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(D) The gravitational potential energy $U$ of an object of mass $m$ at a point $P$ due to a planet of mass $M$ at distance $r$ is given by $U = -\frac

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$10$

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(D) The gravitational potential energy $U$ of an object of mass $m$ at a point $P$ due to a planet of mass $M$ at distance $r$ is given by $U = -\frac{GMm}{r}$.The escape speed $v_e$ is given by the condition that the total energy must be at least zero,i.e.,$\frac{1}{2}mv_e^2 + U = 0$,which implies $v_e = \sqrt{\frac{2|U|}{m}} = \sqrt{\frac{2GM}{r}}$.Given that the escape speed due to planet $B$ alone is $v_{eB} = \sqrt{\frac{2GM}{r}} = 5 \ km/s$.When both planets $B$ and $C$ are considered,the total gravitational potential energy at point $P$ is $U_{total} = U_B + U_C = -\frac{GMm}{r} - \frac{G(6M)m}{2r} = -\frac{GMm}{r} - 3\frac{GMm}{r} = -4\frac{GMm}{r}$.The escape speed $v_{total}$ due to both planets is given by $\frac{1}{2}mv_{total}^2 = |U_{total}| = 4\frac{GMm}{r}$.Thus,$v_{total} = \sqrt{\frac{8GM}{r}} = 2 \sqrt{\frac{2GM}{r}}$.Substituting the value $v_{eB} = 5 \ km/s$,we get $v_{total} = 2 	imes 5 \ km/s = 10 \ km/s$.

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