The escape velocity from the surface of earth is v _ e . A body is...

& If $v$ be the velocity in the inter planetary space and ${U}_{i,}{U}_{f}$ initial and final PE

Then, by the law of conservation of energy,

${U}_{i} + {K}_{i} = {U}_{f} + {K}_{f}$

$\dfrac{- G{M}_{e}m}{Re} + \dfrac{1}{2}m\left(3{v}_{e}\right)^{2} = 0 + \dfrac{1}{2}mv^{2}$

$\dfrac{- G{M}_{e}}{Re} + \dfrac{9}{2}{v}_e^{2} = \dfrac{1}{2}v^{2}$

$⇒ \dfrac{- 2G{M}_{e}}{Re} + 9{V}_e^{2} = v^{2}$

$⇒ \dfrac{- 2g{R}_e^{2}}{Re} + 9{v}_e^{2} = v^{2} \left[∵ G{M}_{e} = g{R}_e^{2}\right]$

$- 2g{R}_{e} + 9 \times2g{R}_{e} = v^{2} \left[∵ {v}_e^{2} = 2g{R}_{e}\right]$

$16g{R}_{e} = v^{2}$
$⇒ v = \sqrt{8} \cdot \sqrt{2g{R}_{e}} ===2\sqrt{2}{v}_{e}</p>$