question_answer

A particle moves with a velocity \[(3i+4j)m/s\]from origin. The displacement of particle along line \[x=y\] after two seconds will be:

A) 10m     

B) \[\frac{7}{\sqrt{2}}\]

C) \[7\sqrt{2}m\]

D) None of these

Correct Answer: C

Solution :

If \[\vec{A}\] is the displacement along the velocity vector,

then  \[\overrightarrow{A}=2\overrightarrow{v}=\left( 6\hat{i}+8\hat{j} \right)\]

Unit vector along line\[y=x,\]

\[\overrightarrow{B}=\cos 45{}^\circ \hat{i}+\sin 45{}^\circ \hat{j}\]

\[=\left( \frac{{\hat{i}}}{\sqrt{2}}+\frac{{\hat{j}}}{\sqrt{2}} \right)\]

Thus the displacement along \[\overrightarrow{B}\]

\[A\cos \theta =\frac{AB\cos \theta }{B}=\frac{\overrightarrow{A}.\overrightarrow{B}}{B}\]

\[^{=}\frac{\left( 6\hat{i}+8\hat{j} \right).\left( \frac{{\hat{i}}}{\sqrt{2}}+\frac{{\hat{j}}}{\sqrt{2}} \right)}{1}\]

\[=7\sqrt{2}\,\operatorname{m}\]