question_answer 65)

                  A girl riding a bicycle with a speed of 5 m/s towards north direction, observes rain falling vertically down. If she increases her speed to 10 m/s, rain appears to meet her at \[{{45}^{o}}\] to the vertical. What is the speed of the rain? In what direction does rain fall a observed by a ground based observer?

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 \[|{{\vec{\upsilon }}_{r}}|\]                 \[=\,\sqrt{25+\,25}\,\,=\,5\sqrt{2}\,\,m{{s}^{-1}}.\]