Two particles A and B of equal mass m each are attached by a string of length 2 ℓ and initially placed over a smooth horizontal table in the position shown in figure. Particle B is projected across th

Question

Two particles A and B of equal mass m each are attached by a string of length $2 ℓ$ and initially placed over a smooth horizontal table in the position shown in figure. Particle B is projected across the table with speed u perpendicular to AB as shown in figure. The ratio $\frac{v _{B}}{v _{A}}$ after the string becomes taut is $\frac{α}{β}$ . Find the least integral value of $(α + β) /2$ . (where $v_{B}$ and $v_{A}$ are speed of B & A respectively after the string become taut.)

TestHub · Accepted Answer

Resolve velocity of B along and perpendicular to the taut string.

$v_{A} = \frac{u c o s 3 0 ^{\circ}}{2} = \frac{3 u}{4}$

Perpendicular component of B remains $u sin 3 0^{\circ} = u /2$ .

$v_{B} = \frac{3 u ^{2}}{16} + \frac{u ^{2}}{4} = \frac{7 u}{4}$

$\frac{v _{B}}{v _{A}} = \frac{7}{3} \Rightarrow \frac{α + β}{2} = 5$

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