Question

Three particles are kept at the three vertices of a right angled triangle as shown:


If each particle starts moving towards the next vertex with the same speed $v$ as shown in the figure. Find the velocity of $COM$ at that instant. (Masses of all particles are same)

• A. $\frac{1}{3}\left[\right(v-\sqrt{3}v)\hat{i}+2v\hat{j}]$
• B. $\frac{1}{3}[(v-\frac{\sqrt{3}v}{2})\hat{i}+\frac{v}{2}\hat{j}]$
• C. $\frac{1}{3}[2v\hat{i}+\sqrt{3}v\hat{j}]$
• D. $\frac{1}{3}[(v-\frac{\sqrt{3}v}{2})\hat{i}-\frac{v}{2}\hat{j}]$

Answer 1

The correct option is D $\frac{1}{3}[(v-\frac{\sqrt{3}v}{2})\hat{i}-\frac{v}{2}\hat{j}]$

From the figure, velocities of particles are
$V_A=v\hat{i}$, $V_B=-v\cos {30}^{\circ }\hat{i}+v\sin {30}^{\circ }\hat{j}=\frac{-\sqrt{3}v}{2}\hat{i}+\frac{v}{2}\hat{j}$
and $V_C=-v\hat{j}$
Let mass of each particle be $m$.
Then,
${\overrightarrow{v}}_{\text{com}}=\frac{m_a\overrightarrow{v_a}+m_b\overrightarrow{v_b}+m_c\overrightarrow{v_c}}{m_a+m_b+m_c}$
$=\frac{m\left(v\hat{i}\right)+m(\frac{-\sqrt{3}v}{2}\hat{i}+\frac{v}{2}\hat{j})+m(-v\hat{j})}{m+m+m}$
$\Rightarrow {\overrightarrow{v}}_{\text{com}}=\frac{1}{3}[(v-\frac{\sqrt{3}v}{2})\hat{i}-\left(\frac{v}{2}\right)\hat{j}]$
$\Rightarrow {\overrightarrow{v}}_{com}=\frac{1}{3}[(v-\frac{\sqrt{3}v}{2})\hat{i}-\frac{v}{2}\hat{j}]$