A projectile is fired upwards from the surface of the earth with a velocity \(kv_e\) where \(v_e\) is the escape velocity and \(k<1\). If \(r\) is the maximum distance from the center of the earth to which it rises and \(R\) is the radius of the earth, then \(r\) equals:
1. \(\frac{R}{k^2}\)
2. \(\frac{R}{1-k^2}\)
3. \(\frac{2R}{1-k^2}\)
4. \(\frac{2R}{1+k^2}\)

If \(A\) is the areal velocity of a planet of mass \(M,\) then its angular momentum is:

Magnitude of potential energy (\(U\)) and time period \((T)\) of a satellite are related to each other as:
1. \(T^2\propto \frac{1}{U^{3}}\)
2. \(T\propto \frac{1}{U^{3}}\)
3. \(T^2\propto U^3\)
4. \(T^2\propto \frac{1}{U^{2}}\)

For the moon to cease as the earth's satellite, its orbital velocity has to be increased by a factor of:

A rocket of mass \(M\) is launched vertically from the surface of the earth with an initial speed \(v\). Assuming the radius of the earth to be \(R\) and negligible air resistance, the maximum height attained by the rocket above the surface of the earth is:
1. \(\frac{R}{\left(\frac{gR}{2v^2}-1\right)}\)
2. \(R\left({\frac{gR}{2v^2}-1}\right)\)
3. \(\frac{R}{\left(\frac{2gR}{v^2}-1\right)}\)
4. \(R{\left(\frac{2gR}{v^2}-1\right)}\)

If two planets are at mean distances \(d_1\) and \(d_2\) from the sun and their frequencies are \(n_1\) and \(n_2\) respectively, then:
1. \(n^2_1d^2_1= n_2d^2_2\)
2. \(n^2_2d^3_2= n^2_1d^3_1\)
3. \(n_1d^2_1= n_2d^2_2\)
4. \(n^2_1d_1= n^2_2d_2\)

The distance of a planet from the sun is \(5\) times the distance between the earth and the sun. The time period of the planet is: 

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:

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. \(\frac{R}{5}\)

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. \(\frac{1}{R}\)