$A$ body is projected vertically upwards from the surface of a planet of radius $R$ with a velocity equal to half the escape velocity of that planet. Then,the maximum height attained by the body is

The escape velocity of a body on an imaginary planet,which has thrice the radius of the Earth and double the mass of the Earth,is (where $v_e$ is the escape velocity of Earth):

$A$ body is projected vertically upwards from the surface of a planet of radius $R$ with a velocity equal to half the escape velocity for that planet. The maximum height attained by the body is

The ratio of accelerations due to gravity $g_{1}:g_{2}$ on the surfaces of two planets is $5:2$ and the ratio of their respective average densities $\rho_{1}:\rho_{2}$ is $2:1$. What is the ratio of respective escape velocities $v_{1}:v_{2}$ from the surface of the planets?

$A$ spaceship is stationed on Mars. How much energy must be expended on the spaceship to launch it out of the solar system? Mass of the spaceship $= 1000 \; kg$,mass of the Sun $= 2 \times 10^{30} \; kg$,mass of Mars $= 6.4 \times 10^{23} \; kg$,radius of Mars $= 3395 \; km$,radius of the orbit of Mars $= 2.28 \times 10^{8} \; km$,$G = 6.67 \times 10^{-11} \; N m^{2} kg^{-2}$.

If ${v_e}$ and ${v_o}$ represent the escape velocity and orbital velocity of a satellite corresponding to a circular orbit of radius $R$,then