Two satellites A and B go around the earth in circular orbits at heights of ${\mathrm{R}}_{\mathrm{A}} \mathrm{and} {\mathrm{R}}_{\mathrm{B}}$ respectively from the surface of the earth. Assuming earth to be a uniform sphere of radius $R_e$, the ratio of the magnitudes of their orbital velocities is:

1.  $\sqrt{\frac{R_B}{R_A}}$

2.  $\frac{R_B+R_e}{R_A+R_e}$

3.  $\sqrt{\frac{R_B+R_e}{R_A+R_e}}$

4.  ${\left(\frac{R_A}{R_B}\right)}^2$

The orbital angular momentum of a satellite revolving at a distance \(r\)from the centre is \(L\). If the distance is increased to 16r, then the new angular momentum will be:

A body of mass \(m\) kg starts falling from a point \(2R\) above the Earth’s surface. Its kinetic energy when it has fallen to a point \(R\) above the Earth’s surface, is:
[\(R\text-\) Radius of Earth, \(M\text-\) Mass of Earth, \(G\text-\) Gravitational Constant]
1. \(\frac{1}{2} \frac{G M m}{R}\)
2. \(\frac{1}{6} \frac{G M m}{R}\)
3. \(\frac{2}{3} \frac{G M m}{R}\)
4. \(\frac{1}{3} \frac{G M m}{R}\)

A body is projected vertically upwards from the surface of a planet of radius R with a velocity equal to half the escape velocity for that planet. The maximum height attained by the body is:

1.  R/3                            

2.  R/2

3.  R/4                            

4.  R/5

A satellite is launched into a circular orbit of radius \(R\) around the Earth while a second satellite is launched into an orbit of radius \(1.02~\text{R}\). The percentage difference in the time periods of the two satellites is: 

If the gravitational force between two objects were proportional to \(\frac{1}{R}\) (and not as $1/R^2$) where \(R\) is the separation between them, then a particle in circular orbit under such a force would have its orbital speed v proportional to:

1. $1/R^2$                        

2. $R^{\circ }$

3. $R^1$                             

4. \(1/R\)

If the acceleration due to gravity at a height \(1\) km above the earth is similar to a depth \(d\) below the surface of the earth, then: 
1. \(d= 0.5\) km
2. \(d=1\) km
3. \(d=1.5\) km
4. \(d=2\) km

Two astronauts are floating in a gravitational free space after having lost contact with their spaceship. The two will:

A remote sensing satellite of the earth revolves in a circular orbit at a height of 0.25 x 10⁶ m above the surface of the earth. If the earth’s radius is 6.38x10⁶ m and g=9.8ms^-1, then the orbital speed of the satellite is:

1.  7.76 kms^-1

2.  8.56 kms^-1

3.  9.13 kms^-1

4.  6.67 kms^-1

The radii of the circular orbits of two satellites A and B of the earth are \(4R\) and \(R,\) respectively. If the speed of satellite A is \(3v,\) then the speed of satellite B will be: