There are two planets. The ratio of the radii of the two planets is $K$,and the ratio of the acceleration due to gravity of both planets is $g$. What will be the ratio of their escape velocities?

If an object is projected vertically upwards with a speed equal to half the escape speed of Earth,then the maximum height attained by it is ($R$ is the radius of Earth).

$A$ rocket has to be launched from the Earth in such a way that it never returns. If $E$ is the minimum energy delivered by the rocket launcher,what should be the minimum energy that the launcher should have if the same rocket is to be launched from the surface of the Moon? Assume that the density of the Earth and the Moon are equal and that the Earth's volume is $64$ times the volume of the Moon.

$A$ mass of $6 \times 10^{24} \ kg$ is to be compressed into a sphere in such a way that the escape velocity from the sphere is $3 \times 10^8 \ m/s$. What should be the radius of the sphere? (Given: $G = 6.67 \times 10^{-11} \ N \cdot m^2/kg^2$)

An object is kept at rest at a distance of $3R$ above the earth's surface,where $R$ is the earth's radius. The minimum speed with which it must be projected so that it does not return to earth is (Assume $M =$ mass of earth,$G =$ Universal gravitational constant).

$A$ 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 $....\,S.$