Three masses m,2m,and 3m are moving in the x- | vedclass

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(C) According to the law of conservation of linear momentum,the total initial momentum equals the total final momentum.Let the velocity of the combine

Vedclass · Accepted Answer

$\frac{u}{12} (-\hat{i} + \sqrt{3} \hat{j})$

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(C) According to the law of conservation of linear momentum,the total initial momentum equals the total final momentum.Let the velocity of the combined mass be $\vec{v}$.The initial momentum vectors are:$\vec{p}_1 = m(3u)\hat{i} = 3mu\hat{i}$$\vec{p}_2 = 2m(2u)(-\cos 60^\circ \hat{i} + \sin 60^\circ \hat{j}) = 4mu(-\frac{1}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{j}) = -2mu\hat{i} + 2\sqrt{3}mu\hat{j}$$\vec{p}_3 = 3m(u)(-\cos 60^\circ \hat{i} - \sin 60^\circ \hat{j}) = 3mu(-\frac{1}{2}\hat{i} - \frac{\sqrt{3}}{2}\hat{j}) = -1.5mu\hat{i} - 1.5\sqrt{3}mu\hat{j}$Total initial momentum $\vec{P}_{total} = \vec{p}_1 + \vec{p}_2 + \vec{p}_3 = (3 - 2 - 1.5)mu\hat{i} + (2\sqrt{3} - 1.5\sqrt{3})mu\hat{j} = -0.5mu\hat{i} + 0.5\sqrt{3}mu\hat{j}$Total mass $M = m + 2m + 3m = 6m$.Using $\vec{P}_{total} = M\vec{v}$:$-0.5mu\hat{i} + 0.5\sqrt{3}mu\hat{j} = 6m\vec{v}$$\vec{v} = \frac{-0.5u\hat{i} + 0.5\sqrt{3}u\hat{j}}{6} = \frac{u}{12}(-\hat{i} + \sqrt{3}\hat{j})$.

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 $P$ and stick together. The velocity of the resulting mass will be

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