The masses of two fixed spheres are M and 2M | vedclass

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(D) Let the masses be $M_1 = M$ and $M_2 = 2M$. The distance between their centres is $d = 10R$. The particle of mass $m = \frac{M}{10}$ is projected

Vedclass · Accepted Answer

$\sqrt{\frac{6 GM}{5 R}}$

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(D) Let the masses be $M_1 = M$ and $M_2 = 2M$. The distance between their centres is $d = 10R$. The particle of mass $m = \frac{M}{10}$ is projected from the mid-point,which is at a distance $r_1 = 5R$ from $M_1$ and $r_2 = 5R$ from $M_2$.By the law of conservation of energy,the total energy at the mid-point must equal the total energy at infinity (where potential energy and kinetic energy are zero).Initial Energy $E_i = K_i + U_i = \frac{1}{2}mv^2 - \frac{GM_1m}{r_1} - \frac{GM_2m}{r_2}$.Substituting the values: $E_i = \frac{1}{2}mv^2 - \frac{GMm}{5R} - \frac{G(2M)m}{5R} = \frac{1}{2}mv^2 - \frac{3GMm}{5R}$.Final Energy $E_f = 0$.Setting $E_i = E_f$: $\frac{1}{2}mv^2 = \frac{3GMm}{5R}$.Solving for $v$: $v^2 = \frac{6GM}{5R}$,so $v = \sqrt{\frac{6GM}{5R}}$.

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