$A$ satellite of mass $m$ is in a circular orbit of radius $2R_E$ around the Earth. The energy required to transfer it to a circular orbit of radius $4R_E$ is (where $M_E$ and $R_E$ are the mass and radius of the Earth,respectively).

$A$ star like the sun has several bodies moving around it at different distances. Consider that all of them are moving in circular orbits. Let $r$ be the distance of the body from the centre of the star and let its linear velocity be $v$,angular velocity $\omega$,kinetic energy $K$,gravitational potential energy $U$,total energy $E$ and angular momentum $l$. As the radius $r$ of the orbit increases,determine which of the above quantities increase and which ones decrease.

An astronaut takes a ball of mass $m$ from Earth to space. He throws the ball into a circular orbit about Earth at an altitude of $318.5 \ km$. From Earth's surface to the orbit,the change in total mechanical energy of the ball is $x \frac{GM_e m}{21 R_e}$. The value of $x$ is (take $R_e = 6370 \ km$).

$A$ satellite is revolving around a planet of mass $M$ in an elliptical orbit of semi-major axis $a$. What is the speed of the satellite when it is at a distance $a/2$ from the planet?

In order to shift a body of mass $m$ from a circular orbit of radius $3R$ to a higher radius $5R$ around the earth,the work done is

The additional kinetic energy to be provided to a satellite of mass $m$ revolving around a planet of mass $M$ to transfer it from a circular orbit of radius $R_1$ to another of radius $R_2$ $(R_2 > R_1)$ is: