A satellite is revolving in a circular orbit at a height \(h\) from the earth's surface (radius of earth \(R\); \(h<<R\)). The minimum increase in its orbital velocity required, so that the satellite could escape from the earth's gravitational field is close to: (Neglect the effect of the atmosphere.)
1. \(\sqrt{2gR}\)
2. \(\sqrt{gR}\)
3. \(\sqrt{\frac{gR}{2}}\)
4. \(\sqrt{gR}\left(\sqrt{2}-1\right)\)

A rocket has to be launched from earth in such a way that it never returns. If \(E\) is the minimum energy delivered by the rocket launcher, what should be the minimum energy that the launcher should have if the same rocket is to be launched from the surface of the moon? Assume that the density of the earth and the moon are equal and that of earth's volume is \(64\) times the volume of the moon.
1. \( \frac{E}{4} \)
2. \(\frac{E}{32} \)
3. \(\frac{E}{16} \)
4. \(\frac{E}{64}\)

Select the correct option based on the statements below:

The initial velocity \(v_i\) required to project a body vertically upward from the surface of the earth to reach a height of \(10R\), where \(R\) is the radius of the earth, may be described in terms of escape velocity \(v_e\) such that \(v_i=\sqrt{\frac{x}{y}} \times v_e \). The value of \(x\) will be:
1. \(10\)
2. \(20\)
3. \(30\)
4. \(40\)