Two bodies of masses $m_1$ and $m_2$ are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is

If a tunnel is cut at any orientation through the Earth,a ball released from one end will reach the other end in time ........ $\text{min}$ (neglect Earth's rotation).

Fill in the blanks:
$(a)$ The value of gravitational intensity at the center of the Earth is ..... .
$(b)$ The potential energy of a satellite is $-8 \times 10^9 \, J$,then its binding energy is ............ .
$(c)$ Kepler's second law regarding the constancy of the areal velocity of a planet is a consequence of the law of conservation of .......... .

Four similar particles of mass $m$ are orbiting in a circle of radius $r$ in the same direction because of their mutual gravitational attractive force. The velocity of a particle is given by

The figure shows the variation of the gravitational acceleration $a_g$ of four planets with the radial distance $r$ from the center of the planet for $r \ge R$ (where $R$ is the radius of the planet). Plots $1$ and $2$ coincide for $r \ge R_2$,and plots $3$ and $4$ coincide for $r \ge R_4$. The sequence of the planets in the descending order of their densities is:

The mass density of a spherical body is given by $\rho(r) = \frac{k}{r}$ for $r \leq R$ and $\rho(r) = 0$ for $r > R$,where $r$ is the distance from the centre. The correct graph that describes qualitatively the acceleration,$a$,of a test particle as a function of $r$ is