Question

A boat crosses a river of width 200m in the shortest time and is found to experience a drift of 100m is reaching the opposite bank. The time taken now is 't'. If the same boat is to cross the river by shortest path, the time taken to cross will be:

• A. $2t$
• B. $\sqrt{2}t$
• C. $3t$
• D. $\frac{2t}{\sqrt{3}}$

Answer 1

The correct option is D $\frac{2t}{\sqrt{3}}$

Let the velocity of the man and river is $v_m$ and $v_r$

when it cross the river in shortest time t

$v_m=\frac{200}{t}$

it drift due the river velocity

$v_r=\frac{100}{t}$

when he cross the river by shortest path than one vector component of v_{m} is cancel with river velocity $v_r$

Let it make a angle as shown in fig

$v_{m}sin\theta =v_r$ from this we get

$\frac{200}{t}sin\theta =\frac{100}{t}$

$\Rightarrow sin\theta =\frac{1}{2}$

$\Rightarrow \theta ={30}^0.$

than man cross the river with velocity $v_{m}cos\theta =v_{m}cos{30}^0$

time taken by the man to cross is $t=\frac{200}{v_m\frac{\sqrt{3}}{2}}=\frac{400}{\frac{200\sqrt{3}}{t}}=\frac{2t}{\sqrt{3}}$