A space probe projected from the earth moves round the moon in a circular orbit at a distance equal to its radius Rₘₒₒₙ= R/4 where R = radius of the earth. Its rocket launcher moves in circular orbit around the earth at a distance equal to R from its surface. The ratio of the times taken for one revolution by the probe and the rocket launcher is (Mₘₒₒₙ = M/80, where M = mass of the earth))

The correct answer is √5:2.

Step-by-Step Explanation:

1. Identify the given parameters:
- Radius of the Earth = $R$
- Mass of the Earth = $M$
- Radius of the Moon = $R_{\mathrm{moon}}=\frac{R}{4}$
- Mass of the Moon = $M_{\mathrm{moon}}=\frac{M}{80}$

2. Determine the orbital radii:
- The space probe moves around the Moon in a circular orbit at a distance equal to its radius (meaning a height of $R_{\mathrm{moon}}$ above the Moon's surface). Therefore, the orbital radius of the probe ($r_p$) from the center of the Moon is:
$r_p=R_{\mathrm{moon}}+R_{\mathrm{moon}}=2R_{\mathrm{moon}}=2\left(\frac{R}{4}\right)=\frac{R}{2}$
- The rocket launcher moves around the Earth in a circular orbit at a distance equal to $R$ from the Earth's surface. Therefore, the orbital radius of the launcher ($r_L$) from the center of the Earth is:
$r_L=R+R=2R$

3. Use the formula for the time period of a circular orbit:
The time period of one revolution of a satellite orbiting a central body of mass $M_c$ at a radius $r$ is given by Kepler's Third Law:
$T=2\pi \sqrt{\frac{r^3}{G{M}_c}}$

4. Calculate the ratio of the time periods:
For the space probe:
$T_p=2\pi \sqrt{\frac{{r_p}^3}{G{M}_{\mathrm{moon}}}}=2\pi \sqrt{\frac{{\left(R/2\right)}^3}{G\left(M/80\right)}}=2\pi \sqrt{\frac{80R^3}{8GM}}=2\pi \sqrt{\frac{10R^3}{GM}}$
For the rocket launcher:
$T_L=2\pi \sqrt{\frac{{r_L}^3}{GM}}=2\pi \sqrt{\frac{{\left(2R\right)}^3}{GM}}=2\pi \sqrt{\frac{8R^3}{GM}}$
Taking the ratio of $T_p$ to $T_L$:
$\frac{T_p}{T_L}=\frac{2\pi \sqrt{\frac{10R^3}{GM}}}{2\pi \sqrt{\frac{8R^3}{GM}}}=\sqrt{\frac{10}{8}}=\sqrt{\frac{5}{4}}=\frac{\sqrt{5}}{2}$

Thus, the ratio of the times taken for one revolution by the probe and the rocket launcher is √5:2.