$A$ double star is a system of two stars rotating about their centre of mass only under their mutual gravitational attraction. Let the stars have masses $m$ and $2m$ and their separation be $l$. Their time period of rotation about their centre of mass will be proportional to:

$A$ planet of mass $M$ has two natural satellites with masses $m_1$ and $m_2$. The radii of their circular orbits are $R_1$ and $R_2$ respectively. Ignore the gravitational force between the satellites. Define $v_1, L_1, K_1$ and $T_1$ to be,respectively,the orbital speed,angular momentum,kinetic energy,and time period of revolution of satellite $1$; and $v_2, L_2, K_2$ and $T_2$ to be the corresponding quantities of satellite $2$. Given $m_1/m_2 = 2$ and $R_1/R_2 = 1/4$,match the ratios in List-$I$ to the numbers in List-$II$.

List-$I$	List-$II$
$P. \frac{v_1}{v_2}$	$1. \frac{1}{8}$
$Q. \frac{L_1}{L_2}$	$2. 1$
$R. \frac{K_1}{K_2}$	$3. 2$
$S. \frac{T_1}{T_2}$	$4. 8$

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

The orbital speed of Jupiter is

Two particles of identical mass are moving in circular orbits under a potential given by $V(r) = K r^{-n}$,where $K$ is a constant. If the radii of their orbits are $r_1$ and $r_2$ and their speeds are $v_1$ and $v_2$,respectively,then:

Two satellites are launched at a distance $R$ from a planet of negligible radius. Both satellites are launched in the tangential direction. The first satellite launches correctly at a speed $v_0$ and enters a circular orbit. The second satellite,however,is launched at a speed $\frac{1}{2}v_0$. What is the minimum distance between the second satellite and the planet over the course of its orbit?