$A$ particle of mass $M$ is situated at the centre of a spherical shell of same mass $M$ and radius $a$. The gravitational potential at a point situated at a distance of $\frac{a}{2}$ from the centre will be:

$A$ rocket is fired vertically from the surface of Mars with a speed of $2 \; km/s$. If $20 \%$ of its initial kinetic energy is lost due to Martian atmospheric resistance,how far will the rocket go from the surface of Mars before returning to it? (Mass of Mars $= 6.4 \times 10^{23} \; kg$,radius of Mars $= 3395 \; km$,$G = 6.67 \times 10^{-11} \; N m^2 kg^{-2}$)

$A$ mass $m$,travelling at speed $V_0$ in a straight line from far away,is deflected when it passes near a black hole of mass $M$ which is at a perpendicular distance $R$ from the original line of flight. $a$,the distance of closest approach between the mass and the black hole,is given by the relation:

$A$ man of mass $m$ starts falling towards a planet of mass $M$ and radius $R.$ As he reaches near to the surface,he realizes that he will pass through a small hole in the planet. As he enters the hole,he sees that the planet is really made of two pieces: a spherical shell of negligible thickness of mass $\frac{2M}{3}$ and a point mass $\frac{M}{3}$ at the centre. The change in the force of gravity experienced by the man is:

The relative uncertainty in the period of a satellite orbiting around the Earth is $10^{-2}$. If the relative uncertainty in the radius of the orbit is negligible,the relative uncertainty in the mass of the Earth is:

$A$ spacecraft which is moving with a speed $u$ relative to the earth in the $x$-direction,enters the gravitational field of a much more massive planet which is moving with a speed $3u$ in the negative $x$-direction. The spacecraft exits following the trajectory as shown below. The speed of the spacecraft with respect to the earth a long time after it has escaped the planet's gravity is given by