The masses and radii of the earth and moon are $\left({M}_{1}, {R}_{1}\right)$ and $\left({M}_{2}, {R}_{2}\right)$ respectively. Their centres are at a distance ' ${r}$ ' apart. Find the minimum escape velocity for a particle of mass ' ${m}$ ' to be projected from the middle of these two masses:

A body of mass $m$ kg. starts falling from a point $2R$ above the Earth’s surface. Its kinetic energy when it has fallen to a point $‘R’$ above the Earth’s surface [$R-$Radius of Earth, $M-$Mass of Earth, $G-$Gravitational Constant]

Two planets have masses $M$ and $16\, M$ and their radii are $a$ and $2\, a$ respectively. The separation between the centres of the planets is $10\, a.$ A body of mass $m$ is fired from the surface of the larger planet towards the smaller planet along the line joining their centres. For the body to be able to reach at the surface of smaller planet, the minimum firing speed needed is

In order to shift a body of mass $m$ from a circular orbit of radius $3R$ to a higher radius $5R$ around the earth, the work done is

A rocket is fired vertically with a speed of $5\; km s^{-1}$ from the earth’s surface. How far from the earth does the rocket go before returning to the earth ? Mass of the earth $=6.0 \times 10^{24} \;kg ;$ mean radius of the earth $=6.4 \times 10^{6} \;m ; G=6.67 \times 10^{-11} \;N m ^{2} kg ^{2}$