Kepler's third law states that the square of the period of revolution (T) of a planet around the sun, is proportional to the third power of the average distance r between the sun and planet i.e. $T^2 = Kr^3$, here K is constant. If the masses of the sun and planet are M and m respectively, then as per Newton's law of gravitation, the force of attraction between them is $F = GMm/r^2$, here G is gravitational constant. The relation between G and K is described as:

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: