If an artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of the escape velocity from the earth,the height of the satellite above the surface of the earth is

$A$ geostationary satellite is revolving around the Earth. If the radius of the Earth is $R$ and the angular speed of the Earth about its own axis is $\omega$,then the radius of the orbit of the geostationary satellite is ($g =$ acceleration due to gravity).

Give the uses of geostationary satellite and polar satellite.

An artificial satellite is revolving around a planet of radius $R$ in a circular orbit of radius $a$. If the time period of revolution of the satellite $T \propto a^{3/2} g^x R^y$,then the values of $x$ and $y$ are respectively. [Note: $g$ is the acceleration due to gravity at the surface of the planet.]

Two point masses of mass $m_1 = fM$ and $m_2 = (1 - f)M$ $(f < 1)$ are in outer space (far from the gravitational influence of other objects) at a distance $R$ from each other. They move in circular orbits about their centre of mass with angular velocities $\omega_1$ for $m_1$ and $\omega_2$ for $m_2$. In that case:

$A$ satellite of mass $m$ is launched vertically upwards with an initial speed $u$ from the surface of the earth. After it reaches height $R$ ($R =$ radius of the earth), it ejects a rocket of mass $\frac{m}{10}$ so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is ($G$ is the gravitational constant, $M$ is the mass of the earth).