$A$ 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

$A$ particle of mass $m$ is kept at rest at a height $3R$ from the surface of the Earth,where $R$ is the radius of the Earth and $M$ is the mass of the Earth. The minimum speed with which it should be projected upward so that it does not return back is ($g$ = acceleration due to gravity on the Earth's surface).

$A$ spaceship is stationed on Mars. How much energy must be expended on the spaceship to launch it out of the solar system? Mass of the spaceship $= 1000 \; kg$,mass of the Sun $= 2 \times 10^{30} \; kg$,mass of Mars $= 6.4 \times 10^{23} \; kg$,radius of Mars $= 3395 \; km$,radius of the orbit of Mars $= 2.28 \times 10^{8} \; km$,$G = 6.67 \times 10^{-11} \; N m^{2} kg^{-2}$.

Why does a rocket not have escape energy?

State two factors that determine whether a planet will have an atmosphere or not.

If $M$ is the mass of a planet and $R$ is its radius,then in order to become a black hole (where $c$ is the speed of light),which condition must be satisfied?