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Three masses m, 2m, 3m are moving in a x-y plane with speed 3u, 2u and u respectively as shown in the figure. The three masses collide at point P and stick together. The velocity of the resulting mass will be:(A)$\dfrac{u}{12}(\hat{i}+\sqrt{3}\hat{j})$ (B)$\dfrac{u}{12}(\hat{i}-\sqrt{3}\hat{j})$ (C)$\dfrac{u}{12}(-\hat{i}+\sqrt{3}\hat{j})$ (D)$\dfrac{u}{12}(-\hat{i}-\sqrt{3}\hat{j})$

Last updated date: 13th Jun 2024
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Hint: This is a problem where collision is involved. In such a case we use the conservation of momentum and energy principles. But since it is not mentioned if it is elastic or an inelastic collision, we only use the conservation of momentum principle to solve this problem.
Formula used:
The conservation of momentum law gives us the following:
${{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}+{{m}_{3}}{{u}_{3}}={{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}+{{m}_{3}}{{v}_{3}}$

Let the masses of the three bodies be ${{m}_{1}}$ , ${{m}_{2}}$ and ${{m}_{3}}$ and the initial velocities be given by u and the final velocities be given by v for the three different masses.
Now, applying the conservation of momentum principle, we have
${{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}+{{m}_{3}}{{u}_{3}}={{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}+{{m}_{3}}{{v}_{3}}$ .................. (1)
Given,
${{m}_{1}}=m$
${{m}_{2}}=2m$
${{m}_{3}}=3m$
And the initial velocities be given by
${{u}_{1}}=3u$
${{u}_{2}}=2u$
${{u}_{3}}=u$
Also, the three masses after colliding stick together, so the final mass is given by the sum of the masses of the three bodies.
Thus, plugging these values of mass and velocities in equation (1), and resolving the velocity into parallel and perpendicular components, we have
$\Rightarrow m\times 3u\hat{i}+2m\times 2u(-\hat{i}\cos 60-\hat{j}\sin 60)+3m\times u(-\hat{i}\cos {{60}^{\circ }}+\hat{j}\sin {{60}^{\circ }})=(m+2m+3m)v$
Where, v is the value of final velocity.
Solving the equation and simplifying we have,
$\Rightarrow \vec{v}=\dfrac{u}{12}(-\hat{i}-\sqrt{3}\hat{j})$
Thus the correct value of velocity after collision is given by $\dfrac{u}{12}(-\hat{i}-\sqrt{3}\hat{j})$.

So the correct option is (D) $\dfrac{u}{12}(-\hat{i}-\sqrt{3}\hat{j})$ .