The relay satellite transmits the television signals continuously from one part of the world to another because its

$A$ 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 ...... $km s^{-1}$.

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?

The radius of the orbit of a geostationary satellite is (mean radius of earth is $R$,angular velocity about own axis is $\omega$,and acceleration due to gravity on earth's surface is $g$).

$A$ satellite of mass $\frac{M}{2}$ is revolving around the Earth in a circular orbit at a height of $\frac{R}{3}$ from the Earth's surface. The angular momentum of the satellite is $M \sqrt{\frac{GMR}{x}}$. The value of $x$ is . . . . . . ,where $M$ and $R$ are the mass and radius of the Earth,respectively. ($G$ is the gravitational constant)

The distance between two stars of masses $3 M_S$ and $6 M_S$ is $9 R$. Here $R$ is the mean distance between the centers of the Earth and the Sun,and $M_S$ is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits with period $n T$,where $T$ is the period of Earth's revolution around the Sun. The value of $n$ is. . . . .