If a rocket is fired with sped $\mathrm{v}=2\sqrt{\mathrm{gR}}$ near the earth’s surface and coast upwards, its speed in the interstellar space is :

According to law of conservation of energy,
$\frac{1}{2}{\mathrm{mv}}^2-\frac{\mathrm{GMm}}{\mathrm{R}}=\frac{1}{2}{\mathrm{mv}}_1^2+0\text{ or }\frac{1}{2}{\mathrm{mv}}_1^2=\frac{1}{2}{\mathrm{mv}}^2-\frac{\mathrm{GMm}}{\mathrm{R}}$
$\mathrm{or} {\mathrm{v}}_1^2={\mathrm{v}}^2-\frac{2\mathrm{GM}}{\mathrm{R}}={\mathrm{v}}^2-{\mathrm{v}}_{\mathrm{e}}^2\therefore {\mathrm{v}}_1=\sqrt{{\mathrm{v}}^2-{\mathrm{v}}_{\mathrm{e}}^2}=\sqrt{4\mathrm{gR}-2\mathrm{gR}}=\sqrt{2\mathrm{gR}}.$