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

Mass $$m$$: $$\vec{v}_1 = 3u \hat{i}$$

Mass $$2m$$: $$\vec{v}_2 = 2u(-\cos 60^\circ \hat{i} - \sin 60^\circ \hat{j}) = -u \hat{i} - u\sqrt{3} \hat{j}$$

Mass $$3m$$: $$\vec{v}_3 = u(-\cos 60^\circ \hat{i} + \sin 60^\circ \hat{j}) = -\frac{u}{2} \hat{i} + \frac{u\sqrt{3}}{2} \hat{j}$$

$$P_x = m(3u) + 2m(-u) + 3m\left(-\frac{u}{2}\right) = 3mu - 2mu - 1.5mu = -0.5mu$$

$$P_y = m(0) + 2m(-u\sqrt{3}) + 3m\left(\frac{u\sqrt{3}}{2}\right) = -2mu\sqrt{3} + 1.5mu\sqrt{3} = -0.5mu\sqrt{3}$$

$$\vec{P}_{initial} = -\frac{mu}{2} \hat{i} - \frac{mu\sqrt{3}}{2} \hat{j}$$

$$\vec{P}_{final} = \vec{P}_{initial}$$

$$6m \cdot \vec{V} = -\frac{mu}{2} \hat{i} - \frac{mu\sqrt{3}}{2} \hat{j}$$

$$\vec{V} = \frac{u}{12} (-\hat{i} - \sqrt{3}\hat{j})$$