If a planet of mass $6.4 \times 10^{23} \ kg$ can be compressed into a sphere such that the escape velocity from its surface is $8 \times 10^4 \ m/s$,then what should be the radius of the sphere (in $km$)? (Gravitational constant,$G = 6.67 \times 10^{-11} \ N \cdot m^2/kg^2$)

$A$ 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.

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

$A$ 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.

$A$ body is thrown with a velocity equal to $n$ times the escape velocity $(v_e)$. What will be the velocity of the body at a large distance away?

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 ...... .