question_answer

                The speed of a boat is 5 km/h in still water. It crosses a river of width 1.0 km along the shortest possible path in 15 min. The velocity of the river water is: (in km/h)                    

A)                 5       

B)                 1             

C)                                            3         

D)                                            4

Correct Answer: C

Solution :

Let \[{{v}_{r}}=\] velocity of river                 \[{{v}_{br}}=\]velocity of boat in still water and                 w = width of river                 Time taken to cross the river = 15 min                 \[=\frac{15}{60}h=\frac{1}{4}\,h\]                                 Shortest path is taken when \[{{v}_{b}}\] is along AB. In this case                 \[v_{br}^{2}=v_{r}^{2}+v_{b}^{2}\]                 Now,     \[t=\frac{w}{{{v}_{b}}}=\frac{w}{\sqrt{v_{br}^{2}-v_{r}^{2}}}\] \[\therefore \frac{1}{4}=\frac{1}{\sqrt{{{5}^{2}}-v_{r}^{2}}}\] \[\Rightarrow {{5}^{2}}-v_{r}^{2}=16\] \[\Rightarrow v_{r}^{2}=25-16=9\] \[\therefore {{v}_{r}}=\sqrt{9}\,=3\,km/h\]                 Note:    If \[{{v}_{r}}\ge {{v}_{br}}\], the boatman can never reach at point B.