Chapter 5, Problem 65P

a)

To determine

The mean distance from the Sun.

Answer to Problem 65P

Solution:

  $179\text{AU}$

Explanation of Solution

Orbital period of the comment Hale-Bopp is 2400 years.

Formula Used:

Accordingto law if $r_{comet}\text{and}{r}_{Earth}$ are the mean distances of comet Hale-Bopp and Earth from the sun and $T_{comet,}{T}_{Earth}$ represent their periods then we write

  ${\left(\frac{r_{comet}}{{\text{r}}_{Earth}}\right)}^3={\left(\frac{T_{comet,}}{T_{Earth}}\right)}^2$

Solve above equations for $r_{comet}$

  ${\left(\frac{r_{comet}}{{\text{r}}_{Earth}}\right)}^3={\left(\frac{T_{comet,}}{T_{Earth}}\right)}^2$

  $\left(\frac{r_{comet}}{{\text{r}}_{Earth}}\right)={\left(\frac{T_{comet,}}{T_{Earth}}\right)}^{2/3}$

  $r_{comet}={\text{r}}_{Earth}{\left(\frac{T_{comet,}}{T_{Earth}}\right)}^{2/3}$

Calculation:

Substitute 1 $\text{AU}\text{for}{\text{r}}_{Earth},2400\text{y}\text{for}{T}_{comet}\text{, and}{T}_{Earth,}$

  $\begin{array}{l}{r}_{comet,}=\left(1\text{AU}\right){\left(\frac{\left(2400y\right)}{\left(1y\right)}\right)}^{\frac{2}{3}}\\ =179\text{AU}\end{array}$

Conclusion: Therefore, the mean distance of comet Hale-Bopp from the sun is $=179\text{AU}$

b)

To determine

The farthest distance

Answer to Problem 65P

Solution: $357\text{AU}$

Explanation of Solution

If the closest distance of the comet Hale-Bopp from the sun is $r_{\min }$ farthest distance $r_{\max }$ then the expression for mean distance $r_{\text{constant}}$ can be written as follows

  $\begin{array}{l}{r}_{\text{constant}}=\frac{r_{\min }+r_{\max }}{2}\\ \text{solve}\text{above}\text{equation}\text{for}{r}_{\max }\end{array}$

  $r_{\text{constant}}=\frac{r_{\min }+r_{\max }}{2}$

  $\begin{array}{l}{r}_{\max }=2r_{\text{constant}}-r_{\min }\\ =2\left(179\right)-1\\ =357\text{AU}\end{array}$

Conclusion:

The farthest distance is $357\text{AU}$ .

c)

To determine

The ratio speed at the point to the speed at the farthest point

Answer to Problem 65P

Solution: $\frac{357}{1}$

Explanation of Solution

Formula Used:

  $\begin{array}{l}\frac{1}{2}\left(v_{\max }\Delta t\right)v_{\min }=\frac{1}{2}\left(v_{\min }\Delta t\right)r_{\max }\\ \frac{v_{\max }}{v_{\min }}=\frac{r_{\max }}{r_{\min }}\end{array}$

Calculation:

Substitute $357\text{AU}\text{for}{r}_{\max },\text{and}1\text{AU}\text{for}{r}_{\min }$

  $\begin{array}{l}\frac{v_{\max }}{v_{\min }}=\frac{\text{357AU}}{\text{1AU}}\\ =\frac{357}{1}\end{array}$

Conclusion: Therefore, the ratio of the speed of the planet at the closest point to the farthest point $\frac{357}{1}$ .