$A$ boy can jump to a height $h$ on the surface of a planet. If the density of the planet is $d$,what should be its radius $R$ so that the boy can escape from the gravitational pull of the planet?

$A$ small asteroid is orbiting around the sun in a circular orbit of radius $r_0$ with speed $v_0$. $A$ rocket is launched from the asteroid with speed $v = \alpha v_0$,where $v$ is the speed relative to the sun. The highest value of $\alpha$ for which the rocket will remain bound to the solar system is (ignoring gravity due to the asteroid and effects of other planets).

The escape velocity of a rocket launched from the surface of the earth:

The condition for a uniform spherical mass $m$ of radius $r$ to be a black hole is [$G=$ gravitational constant and $c=$ speed of light]

Planet $A$ has mass $M$ and radius $R$. Planet $B$ has half the mass and half the radius of Planet $A$. If the escape velocities from the Planets $A$ and $B$ are $v_{A}$ and $v_{B}$ respectively,then $\frac{v_{A}}{v_{B}}=\frac{n}{4}$. The value of $n$ is

$A$ body is projected vertically upwards from the surface of the Earth with a velocity sufficient enough to carry it to infinity. The time taken by it to reach height $h$ is $....\,S.$