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 same mass density is: 

1. 3v 

2. 4v 

3. v 

4. 2v 

A particle of mass 'm' is projected with a velocity $\mathrm{v}={\mathrm{kV}}_{\mathrm{e}} \left(\mathrm{k}<1\right)$ from the surface of the earth.

$\left({\mathrm{V}}_{\mathrm{e}}=\mathrm{escape} \mathrm{velocity}\right)$

The maximum height above the surface reached by the particle is:

1. $\frac{{\mathrm{R}}^2\mathrm{k}}{1+\mathrm{k}}$

2. $\frac{{\mathrm{Rk}}^2}{1-{\mathrm{k}}^2}$

3. $\mathrm{R}{\left(\frac{\mathrm{k}}{1-\mathrm{k}}\right)}^2$

4. $\text{R}{\left(\frac{\mathrm{k}}{1+\mathrm{k}}\right)}^2$

A body weighs 72 N on the surface of the earth. What is the gravitational force on it at a height equal to half the radius of the earth?

1. 32 N

2. 30 N

3. 24 N

4. 48 N

What is the depth at which the value of acceleration due to gravity becomes 1/n times the value at the surface of the earth? (radius of the earth = R) 

1.  $\frac{\mathrm{R}}{{\mathrm{n}}^2}$

2.  $\frac{\mathrm{R}(\mathrm{n}-1)}{\mathrm{n}}$

3. $\frac{\mathrm{Rn}}{(\mathrm{n}-1)}$

4. R/n 

The work done to raise a mass m from the surface of the earth to a height h, which is equal to the radius of the earth, is:

1. $\frac{3}{2}mgR$

2. mgR

3. 2mgR

4. $\frac{1}{2}mgR$

A body weighs 200 N on the surface of the earth. How much will it weigh halfway down to the centre of the earth?

1. 100 N

2. 150 N

3. 200 N

4. 250 N

A mass falls from a height 'h' and its time of fall 't' is recorded in terms of time period T of a simple pendulum. On the surface of the earth, it is found that t=2T. The entire set up is taken on the surface of another planet whose mass is half of that of the earth and radius is same. The same experiment is repeated and corresponding times noted as t' and T'. Then we can say:

1. t' = $\sqrt{2}T$

2. t' > 2T'

3. t' < 2T'

4. t' = 2T'

The time period of a geostationary satellite is 24 h at a height 6$R_E$ ($R_E$ is the radius of the earth) from the surface of the earth. The time period of another satellite whose height is 2.5$R_E$ from the surface, will be:

1. 6$\sqrt{2}h$

2. 12$\sqrt{2}h$

3. $\frac{24}{2.5}h$

4.  $\frac{12}{2.5}h$

Assuming that the gravitational potential energy of an object at infinity is zero, the change in potential energy (final - initial) of an object of mass m when taken to a height h from the surface of the earth (of radius R and mass M), is given by:

1. $\frac{GMm}{R+h}$

2. $\frac{GMmh}{R(R+h)}$

3. mgh

4. $\frac{GMm}{R+h}$

At what temperature will the RMS speed of oxygen molecules become just sufficient for escaping from the earth's atmosphere? 
(Given : Mass of oxygen molecule (m) = 2.76 x 10^-26 kg, Boltzmann's constant k_B = 1.38 × 10^-23 J K^-1):

1. $2.508\times {10}^4 K$

2. $8.360\times {10}^4 K$

3. $5.016\times {10}^4 K$

4. $1.254\times {10}^4 K$