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

1.	keep floating at the same distance between them
2.	move towards each other
3.	move away from each other
4.	will become stationary

A remote sensing satellite of the earth revolves in a circular orbit at a height of \(0.25\times 10^{6}\) m above the surface of the earth. If the earth’s radius is \(6.38\times 10^{6}\) m and \(g = 9.8\) ms^-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:

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

If the gravitational force between two objects were proportional to \(\frac{1}{R}\) (and not as\(\frac{1}{R^2}\)${\mathrm{}}^{}$) 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. \(\frac{1}{R^2}\)
2. \(R^{0}\)
3. \(R^{1}\)
4. \(\frac{1}{R}\)

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: 

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. \(\frac{R}{3}\)
2. \(\frac{R}{2}\)
3. \(\frac{R}{4}\)
4. \(\frac{R}{5}\)

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. \(\frac{1}{3} \frac{G M m}{R}\)

Two satellites \(A\) and \(B\) go around the earth in circular orbits at heights of \(R_A ~\text{and}~R_B\) 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. \(\left(\frac{R_{A}}{R_{B}}\right)^{2}\)