The escape velocity of an object from a planet is $16 \ km/s$. If the escape velocity of the object from another planet having twice the density and three times the radius of the planet is $v \sqrt{2} \ km/s$,then the value of $v$ is

The masses of two fixed spheres are $M$ and $2M$ and the radius of each sphere is $R$. Their centres are $10R$ apart. The minimum speed with which a particle of mass $\frac{M}{10}$ be projected from the mid-point of the line joining the centres of the two spheres so that it escapes to infinity is . . . . . . .

The initial velocity $v_{i}$ required to project a body vertically upward from the surface of the earth to reach a height of $10 R$,where $R$ is the radius of the earth,may be described in terms of escape velocity $v_{e}$ such that $v_{i} = \sqrt{\frac{x}{y}} \times v_{e}$. The value of $x$ will be ...... .

To project a body of mass $m$ from Earth's surface to infinity,the required kinetic energy is (assume,the radius of Earth is $R_E$,$g =$ acceleration due to gravity on the surface of Earth):

Earth binds the atmosphere because of

The escape velocity of a body from a planet of mass $M$ and radius $R$ is $14 \,km \,s^{-1}$. The escape velocity of the body from another planet having same mass and diameter $8R$ (in $km \,s^{-1}$) is