An artificial satellite is moving around the Earth in a circular orbit with a speed equal to one-fourth the escape speed of a body from the surface of the Earth. The height of the satellite above the Earth's surface is ............ ($R$ is the radius of the Earth). (in $R$)

$A$ body $A$ of mass $m$ is moving in a circular orbit of radius $R$ about a planet. Another body $B$ of mass $\frac{m}{2}$ collides with $A$ with a velocity which is half $\left(\frac{\overrightarrow{v}}{2}\right)$ the instantaneous velocity $\overrightarrow{v}$ of $A$. The collision is completely inelastic. Then,the combined body

The orbital velocity of an artificial satellite in a circular orbit just above the Earth's surface is $v$. For a satellite orbiting at an altitude of half of the Earth's radius,the orbital velocity is

The largest and the shortest distance of the earth from the sun are ${r_1}$ and ${r_2}$ respectively. What is its distance from the sun when it is at a position perpendicular to the major axis of the orbit drawn from the sun?

The mean radius of Earth is $R$,and its angular speed on its axis is $\omega$. What will be the radius of orbit of a geostationary satellite?

$A$ satellite $S$ is moving in an elliptical orbit around the Earth. The mass of the satellite is very small compared to the mass of the Earth.