\[\begin {array} {1 1} (a)\;\frac{u \sqrt 3}{4} & \quad (b)\;u \sqrt 3 \\ (c)\;\frac{u \sqrt 3}{2} & \quad (d)\;\frac{u}{2} \end {array}\]

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Initial velocity of ball along $AB= u \cos \theta$

and perpendicular to to $AB= u \sin \theta$

When the string becomes , but the two bodies would acquire the same velocity along AB

By momentum conservation

$2 mv =mu \cos \theta$

$v= \large\frac{u \cos \theta}{2}$

$\quad= \large\frac{u}{2}$$ \times \large \frac{4 l^2 -l^2}{2l}$

$\quad= \large\frac{u}{2}$$ \times \large\frac{ \sqrt 3 l} {2l}$

$\quad= \large\frac{\sqrt 3}{4}$$u$

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