Answer:
Let velocity of
rain, \[{{\vec{\upsilon }}_{r}}\,=\,p\hat{i}\,+q\hat{j}\] ?.. (i)
Ist case: Velocity
of girl, \[{{\vec{\upsilon }}_{g}}=\,(5\,\,m{{s}^{-1}})\,\hat{i}\]
\ Velocity of rain w.r.t. girl, \[{{\vec{\upsilon
}}_{rg}}=\,{{\vec{\upsilon }}_{r}}-{{\vec{\upsilon }}_{g}}\]
or \[{{\vec{\upsilon }}_{rg}}\,=\,(p\hat{i}+q\hat{j})-5i=(p-5)\,i+q\hat{j}\]
Sine rain appears
to fall vertically downward, so \[(p5)=0\] or \[p=5\]
2nd case : \[{{\vec{\upsilon
}}_{g}}=\,(10\,m{{s}^{-1}})\hat{i}\]
\[\therefore \] \[{{\vec{\upsilon
}}_{{{r}_{g}}}}\,={{\vec{\upsilon
}}_{r}}-10\,\hat{i}\,=\,p\hat{i}+\,qj-10\hat{i}\]
\[=(p-10)\,i\,+\,q\hat{j}=5\hat{i}+\,q\hat{j}\]
Since rain appears
to fall at \[{{45}^{o}}\] to the vertical \[\therefore \,\,q=-5\]
Hence, \[{{\vec{\upsilon
}}_{r}}=5\hat{i}-5\hat{j}\] and \[|{{\vec{\upsilon }}_{r}}|\]
\[=\,\sqrt{25+\,25}\,\,=\,5\sqrt{2}\,\,m{{s}^{-1}}.\]
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