Question

A space vehicle of mass 300 kg is in circular orbit of radius $2R_e$ about the earth. Energy required to shift vehicle from orbit of radius $2R_e$ to orbit of radius $4R_e$ is $n\times {10}^{9}J$. Here $R_e=6.4\times {10}^6$ is radius of earth and $g=10m/s^2$ then n is (Answer upto two digits after the decimal point.)

Answer 1

$\Delta E=E_F-E_I$;
where $E=\frac{-GMm}{r}+\frac{1}{2}m{v}^2=\frac{-GMm}{2r}$

$\Delta E=\frac{-GMm}{2\times 4R_e}+\frac{GMm}{2\times 2R_e}=\frac{GMm}{8R_e}$

$\Delta E=\frac{GM}{R_e^2}\frac{m{R}_e}{8}=g\frac{m{R}_e}{8}=\frac{10\times 300\times 6.4\times {10}^6}{8}n=2.40$