The orbit of a geostationary satellite is circular. The time period of the satellite depends on $(i)$ mass of the satellite,$(ii)$ mass of the earth,$(iii)$ radius of the orbit,and $(iv)$ height of the satellite from the surface of the earth.

$A$ remote-sensing satellite of Earth revolves in a circular orbit at a height of $0.25 \times 10^6 \, m$ above the surface of Earth. If Earth's radius is $6.38 \times 10^6 \, m$ and $g = 9.8 \, m s^{-2}$,then the orbital speed of the satellite is ...... $km s^{-1}$.

Consider a binary star system consisting of two stars of masses $M_1$ and $M_2$ separated by a distance of $30 \text{ AU}$ with a period of revolution equal to $30 \text{ years}$. If one of the two stars is $5$ times farther from the centre of mass than the other, find the masses of the two stars in terms of solar masses.

An earth satellite of mass $m$ revolves in a circular orbit at a height $h$ from the surface of the earth. $R$ is the radius of the earth and $g$ is acceleration due to gravity at the surface of the earth. The velocity of the satellite in the orbit is given by

$A$ planet revolves around the sun in an elliptical orbit. When it is closest to the sun,its distance from the sun is $d_1$ and its velocity is $v_1$. When it is farthest from the sun,its distance from the sun is $d_2$ and its velocity is $v_2$. Find the value of $v_2$.

Two satellites $A$ and $B$ of masses $200 \, kg$ and $400 \, kg$ are revolving around the Earth at heights of $600 \, km$ and $1600 \, km$ respectively. If $T_{A}$ and $T_{B}$ are the time periods of $A$ and $B$ respectively,then find the value of $T_{B} - T_{A}$.
[Given: Radius of Earth $R = 6400 \, km$,Mass of Earth $M = 6 \times 10^{24} \, kg$,$G = 6.67 \times 10^{-11} \, Nm^{2}/kg^{2}$]