A particle of mass \(\mathrm{m}\) is thrown upwards from the surface of the earth, with a velocity \(\mathrm{u}\). The mass and the radius of the earth are, respectively, \(\mathrm{M}\) and \(\mathrm{R}\). \(\mathrm{G}\) is the gravitational constant and \(\mathrm{g}\) is the acceleration due to gravity on the surface of the earth. The minimum value of \(\mathrm{u}\) so that the particle does not return back to earth is:
1. \(\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}^2}} \)
2. \(\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}}} \)
3.\(\sqrt{\frac{2 \mathrm{gM}}{\mathrm{R}^2}} \)
4. \(\sqrt{ \mathrm{2gR^2}}\)

The escape velocity of a particle of mass \(m\) varies as:

If the radius of a planet is R and its density is $\mathrm{\rho }$, the escape velocity from its surface will

be

1. $V_e<mspace\; width="1em"></mspace>\propto <mspace\; width="1em"></mspace>pR$                         

2. $V_e<mspace\; width="1em"></mspace>\propto <mspace\; width="1em"></mspace>R\sqrt{\rho }$

3. $V_e\propto \frac{\sqrt{p}}{R}$                        

4. $V_e<mspace\; width="1em"></mspace>\propto \frac{1}{\sqrt{p}R}$

A particle of mass m is thrown upwards from the surface of the earth with a velocity u. The mass and the radius of the earth are, respectively, M and R. G is gravitational constant and g is acceleration due to gravity on the surface of the earth. The minimum value of u so that the particle does not return back to earth is

(1) $\sqrt{\frac{2GM}{R}}$                                       

(2) $\sqrt{\frac{2GM}{R^{\mathit{2}}}}$

(3) $\sqrt{2g{R}^{\mathit{2}}}$                                       

(4) $\sqrt{\frac{2GM}{R^3}}$

A body is moving in a low circular orbit about a planet of mass \(M\) and radius \(R\). The radius of the orbit can be taken to be \(R\) itself. The ratio of the speed of this body in the orbit to the escape velocity from the planet is:
1. \(\sqrt{2}\)
2. \(\dfrac{1}{\sqrt{2}}\)
3. \(2\)
4. \(1\)