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Sedaia [141]
3 years ago
13

"Opportunity" and "Phoenix" are two of the robotic explorers on Mars. Opportunity landed at 2° south latitude, where Mars’ radiu

s is about 2110 miles. Phoenix landed at 68° north latitude, where Mars’ radius is about 790 miles. Mars rotates on its axis once every 24.6 Earth-hours. How far does each explorer travel as Mars rotates by 1 radian? How many hours does it take Mars to rotate 1 radian? Using this answer, how fast is each explorer traveling around Mars’ axis in miles per hour?
Mathematics
1 answer:
kipiarov [429]3 years ago
4 0

Answer:

(a)Distance traveled by each explorer travel as Mars rotates by 1 radian

  • Opportunity=2108.71 miles
  • Phoenix =295.94 miles

(b)Number of hours it takes Mars to rotate 1 radian=3.9152 hours

(c)Speed of each explorer around Mars.

  • Opportunity=538.59 miles per hour
  • Phoenix =75.59 miles per hour

Step-by-step explanation:

<u>Part A</u>

For any given parallel of latitude\text{Circumference}=2\pi R \cos \beta$ where \beta$ is the angle of latitude.

Opportunity landed at 2° south latitude, where Mars’ radius is about 2110 miles.

\text{Circumference at 2\°S latitude}=2\pi*2110* \cos 2^\circ\\=13249.44$ miles

Phoenix landed at 68° north latitude, where Mars’ radius is about 790 miles.

\text{Circumference at 68\°N latitude}=2\pi*790* \cos 68^\circ\\=1859.44$ miles

<u>Part B</u>

Next, we determine the distance (Length of arc) covered by each explorer as Mars rotates by 1 radian.

\text{Length of arc (in radian)}=\dfrac{\theta}{2\pi} \times $Circumference

<u>Opportunity's Distance</u>

=\dfrac{1}{2\pi} \times 13249.44\\\\ =2108.71$ miles

<u>Phoenix's Distance</u>

=\dfrac{1}{2\pi} \times 1859.44\\\\ =295.94$ miles

<u>Part C</u>

Mars rotates on its axis once every 24.6 Earth-hours.

Therefore:

\dfrac{24.6}{2\pi} \approx $ 3.9152 hour per radian

<u>Part D:</u>Speed of each explorer

\text{Speed}=\dfrac{\text{Distance}}{\text{Time}}

\text{Speed of Opportunity = }\dfrac{2108.71}{3.9152}\\$=538.59 miles per hour\\\\Speed of Phoenix =\dfrac{295.94}{3.9154}\\=75.59$ miles per hour

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