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(A) According to the law of conservation of mechanical energy,the total energy at the surface is equal to the total energy at the maximum height $h$.A

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$R/3$

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(A) According to the law of conservation of mechanical energy,the total energy at the surface is equal to the total energy at the maximum height $h$.At the surface: $E_i = K_i + U_i = \frac{1}{2}mv^2 - \frac{GMm}{R}$Given $v = \frac{v_e}{2}$,where $v_e = \sqrt{\frac{2GM}{R}}$. Thus,$v^2 = \frac{v_e^2}{4} = \frac{2GM}{4R} = \frac{GM}{2R}$.$E_i = \frac{1}{2}m(\frac{GM}{2R}) - \frac{GMm}{R} = \frac{GMm}{4R} - \frac{GMm}{R} = -\frac{3GMm}{4R}$.At maximum height $h$,the velocity is zero,so $E_f = K_f + U_f = 0 - \frac{GMm}{R+h}$.Equating $E_i = E_f$: $-\frac{3GMm}{4R} = -\frac{GMm}{R+h}$.$\frac{3}{4R} = \frac{1}{R+h} \implies 3(R+h) = 4R \implies 3R + 3h = 4R \implies 3h = R \implies h = \frac{R}{3}$.

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