Question

A man wishes to cross a river in a boat. If he crosses the river in minimum time, he takes

10

\min

with a drift of

120 m

. If he crosses the river taking the shortest route, he takes

12.5 \min

. Find the velocity of the boat with respect to water.

Accepted Answer

Given, minimum time of crossing, $t_{1} = 10$ $\min$ ,

drift at minimum time of crossing, $x = 120 m$ ,

time of crossing for shortest route, $t_{2} = 12.5 \min$ .

Case $1$ : If the man crosses the river in minimum time, he should move perpendicular to bank or normal to the direction of water flow.

Let $|{\vec{v}}_{m, w}| = v$ and $|{\vec{v}}_{w}| = u$ .

Time taken to cross river,

$t_{1} = \frac{d}{v}, \Rightarrow 10 = \frac{d}{v} \Rightarrow d = 10 v$

Drift,

$x = u t_{1}, \Rightarrow 120 = u \times 10 \Rightarrow u = 12 m \min^{- 1}$

Case $2$ : If the man crosses river taking the shortest route, the drift should be zero. Time taken to cross river,

$t_{2} = \frac{d}{\sqrt{v^{2} - u^{2}}} \Rightarrow 12.5 = \frac{d}{\sqrt{v^{2} - u^{2}}}$

From case $1, d = 10 v$ and $u = 12 {m \min}^{- 1}$
$12.5 = \frac{10 v}{\sqrt{v^{2} - 12^{2}}} \Rightarrow v = 20 m \min^{- 1}$

B M Sharma Solutions for Chapter: Kinematics II, Exercise 54: ILLUSTRATION 5.50

B M Sharma Physics Solutions for Exercise - B M Sharma Solutions for Chapter: Kinematics II, Exercise 54: ILLUSTRATION 5.50

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