The escape velocity of a body on the surface of the earth is $11.2 \, km/s$. If the earth's mass increases to twice its present value and the radius of the earth becomes half,the escape velocity would become ......... $km/s$.

$A$ 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.

$A$ very small groove is made in the earth,and a particle of mass $m_0$ is placed at $\frac{R}{2}$ distance from the centre. Find the escape speed of the particle from that place.

$A$ bullet is fired vertically upwards with velocity $v$ from the surface of a spherical planet. When it reaches its maximum height,its acceleration due to the planet's gravity is $1/4$ of its value at the surface of the planet. If the escape velocity from the planet is $v_{esc} = v \sqrt{N}$,then the value of $N$ is (ignore energy loss due to atmosphere).

The escape velocity from a spherical planet $A$ is $10 \ km/s$. The escape velocity from another planet $B$ whose density and radius are $10\%$ of those of planet $A$,is . . . . . . $m/s$.

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).