A) \[2.62\,\,m{{s}^{-1}}\]
B) \[4.6\,\,m{{s}^{-1}}\]
C) \[0.89\,\,m{{s}^{-1}}\]
D) \[1.414\,\,m{{s}^{-1}}\]
Correct Answer: C
Solution :
Let the man start crossing the road at an angle \[\theta \] with the road side. For safe crossing, the condition is that the man must cross the road by the time, the truck describes the distance\[(4+2\cot \theta )\]. So,\[\frac{4+2\cot \theta }{2}=\frac{2/\sin \theta }{v}or\,\,v=\frac{2}{2\sin \theta +\cos \theta }\] For minimum\[v,\,\,\frac{dv}{d\theta }=0\] \[\Rightarrow \] \[\frac{-2(2\cos \theta -\sin \theta )}{{{(2\sin \theta +\cos \theta )}^{2}}}=0\] \[\Rightarrow \] \[2\cos \theta -\sin \theta =0\] \[\Rightarrow \] \[\tan \theta =2,\,\,so\,\,\sin \theta =\frac{2}{\sqrt{5}},\,\,\cos \theta =\frac{1}{\sqrt{5}}\] \[{{v}_{\min }}=\frac{2}{2(2/\sqrt{5})+(1/\sqrt{5})}=\frac{2}{\sqrt{5}}=0.89\,\,m/s\]You need to login to perform this action.
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