During an Apollo lunar landing mission, the command module continued to orbit the Moon at an altitude of about 100 km. How long it take to go around the Moon once?

The speed of an object which is going in the orbit of a planet depends upon the radius of orbital. This can be used to find the expression for the time period.

The altitude of the Moon is \(R' = 100\;{\rm{km}}\).

The velocity of rotation can be calculated as,

\(v = \frac{{2\pi r}}{T}\)…… (i)

Here, r is the radius and T is the orbitalperiod.

The expression for the orbital speed is given as,

\(v = \sqrt {\frac{{G{M_{Moon}}}}{r}} \)…… (ii)

Here, G is the universal gravitational constant and its value is \(6.67 \times {10^{ - 11}}\;{\rm{N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/k}}{{\rm{g}}^{\rm{2}}}\).

On combining equation (i) and equation (ii),

\(\begin{aligned}\sqrt {\frac{{G{M_{{\rm{Moon}}}}}}{r}} &= \frac{{2\pi r}}{T}\\T &= 2\pi \sqrt {\frac{{{r^3}}}{{G{M_{{\rm{Moon}}}}}}} \end{aligned}\)

The radius of the Moon is,

\({R_{Moon}} = 1.74 \times {10^6}\;{\rm{m}}\)

The Mass of the Moon is,

\({M_{Moon}} = 7.35 \times {10^{22}}\;{\rm{kg}}\)

The time taken by the moon to around is,

\(\begin{aligned}T &= 2\pi \sqrt {\frac{{{{\left( {{R_{{\rm{Moon}}}} + 100\,{\rm{km}}} \right)}^3}}}{{G{M_{{\rm{Moon}}}}}}} \\ &= 2\pi \sqrt {\frac{{{{\left( {1.74 \times {{10}^6}\;{\rm{m}} + {\rm{100}}\;{\rm{km}}\left( {\frac{{1000\;{\rm{m}}}}{{1\;{\rm{km}}}}} \right)} \right)}^3}}}{{\left( {6.67 \times {{10}^{ - 11}}\;{\rm{N}} \cdot {{\rm{m}}^2}/{\rm{k}}{{\rm{g}}^2}} \right)\left( {7.35 \times {{10}^{22}}\;{\rm{kg}}} \right)}}} \\ &= 7.1 \times {10^3}\;{\rm{s}} \times \frac{{1\;{\rm{hr}}}}{{3600\;{\rm{s}}}}\\ \approx 2\;{\rm{hrs}}\end{aligned}\)

Thus, moon will take nearly two hours to complete the round.