A particle of mass \(m\) is projected with a velocity, \(v=kv_{e} ~(k<1)\) from the surface of the earth. The maximum height, above the surface, reached by the particle is:
(Where \(v_e=\) escape velocity, \(R=\) the radius of the earth)

1.	\(\dfrac{R^{2}k}{1+k}\)	2.	\(\dfrac{Rk^{2}}{1-k^{2}}\)
3.	\(R\left ( \dfrac{k}{1-k} \right )^{2}\)	4.	\(R\left ( \dfrac{k}{1+k} \right )^{2}\)

The escape velocity from the Earth's surface is \(v\). The escape velocity from the surface of another planet having a radius, four times that of Earth and the same mass density is: 

Two spheres of masses \(m\) and \(M\) are situated in air and the gravitational force between them is \(F.\) If the space around the masses is filled with a liquid of specific density \(3,\) the gravitational force will become:
1. \(3F\)
2. \(F\)
3. \(F/3\)
4. \(F/9\)

With what velocity should a particle be projected so that its height becomes equal to the radius of the earth?

1.  ${\left(\frac{GM}{R}\right)}^{1/2}$

2. ${\left(\frac{8GM}{R}\right)}^{1/2}$

3.  ${\left(\frac{2GM}{R}\right)}^{1/2}$

4.  ${\left(\frac{4GM}{R}\right)}^{1/2}$

If a body of mass \(m\) placed on the earth's surface is taken to a height of \(h = 3R,\) then the change in gravitational potential energy is:

1. \(\dfrac{mgR}{4}\)

2. \(\dfrac{2}{3} mgR\)

3. \(\dfrac{3}{4} mgR\)

4. \(\dfrac{mgR}{2}\)

For moon, its mass is \(\frac{1}{81}\) of Earth's mass and its diameter is \(\frac{1}{3.7}\) of Earth's diameter. If acceleration due to gravity at Earth's surface is \(9.8~\text{m/s}^2,\) then at the moon, its value is: 

Rohini satellite is at a height of \(500\) km and Insat-B is at a height of \(3600\) km from the surface of the earth. The relation between their orbital velocity (\(v_R,~v_i\)) is:
1. \(v_R>v_i\)
2. \(v_R<v_i\)
3. \(v_R=v_i\)
4. no specific relation 

If the radius of the earth shrinks by 1%, then for acceleration due to gravity, there would be:
1. No change at the poles
2. No change at the equator
3. Maximum change at the equator
4. Equal change at all locations