Question

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

Answer 1

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:

Part A

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.

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

Part B

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

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

Opportunity's Distance

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

Phoenix's Distance

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

Part C

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

Therefore:

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

Part D:Speed of each explorer

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

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