Question

Sol. $Initial\left\{ \begin{matrix} m_{1} = 0.1, & {\bar{v}}_{1} = 20\widehat{i} \\ m_{2} = 0.2kg, & {\bar{v}}_{2} = 20\widehat{j} \\ m_{3} = 0.4kg, & {\bar{v}}_{3} = 20\widehat{k} \end{matrix} \right.\ $

$final\left\{ \begin{aligned} & {\bar{v}}_{3} = 0 \\ & {\bar{v}}_{2} = 10\widehat{j} + 5\widehat{k} \\ & {\bar{v}}_{1} = v_{x}\widehat{i} + v_{y}\widehat{j} + v_{z}\widehat{k} \end{aligned} \right.\ $

∴ f_ext = 0 ⇒ $({\bar{v}}_{cm})_{f} = ({\bar{v}}_{cm})_{i}$

⇒ $(v_{x})_{cm\ i} = (v_{x})_{cm\text{ f}} \Rightarrow \frac{0.1 \times 20}{0.7} = \frac{0.1 \times v_{x}}{0.7}$

⇒$(v_{y})_{cm\text{ i}} = (v_{y})_{cm\text{ f}}$

⇒ $\frac{0.2 \times 20}{0.7} = \frac{0.20 \times 10 + 0.1 \times v_{y}}{0.7}$

⇒ v_y = 20 m/s

(v₂)cm i = (v₂)_{cm f}

⇒ $\frac{0.4 \times 20}{0.7} = \frac{0.2 \times 5 + 0.1 \times v_{z}}{0.7}$

⇒ v₂ =70 $\widehat{k}$

= ${\overrightarrow{v}}_{1} = 20\widehat{i} + 20\widehat{j} + 70\widehat{k}$