A rocket is fired vertically upward with a speed of $v = \frac{v_e}{\sqrt{2}}$ from the Earth's surface, where $v_e$ is escape velocity on the surface of Earth. The distance from the surface of Earth upto which the rocket can go before returning to the Earth is: (given, the radius of Earth $R = 6400$ km)

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

A black hole is an object whose gravitational field is so strong that even light cannot escape from it. To what approximate radius would earth (mass = 5.98 × 10²⁴ kg) have to be compressed to be a black hole?

A body projected vertically from the earth reaches a height equal to earth's radius before returning to the earth. The power exerted by the gravitational force is greatest:

The figure shows elliptical orbit of a planet m about the sun S. The shaded area SCD is twice the shaded area SAB. If $t_1$ is the time for planet to move from C to D and $t_2$ is the time to move from A to B, then:

If the mass of the sun were ten times smaller and the universal gravitational constant were ten times larger in magnitude, which of the following statements would not be correct?

The kinetic energies of a planet in an elliptical orbit around the Sun, at positions A, B and C are $K_A$, $K_B$ and $K_C$ respectively. AC is the major axis and SB is perpendicular to AC at the position of the Sun S, as shown in the figure. Then:

At what height from the surface of the earth, are the gravitation potential and the value of $g$ are: $-5.4 \times 10^7 \text{ J/kg}$ and $6.0 \text{ ms}^{-2}$ respectively? (Take, the radius of the earth as $6400 \text{ km}$)

A particle is released from a height $S$ above the surface of the earth. At a certain height, its kinetic energy is three times its potential energy. The height from the earth's surface and the speed of the particle at that instant are respectively: