Physics - Gravitation Question with Solution | TestHub

Let escape velocity be$v_e$, then kinetic energy is

$=\frac{1}{2}{mv}_e^2 \dots .\left(\mathrm{i}\right)$

$\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{e}\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{ }\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}=+\frac{{GM}_{\mathrm{e}}m}{R_e} \dots \left(\mathrm{i}\mathrm{i}\right)$

Equating Eqs. (i) and (ii), we get

$\frac{1}{2}{\mathit{\text{m}}}_{\mathit{\text{e}}}{v}_e^2=\frac{G{M}_{e}m}{R_e}$

$⟹ v_e=\sqrt{\frac{2{GM}_e}{R_e}}$

$⟹ R=\frac{{2GM}_e}{v_e^2}$

Given,

$G=\text{6.67}\times 10^{-11}N {\text{m}}^2{\text{ kg}}^{\text{-2}}$

$R=\frac{2\times 6.67\times {10}^{-11}\times 6\times {10}^{24}}{(3\times {10}^8{)}^2}$

$R=8.89\times {10}^{-3}$

$R\approx 9\times {10}^{-\mathrm{3 }}\mathrm{m}=\mathrm{9 }\mathrm{m}\mathrm{m}$