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(D) According to the law of conservation of energy, the total energy at the surface of the Earth is equal to the total energy at infinity (or a very l

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$\sqrt{2} v_e$

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(D) According to the law of conservation of energy, the total energy at the surface of the Earth is equal to the total energy at infinity (or a very large distance where the potential energy is zero).Total energy at the surface: $E_i = K_i + U_i = \frac{1}{2} m(\sqrt{3} v_e)^2 - \frac{GMm}{R}$.We know that the escape velocity $v_e = \sqrt{\frac{2GM}{R}}$, so $v_e^2 = \frac{2GM}{R}$, which implies $\frac{GM}{R} = \frac{v_e^2}{2}$.Substituting this into the energy equation: $E_i = \frac{1}{2} m(3 v_e^2) - m(\frac{v_e^2}{2}) = \frac{3}{2} m v_e^2 - \frac{1}{2} m v_e^2 = m v_e^2$.At a very large distance, the potential energy is $0$, so the final energy is $E_f = \frac{1}{2} m v^2$.Equating $E_i = E_f$: $m v_e^2 = \frac{1}{2} m v^2$.$v^2 = 2 v_e^2 \implies v = \sqrt{2} v_e$.

$A$ body of mass $m$ is thrown vertically upward with speed $\sqrt{3} v_e$, where $v_e$ is the escape velocity of a body from the Earth's surface. The final velocity of the body is

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