Chapter 12, Problem 41QP

(a)

To determine

The number of times the Halley’s Comet reappeared in the sky since its early sighting.

Answer to Problem 41QP

The Halley’s Comet has reappeared for $29$ times since its first observed appearance in the sky.

Explanation of Solution

The Halley’s Comet was first observed in the sky in $240\text{BCE}$. The time since the first appearance of the Halley’s Comet is the sum of the years passed and the $240\text{years}$ BCE.

Write the expression for the time elapsed since the first appearance of comet.

  $T=n+240\text{years}$        (I)

Here, $T$ is the time elapsed and $n$ is the number of years passed after BCE.

Write the expression for the number of reappearance of the Halley’s Comet.

  $N=\frac{T}{T_{Halley}}$        (II)

Here, $N$ is the number of reappearance of the comet and $T_{Halley}$ is the orbital time period of the comet Halley.

Conclusion:

Substitute $2019$ for $n$ in equation (I).

  $\begin{array}{c}T=2019+240\text{years}\\ =\text{2259}\text{years}\end{array}$

Substitute $2259\text{years}$ for $T$ and $76.4\text{years}$ for $T_{Halley}$ in equation (II).

  $\begin{array}{c}N=\frac{2259\text{years}}{76.4\text{years}}\\ =29.56\\ \approx 29\end{array}$

Thus, the Halley’s Comet has reappeared for $29$ times since its first observed appearance in the sky.

(b)

To determine

The mass lost by the comet since its first appearance in $240\text{BCE}$.

Answer to Problem 41QP

The total mass lost by the comet since its early sighting is $8.7\times {10}^{12}\text{kg}$.

Explanation of Solution

The every comet when comes near the sun, the comet action takes place and the body losses some of material as the action occurs.

The Halley’s Comet has appeared for $29$ times since its first appearance in the sky.

Write the expression for the mass lost by the comet during all its appearance.

  $m=N\cdot m_L$        (III)

Here, $m$ is the total mass lost by the comet and $m_L$ is the mass lost by the comet each time it passes near the sun.

Conclusion:

Substitute $29$ for $N$ and $3\times {10}^{11}\text{kg}$ for $m_L$ in equation (III).

  $\begin{array}{c}m=29\left(3\times {10}^{11}\text{kg}\right)\\ =8.7\times {10}^{12}\text{kg}\end{array}$

Thus, the total mass lost by the comet since its early sighting is $8.7\times {10}^{12}\text{kg}$.

(c)

To determine

The percentage of the comet’s total mass lost by the comet.

Answer to Problem 41QP

The mass lost by the Halley’s Comet is about $4\%$ of the total mass of comet.

Explanation of Solution

Write the expression for the total mass of the comet at its first sight.

  $M=m+m_H$        (IV)

Here, $M$ is the total mass of the comet and $m_H$ is the current mass of the Halley’s Comet.

Write the expression for the percentage of mass of the comet as of its initial mass.

  $x=\frac{m}{M}\times 100\%$        (V)

Here, $x$ is the percentage of mass.

Conclusion:

Substitute $8.7\times {10}^{12}\text{kg}$ for $m$ and $2.2\times {10}^{14}\text{kg}$ for $m_H$ in equation (IV).

  $\begin{array}{c}M=8.7\times {10}^{12}\text{kg}+2.2\times {10}^{14}\text{kg}\\ =2.287\times {10}^{14}\text{kg}\end{array}$

Substitute $2.287\times {10}^{14}\text{kg}$ for $M$ and $8.7\times {10}^{12}\text{kg}$ for $m$ in equation (V).

  $\begin{array}{c}x=\frac{8.7\times {10}^{12}\text{kg}}{2.287\times {10}^{14}\text{kg}}\times 100\%\\ =3.8\%\\ \approx 4\%\end{array}$

Thus, the mass lost by the Halley’s Comet is about $4\%$ of the total mass of comet.