Question

Two particles $A$ and $B$ of equal mass $m$ are attached by a string of length $2l$ and initially placed over a smooth horizontal table in the position shown in Fig. Particle $B$ is projected across the table with speed $u$ perpendicular to $AB$ as shown in the figure. Find the velocities of each particle after the string becomes taut and the magnitude of the impulse tension.

Answer 1

From the impulse diagram of $A$
$J=m\nu ⟹J=\frac{1}{2}mucos\theta$
$\frac{1}{2}mu\frac{\sqrt{3}}{2}=\frac{\sqrt{3}mu}{4}$
Velocity of ball $B$ in the direction perpendicular to the string will remain constant. Hence,
${\nu }^{\prime }=usin\theta =\frac{u}{2}$
Hence, net velocity of $B$ is
${\nu }_B=\sqrt{\left(\frac{u}{2}\right)+(\nu {)}^2}=\frac{\sqrt{7}u}{4}$ and ${\nu }_A=\frac{\sqrt{3}u}{4}$