The orbital velocity of an artificial satellite in a circular orbit very close to the Earth is $v$. The velocity of a geostationary satellite orbiting in a circular orbit at an altitude of $3R$ from the Earth's surface will be

Suppose the gravitational force varies inversely as the $n^{th}$ power of the distance. Then,the time period of a planet in circular orbit of radius $R$ around the sun will be proportional to

Express the constant $k$ of $T^{2}=k(R_{E}+h)^{3}$ in days and kilometres. Given $k = 10^{-13} \; s^{2} \; m^{-3}$. The moon is at a distance of $3.84 \times 10^{5} \; km$ from the earth. Obtain its time period of revolution in days.

$A$ spaceship orbits around a planet at a height of $20 \, km$ from its surface. Assuming that only the gravitational field of the planet acts on the spaceship, what will be the number of complete revolutions made by the spaceship in $24 \, hours$ around the planet? [Given: Mass of planet $= 8 \times 10^{22} \, kg$, Radius of planet $= 2 \times 10^6 \, m$, Gravitational constant $G = 6.67 \times 10^{-11} \, N \cdot m^2/kg^2$]

$A$ satellite is to be placed in an equatorial geostationary orbit around the Earth for communication purposes. The height of such a satellite is $(M_{E} = 6 \times 10^{24} \,kg, R_{E} = 6400 \,km)$.

What is the ratio of the time periods of a satellite $A$ of mass $m$ at a distance $r$ from the center of the Earth and a satellite $B$ of mass $2m$ at a distance $2r$ from the center of the Earth?