$A$ satellite of mass $m$ revolves around the Earth of radius $R$ at a height $x$ from its surface. If $g$ is the acceleration due to gravity on the surface of the Earth,the orbital speed of the satellite is:

$A$ 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 $d_2$. What is its velocity at that point?

If a satellite has to orbit the earth in a circular path every $6 \text{ hrs}$, at what distance from the surface of the earth should the satellite be placed (in $\text{ km}$)? (Radius of earth $R_e = 6400 \text{ km}$)
(Assume $\frac{GM}{4\pi^2} = 8 \times 10^{12} \text{ m}^3\text{s}^{-2}$, where $G$ and $M$ are the gravitational constant and mass of earth, and $10^{1/3} = 2.1$)

$A$ planet is revolving around the sun in an elliptical orbit. Its closest distance from the sun is $r_{min}$,and the farthest distance from the sun is $r_{max}$. If the orbital angular velocity of the planet when it is nearest to the sun is $\omega$,then the orbital angular velocity at the point when it is at the farthest distance from the sun is:

Two satellites,$A$ and $B,$ have masses $m$ and $2m$ respectively. $A$ is in a circular orbit of radius $R,$ and $B$ is in a circular orbit of radius $2R$ around the earth. The ratio of their kinetic energies,$K.E._A / K.E._B ,$ is

$A$ particle of mass $m$ is under the influence of the gravitational field of a body of mass $M (\gg m)$. The particle is moving in a circular orbit of radius $r_0$ with time period $T_0$ around the mass $M$. Then,the particle is subjected to an additional central force,corresponding to the potential energy $V_c(r) = m \alpha / r^3$,where $\alpha$ is a positive constant of suitable dimensions and $r$ is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius $r_0$ in the combined gravitational potential due to $M$ and $V_c(r)$,but with a new time period $T_1$,then $(T_1^2 - T_0^2) / T_1^2$ is given by [ $G$ is the gravitational constant.]