\times {10}^{25}}{10\text{million}\text{years}\times \left(\frac{{10}^6\text{year}}{\text{1}\text{million}\text{years}}\right)\times \left(\frac{3.15\times {10}^7\text{sec}}{1\text{year}}\right)}\\ =\frac{5.97\times {10}^{25}}{3.15\times {10}^{14}\sec }\\ =1.9\times {10}^{11}\text{particles}/\text{sec}\end{array}$

Substitute $6.38\times {10}^6\text{m}$ for $R_{\text{E}}$ in equation (III) to find the surface area of the Earth.

    $\begin{array}{c}A=4\pi {\left(6.38\times {10}^6\text{m}\right)}^2\\ =4\pi \times 40.7044\times {10}^{12}{\text{m}}^{\text{2}}\\ =5.12\times {10}^{14}{\text{m}}^{\text{2}}\end{array}$

Substitute $100{\text{m}}^{\text{2}}$ for $S_{\text{r}}$ and $5.12\times {10}^{14}{\text{m}}^{\text{2}}$ for $S_{\text{E}}$ in equation (III) to find the fraction of the total surface area.

    $\begin{array}{c}\text{fraction}\text{of}\text{total}\text{surface}\text{area}=\frac{100{\text{m}}^{\text{2}}}{5.12\times {10}^{14}{\text{m}}^{\text{2}}}\\ =1.95\times {10}^{-13}\end{array}$

Substitute $1.95\times {10}^{-13}$ for fraction of the total surface area and $1.9\times {10}^{11}\text{particles}/\text{sec}$ for $\text{particles}\text{captured}\text{by}\text{Earth}/\sec$ in equation (V) to find $\text{number}\text{of}\text{impacts}\text{in}\text{100}{\text{m}}^2/\sec$.

    $\begin{array}{c}\text{number}\text{of}\text{impacts}\text{in}\text{100}{\text{m}}^2/\sec =\left[1.95\times {10}^{-13}\times 1.9\times {10}^{11}\text{particles}/\text{sec}\right]\\ =3.705\times {10}^{-2}\text{particles}/\text{sec}\\ =\frac{3.705\times {10}^{-2}\text{particles}}{\mathrm{s}\text{econd}}\times \left(\frac{1\mathrm{s}\text{econd}}{\frac{1}{3600}\text{hr}}\right)\\ =133.38\text{particles}/\text{hr}\\ \approx 133\text{particles}/\text{hr}\end{array}$

Therefore, the number of impacts expected to strike a $100{\text{m}}^{\text{2}}$ region in one hour during Earth’s formation is $133$.