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The width of river is 1 km. The velocity of boat is 5 km/hr. The boat covered the width of river with shortest possible path in 15 min. Then the velocity of river stream is :
A
3 km/hr
B
4 km/hr
C
$\sqrt{29}$ km/hr
D
$\sqrt{41}$ km/hr
Detailed Solution
For the shortest path, the boat is steered upstream at an angle so that its resultant velocity is straight across the river.
Then the speed across the river is $\sqrt{u^2 - v^2}$, where u = 5 km/h is the boat speed in still water and v the river speed.
Time: $t = \dfrac{d}{\sqrt{u^2 - v^2}}$
$\dfrac{1}{4}$ h $= \dfrac{1}{\sqrt{25 - v^2}} \Rightarrow \sqrt{25 - v^2} = 4$
$25 - v^2 = 16 \Rightarrow v = 3$ km/hr
Then the speed across the river is $\sqrt{u^2 - v^2}$, where u = 5 km/h is the boat speed in still water and v the river speed.
Time: $t = \dfrac{d}{\sqrt{u^2 - v^2}}$
$\dfrac{1}{4}$ h $= \dfrac{1}{\sqrt{25 - v^2}} \Rightarrow \sqrt{25 - v^2} = 4$
$25 - v^2 = 16 \Rightarrow v = 3$ km/hr