Three small metallic charged balls (radius tends to zero) have equal charges q and masses m, 4m and m as shown in the figure. The balls are connected by light string (Non-conducting) of length l each, and placed on horizontal frictionless, non-conducting table. Initially balls are at rest and form a straight line. A sharp horizontal impulse gives the central ball a speed v directed perpendicular to strings connecting the balls. Choose correct option(s):

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Three small metallic charged balls (radius tends to zero) have equal charges q and masses m, 4m and m as shown in the figure. The balls are connected by light string (Non-conducting) of length $l$ each, and placed on horizontal frictionless, non-conducting table. Initially balls are at rest and form a straight line. A sharp horizontal impulse gives the central ball a speed v directed perpendicular to strings connecting the balls. Choose correct option(s):

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The minimum distance during subsequent motion between the balls of mass m is $\frac{6 q^{2} l}{3 q^{2} + 16 m ε_{0} v^{2} l}$

The minimum distance during subsequent motion between the balls of mass m is $\frac{3 q^{2} l}{3 q^{2} + 16 π m ε_{0} v^{2} l}$

The maximum difference in electrostatic potential energy of the system during subsequence motion is $\frac{2 m v^{2}}{3}$

The maximum difference in electrostatic potential energy of the system during subsequent motion is $\frac{4 m v^{2}}{3}$

answer is A, C.

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Detailed Solution

$V_{C M} = \frac{4 m v}{6 m} = \frac{2 v}{3}$

In centre of mass frame Gain in P.E = Loss in K.E.
$\frac{q^{2}}{4 π ε_{0}} [\frac{1}{D} - \frac{1}{2 l}] = 2 m v^{2} - \frac{4 m v^{2}}{3} \Rightarrow D = \frac{6 q^{2} l}{3 q^{2} + 16 m π ε_{0} v^{2} l}$