A satellite moving with velocity $v$ in a force free space collects stationary interplanetary dust at a rate of $\frac{{dM}}{{dt}} = \alpha v$ where $M$ is the mass (of satellite + dust) at that instant . The instantaneous acceleration of the satellite is
$ - \frac{{\alpha {v^2}}}{{2M}}$
$ - \frac{{\alpha {v^2}}}{{M}}$
$ - \alpha {v^2}$
$ - \frac{{2\alpha {v^2}}}{{M}}$
The masses and radii of the earth and the moon are $M_1, R_1$ and $M_2, R_2$ respectively. Their centres are distance $d$ apart. The minimum speed with which particle of mass $m$ should be projected from a point midway between the two centres so as to escape to infinity is
A spherical part of radius $R/2$ is excavated from the asteroid of mass $M$ as shown in the figure. The gravitational acceleration at a point on the surface of the asteroid just above the excavation is
Two spheres of masses $m$ and $M$ are situated in air and the gravitational force between them is $F.$ The space around the masses is now filled with a liquid of specific gravity $3.$ The gravitational force will now be
The variation of acceleration due to gravity $g$ with distance $d$ from centre of the earth is best represented by ($R =$ Earth's radius)
A particle of mass $M$ is situated at the centre of a spherical shell of same mass and radius $a$. The gravitational potential at a point situated at $\frac{a}{2}$ distance from the centre, will be