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Two spheres each of mass ‘m’, each moving with velocity u moving in directions at 30° S of E and 30° N of E respectively collide at point O and stick to each other. The velocity of the combination later becomes

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\frac{\sqrt{3} u}{2} due East

\sqrt{3} u due East

\frac{\sqrt{3} u}{2} due West

\frac{2 u}{\sqrt{3}} due North

answer is A.

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Detailed Solution

$u_{1} = u \cos θ i + u \sin θ j$

$u_{2} = u \cos θ i + u \sin θ j$

$v = \frac{m_{1} u_{1} + m_{2} u_{2}}{m_{1} + m_{2}}$

$v = \frac{m (u \cos θ i + u \sin θ j) + m (u \cos θ i - u \sin θ j)}{m + m}$

$v = \frac{m (u \cos θ i + u \sin θ j + u \cos θ i - u \sin θ j)}{2 m}$

$v = \frac{2 u \cos θ i}{2} = u \cos θ i = u \cos 30 ° i$

$= u \frac{\sqrt{3}}{2} i \Rightarrow \frac{\sqrt{3}}{2} u due east$

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