\[\left( \frac{2+2\sqrt{3}}{3} \right)\,\overset{\hat{\ }}{\mathop{i}}\,-\frac{2}{3}\overset{\hat{\ }}{\mathop{j}}\,\] \[4\overset{\hat{\ }}{\mathop{i}}\,\]
A) \[\left( \frac{2+2\sqrt{3}}{3} \right)\hat{i}-\frac{2}{3}\hat{j}\]
B) \[4\hat{i}\]
C) \[\left( \frac{2-2\sqrt{3}}{3} \right)\,\overset{\hat{\ }}{\mathop{i}}\,-\frac{2}{3}\overset{\hat{\ }}{\mathop{j}}\,\]
D) None of the above
Correct Answer: A
Solution :
Here, \[{{m}_{1}}=1\,kg,\,{{\vec{v}}_{1}}=2\,\hat{i}\] \[{{m}_{2}}=2\,kg,\,{{\vec{v}}_{2}}=2\cos {{30}^{o}}\hat{i}-2\sin {{30}^{o}}\hat{j}\] \[{{\vec{v}}_{CM}}=\frac{{{m}_{1}}\,{{{\vec{v}}}_{1}}+{{m}_{2}}{{{\vec{v}}}_{2}}}{{{m}_{1}}+{{m}_{2}}}\] \[=\frac{1\times 2\,\hat{i}+2(2\cos {{30}^{o}}\hat{i}-2\sin {{30}^{o}}\hat{j})}{1+2}\] \[=\left( \frac{2\,i+2\sqrt{3}}{3} \right)\hat{i}-\frac{2}{3}\hat{j}\]
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