Is it necessary to provide an initial speed of $11.2 \, km/s$ to a rocket launched from Earth?

An object $A$ of mass '$m$' is located at a point '$P$' at distances '$r$' and '$2r$' from two planets $B$ and $C$ of masses '$M$' and '$6M$' respectively,as shown in the figure. If the escape speed of the object $A$ from point '$P$' due to the gravitational influence of only planet $B$ is $5 \ km/s$,then the escape speed of the object $A$ from point '$P$' due to the gravitational influence of both the planets is . . . . . . $km/s$.

$A$ 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: $G = 6.67 \times 10^{-11} \ N \cdot m^2/kg^2$)

$A$ body is projected vertically from the Earth's surface of radius $R$ with a velocity equal to half the escape velocity. The maximum height reached by the body is

The escape velocity of a body on a planet $A$ is $12 \, km/s$. The escape velocity of the body on another planet $B$,whose density is four times and radius is half of the planet $A$,is ................... $km/s$.

Explain the escape energy of a body of mass $m$ lying on the surface of the Earth.