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, \(R=\) radius of the earth)

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

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. \(F/9\)

The acceleration due to gravity on planet A is 9 times the acceleration due to gravity on planet B. A man jumps to a height of 2 m on the surface of A. What is the height of a jump by the same person on planet B?

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. $\frac{\mathrm{mgR}}{4}$

2. $\frac{2}{3}\mathrm{mgR}$

3. $\frac{3}{4}\mathrm{mgR}$

4. $\frac{\mathrm{mgR}}{2}$

For moon, its mass is 1/81 of Earth's mass and its diameter is 1/3.7 of Earth's diameter. If acceleration due to gravity at Earth's surface is 9.8 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 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