A rocket is fired 'vertically' from the surface of mars with a speed of $2\; km s ^{-1}$. If $20 \%$ of its initial 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 $=3395\; km ; G=6.67 \times 10^{-11}\; N m ^{2} kg ^{-2}$

What is the minimum energy required to launch a satellite of mass $m$ from the surface of a planet of mass $M$ and radius $R$ in a circular orbit at an altitude of $2R$?

A rocket is launched normal to the surface of earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is $3 \times 10^5$ times heavier than the Earth and is at a distance $2.5 \times 10^4$ times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is $\mathrm{ve}_{\mathrm{e}}=11.2 \mathrm{~km} \mathrm{~s} \mathrm{~s}^{-1}$. The minimum initial $\left(\mathrm{v}_{\mathrm{s}}\right)$ required for the rocket to be able to leave the Sun-earth system is closest to

(Ignore the rotation and revolution of the Earth and the presence of any other planet)

What is binding energy of satellite ? Write its equation.

Calculate energy needed for moving a mass of $4\,\,kg$ from the centre of the earth to its surface (in joule), if radius of the earth is $6400\,\, km$ and acceleration due to gravity at the surface of the earth is $g = 10 \,\,m/sec^2.$