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(D) Using the principle of conservation of mechanical energy: $U_1 + K_1 = U_2 + K_2$Here,$U = -\frac{GM_e m}{r}$ and $K = \frac{1}{2}mv^2$. The escap

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$16$

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(D) Using the principle of conservation of mechanical energy: $U_1 + K_1 = U_2 + K_2$Here,$U = -\frac{GM_e m}{r}$ and $K = \frac{1}{2}mv^2$. The escape velocity is given by $v_e = \sqrt{\frac{2GM_e}{R}}$,so $\frac{GM_e}{R} = \frac{v_e^2}{2}$.At distance $r_1 = 10R$,speed $v_1 = 12 \; km/s$. At surface $r_2 = R$,speed is $v_2$.$-\frac{GM_e m}{10R} + \frac{1}{2}mv_1^2 = -\frac{GM_e m}{R} + \frac{1}{2}mv_2^2$$\frac{1}{2}v_2^2 = \frac{1}{2}v_1^2 + \frac{GM_e}{R} - \frac{GM_e}{10R} = \frac{1}{2}v_1^2 + \frac{9}{10} \left( \frac{GM_e}{R} ight)$Substituting $\frac{GM_e}{R} = \frac{v_e^2}{2}$:$v_2^2 = v_1^2 + \frac{9}{10} v_e^2$$v_2^2 = (12)^2 + 0.9 	imes (11.2)^2 = 144 + 0.9 	imes 125.44 = 144 + 112.896 = 256.896$$v_2 = \sqrt{256.896} \approx 16.028 \; km/s$.The nearest integer is $16 \; km/s$.

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