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(D) For ball $A$ falling from infinity,the velocity $v_A$ upon reaching the earth's surface is the escape velocity:$v_A = \sqrt{\frac{2GM}{R}}$For bal

Vedclass · Accepted Answer

$\sqrt{\frac{7}{6}}$

Vedclass · Answer

(D) For ball $A$ falling from infinity,the velocity $v_A$ upon reaching the earth's surface is the escape velocity:$v_A = \sqrt{\frac{2GM}{R}}$For ball $B$ falling from a height $h = 6R$,we use the conservation of energy:$\frac{1}{2}mv_B^2 - \frac{GMm}{R+h} = 0 - \frac{GMm}{R}$$\frac{1}{2}v_B^2 = GM \left( \frac{1}{R} - \frac{1}{7R} ight) = GM \left( \frac{6}{7R} ight)$$v_B = \sqrt{\frac{12GM}{7R}}$Now,the ratio of the velocities of $A$ and $B$ is:$\frac{v_A}{v_B} = \frac{\sqrt{\frac{2GM}{R}}}{\sqrt{\frac{12GM}{7R}}} = \sqrt{\frac{2GM}{R} 	imes \frac{7R}{12GM}} = \sqrt{\frac{14}{12}} = \sqrt{\frac{7}{6}}$Thus,the ratio of the velocity of $A$ to $B$ is $\sqrt{\frac{7}{6}}$.

$A$ ball $A$ of mass $m$ falls to the surface of the earth from infinity. Another ball $B$ of mass $2m$ falls to the earth from a height equal to six times the radius of the earth. The ratio of the velocities of $A$ and $B$ on reaching the earth is:

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