A particle of mass \(m\) is kept at rest at a height \(3R\) from the surface of the Earth, where \(R\) is the radius of the Earth and \(M\) is the mass of the Earth. The minimum speed with which it should be projected, so that it does not return, is:
(where \(g\) is the acceleration due to gravity on the surface of the Earth)

1.	\(\left(\dfrac{{GM}}{2 {R}}\right)^{\frac{1}{2}} \)	2.	\(\left(\dfrac{{g} R}{4}\right)^{\frac{1}{2}} \)
3.	\( \left(\dfrac{2 g}{R}\right)^{\frac{1}{2}} \)	4.	\(\left(\dfrac{G M}{R}\right)^{\frac{1}{2}}\)

The additional kinetic energy to be provided to a satellite of mass \(m\) revolving around a planet of mass \(M,\) to transfer it from a circular orbit of radius \(R_1\) to another of radius \(R_2\) (\(R_2>R_1\)) is:

1. \(GmM\) $\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

2. \(2GmM\) $\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

3. $\frac{1}{2}GmM$ $\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

4. \(GmM\) $\left(\frac{1}{R_1^2}-\frac{1}{R_2^2}\right)$

The dependence of acceleration due to gravity 'g' on the distance 'r' from the centre of the earth, assumed to be a sphere of radius R of uniform density, is as shown in figure below:
   
The correct figure is:
1. a

2. b

3. c

4. d

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. \(\sqrt{ \mathrm{2gR^2}}\)

Which one of the following plots represents the variation of a gravitational field on a particle with distance \(r\) due to a thin spherical shell of radius \(R?\)
(\(r\) is measured from the centre of the spherical shell)

1.		2.
3.		4.

The earth is assumed to be a sphere of radius \(R\). A platform is arranged at a height \(R\) from the surface of the earth. The escape velocity of a body from this platform is \(fv_e\), where \(v_e\) is its escape velocity from the surface of the earth. The value of \(f\) is:
1. \(\sqrt{2}\)
2. \(\frac{1}{\sqrt{2}}\)
3. \(\frac{1}{3}\)
4. \(\frac{1}{2}\)

Two satellites of Earth, \(S_1\)_, and \(S_2\), are moving in the same orbit. The mass of \(S_1\) is four times the mass of \(S_2\). Which one of the following statements is true?

The figure shows the elliptical orbit of a planet \(m\) about the sun \({S}.\) The shaded area \(SCD\) is twice the shaded area \(SAB.\) If \(t_1\) is the time for the planet to move from \(C\) to \(D\) and \(t_2\) is the time to move from \(A\) to \(B,\) then: