First slide

Relative motion in 2 dimension

Question

A man wishes to cross a river in a boat. If he crosses the river in minimum time he takes 10 minutes with a drift of 120 m. If he crosses the river taking shortest route, he
takes 12.5 minutes, find velocity of the boat with respect to water.

Difficult

A

20 m/min
B

12 m/min
C

10 m/min
D

8 m/min

Solution

$\begin{array}{l} 10 = \frac{L}{v} . . . . . . (1) \\ 12.5 = \frac{L}{\sqrt{v^{2} - u^{2}}} = \frac{L}{v \sqrt{1 - u^{2} / v^{2}}} . . (2) \end{array}$
from (i) and (ii) $\frac{10}{12.5} = \frac{L}{v} \times \frac{v \sqrt{1 - u^{2} / v^{2}}}{L}$
$\begin{array}{l} \frac{4}{5} = \sqrt{1 - \frac{12^{2}}{v^{2}}} \\ \frac{16}{25} = 1 - \frac{12^{2}}{v^{2}} \Rightarrow \frac{12^{2}}{v^{2}} = 1 - \frac{16}{25} = \frac{9}{25} \\ \frac{12}{v} = \frac{3}{5} \Rightarrow v = \frac{12 \times 5}{3} = 20 m / \min \end{array}$

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