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) [AIPMT 1998]

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\[=15\text{ }min\]

\[=\frac{15}{60}h=\frac{1}{4}\,h\]

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.