Three masses \(m,\,2\,m\) and \(3\,m\) are moving in \(x-y\) plane with speed \(3\,u,\,2\,u\,\) and \(u\) respectively as shown in figure. The three masses collide at the same point at \(P\) and stick together. The velocity of resulting mass will be

\(\begin{array}{l}
{m_1}{u_1} + {m_2}{u_2} + ...... = {m_1}{v_1} + {m_2}{v_2} + ....\\
Given\,{m_1}\, = \,m,{m_2} = 2m\,and\,{m_3} = 3m\\
and\,{u_1} = 3u,\,{u_2} = 2u\,and\,{u_3} = u
\end{array}\)

Let the velocity when the stick \(= \overrightarrow v \)

Then, according to question,

\(\begin{array}{l}
m \times 3u\left( i \right) + 2m \times 2u\left( { - \hat i\cos {{60}^ \circ } - \hat j\sin {{60}^ \circ }} \right)\\
 + 3\,m \times u\left( { - \hat i\cos {{60}^ \circ } + \hat j\sin {{60}^ \circ }} \right)\\
 = \left( {m + 2m + 3m} \right)\,\overrightarrow v \\
 \Rightarrow \,3mu\hat i - 4mu\frac{{\hat i}}{2} - 4mu\left( {\frac{{\sqrt 3 }}{2}\hat j} \right) - 3mu\frac{{\hat i}}{2}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 3mu\left( {\frac{{\sqrt 3 }}{2}\hat j} \right) = 6m\,\overrightarrow v \\
 \Rightarrow \,mu\hat i\, - \frac{3}{2}mu\hat i - \frac{{\sqrt 3 }}{2}mu\hat j = 6m\,\overrightarrow v \\
 \Rightarrow \frac{1}{2}mu\hat i - \frac{{\sqrt 3 }}{2}mu\hat j = 6m\,\overrightarrow v \\
 \Rightarrow \,\overrightarrow v  = \frac{u}{{12}}\,\left( { - \hat i - \sqrt 3 \hat j} \right)
\end{array}\)