Question

On July 19, 1969, the lunar orbit of Apollo 11 was adjusted to an average height of 122 kilometers above the Moon's surface. The radius of the Moon is 1840 kilometers, and the mass of the Moon is 7.3 x 1022 kilograms.a. At what speed did the spacecraft orbit the Moon? Include units in your answer.b. How long did it take to orbit once? Include units in your answer.All answers must be in 3 significant digits.

Answer 1

Given,

The height at which the satellite was orbiting the moon, h=122 km=122×10³ m

The radius of the moon, R=1840×10³ m

The mass of the moon, m=7.3×10²² kg

a.

The orbital velocity of the Apollo 11 is given by,

`v=\sqrt[]{(GM)/(R+h)}`

Where G=6.67×10⁻¹¹ m³ kg⁻¹ s⁻² is the gravitational constant.

On substituting the known values,

`\begin{gathered} v=\sqrt[]{(6.67*10^(-11)*7.3*10^(22))/(1840*10^3+122*10^3)} \\ =1.58*10^3\text{ m/s} \end{gathered}`

Thus the orbital speed of the spacecraft is 1.58×10³ m/s

b.

The time it takes for the spacecraft to orbit once is called the time period of the spacecraft.

The time period of the spacecraft is given by,

`T=2\pi\sqrt[]{((R+h)^3)/(GM)}`

On substituting the known values,

`\begin{gathered} T=2\pi*\sqrt{\frac{(1840*10^3+122*10^3)^3}{^{}6.67*10^(-11)*7.3*10^(22)}} \\ =7.83*10^(13)\text{ s} \end{gathered}`

Converting the period to hours,

`\begin{gathered} T=(7.83*10^(13))/(3600) \\ =2.18*10^(10)\text{ hr} \end{gathered}`

Thus the time it takes for the spacecraft to orbit once is 7.83×10¹³ s, that is 2.18×10¹⁰ hr