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 $318.5 \mathrm{~km}$. From earth's surface to the orbit, the change in total mechanical energy of the ball is $x \frac{\mathrm{GM}_e \mathrm{~m}}{21 R_e}$. The value of $x$ is (take $R_{\mathrm{e}}=6370 \mathrm{~km}$ ):

A certain object is projected vertically from the surface of the earth of radius $R$ with a velocity equal to half the escape velocity. The maximum height attained by the object will be

A star is modeled as a uniform spherical distribution of matter keeping mass of star constant. How gravitational pressure on surface depends on volume of the star?

Four particles each of mass $m$ are placed at the vertices of a square of side $l.$ The potential energy of the system is

The earth revolves about the sun in an elliptical orbit with mean radius $9.3 \times {10^7}\,m$ in a period of $1$ year. Assuming that there are no outside influences