Chapter 10, Problem 65QAP

To determine

(a)

The semimajor axis of Apophis

Answer to Problem 65QAP

Semimajor axis $=0.9225au=1.38\times {10}^{11}m$

Explanation of Solution

Given info:

Aphelion = $1.099au$

Perihelion = $0.746au$

Mass of Apophis = $2.7\times {10}^{10}kg$

Formula used:

Semimajor axis $=\frac{R_a+R_p}{2}$

  $R_a=$ Aphelion

  $R_p=$ Perihelion

  $T=2\pi \sqrt{\frac{a^3}{GM}}$

  $T=$ Time

  $G=$ Gravitational constant

  $M=$ Mass

  $a=$ Semimajor axis

  $v=\sqrt{\frac{2GM}{R}}$

  $v=$ Velocity

  $R=$ Radius

Calculation:

Semimajor axis,

Semimajor axis $=\frac{R_a+R_p}{2}=\frac{1.099au+0.746au}{2}=0.9225au$

Semimajor axis $=0.9225au\times 1.496\times {10}^{11}=1.38\times {10}^{11}m$

Conclusion:

Semimajor axis $=0.9225au=1.38*{10}^{11}m$

To determine

(b)

Time taken by Apophis to orbit the Sun

Answer to Problem 65QAP

Time taken by Apophis to orbit the Sun $=323.5days$

Explanation of Solution

Given info:

Aphelion = $1.099au$

Perihelion = $0.746au$

Mass of Apophis = $2.7\times {10}^{10}kg$

Formula used:

Semimajor axis $=\frac{R_a+R_p}{2}$

  $R_a=$ Aphelion

  $R_p=$ Perihelion

  $T=2\pi \sqrt{\frac{a^3}{GM}}$

  $T=$ Time

  $G=$ Gravitational constant

  $M=$ Mass

  $a=$ Semimajor axis

  $v=\sqrt{\frac{2GM}{R}}$

  $v=$ Velocity

  $R=$ Radius

Calculation:

Time taken by Apophis to orbit the Sun,

  $T=2\pi \sqrt{\frac{a^3}{GM}}=2\pi \sqrt{\frac{{(1.38\times {10}^{11}m)}^3}{(6.67\times {10}^{-11}N{m}^2/k{g}^2)\times 1.989\times {10}^{30}kg}}=2.80\times {10}^{7}s=323.5\text{days}$

Conclusion:

Time taken by Apophis to orbit the Sun $=323.5days$

To determine

(c)

Points of slowest and fastest of Apophis

Answer to Problem 65QAP

Apophis travels fastest at perihelion and slowest at aphelion

Explanation of Solution

Given info:

Aphelion = $1.099au$

Perihelion = $0.746au$

Mass of Apophis = $2.7\times {10}^{10}kg$

Formula used:

Semimajor axis $=\frac{R_a+R_p}{2}$

  $R_a=$ Aphelion

  $R_p=$ Perihelion

  $T=2\pi \sqrt{\frac{a^3}{GM}}$

  $T=$ Time

  $G=$ Gravitational constant

  $M=$ Mass

  $a=$ Semimajor axis

  $v=\sqrt{\frac{2GM}{R}}$

  $v=$ Velocity

  $R=$ Radius

Calculation:

Point Apophis travel fastest = at perihelion

Point Apophis travel slowest = at aphelion

Conclusion:

Apophis travels fastest at perihelion and slowest at aphelion

To determine

(d)

Ratio of its maximum speed to minimum speed

Answer to Problem 65QAP

Ratio of the maximum speed to minimum speed $=1.74$

Explanation of Solution

Given info:

Aphelion = $1.099au$

Perihelion = $0.746au$

Mass of Apophis = $2.7\times {10}^{10}kg$

Formula used:

Semimajor axis $=\frac{R_a+R_p}{2}$

  $R_a=$ Aphelion

  $R_p=$ Perihelion

  $T=2\pi \sqrt{\frac{a^3}{GM}}$

  $T=$ Time

  $G=$ Gravitational constant

  $M=$ Mass

  $a=$ Semimajor axis

  $v=\sqrt{\frac{2GM}{R}}$

  $v=$ Velocity

  $R=$ Radius

Calculation:

Ratio of the maximum speed to minimum speed,

  $\begin{array}{l}v\alpha \frac{1}{R}\\ \frac{v_{{}_p}}{v_a}=\frac{R_{{}_a}}{R_p}=\frac{1.099au}{0.746au}=1.74\end{array}$

Conclusion:

Ratio of the maximum speed to minimum speed $=1.74$