Question

A boat can go at a speed $40\text{km/h}$ in still water. If this boat is crossing a river when the speed of current is $20\sqrt{3}\text{km/h}$ and the width of the river is $4\text{km}$, how long will the boat take to cross the river at a point directly opposite to starting point?

• A. $15$ minutes
• B. $1$ hour
• C. $20$ minutes
• D. $12$ minutes

Answer 1

The correct option is D $12$ minutes


When the boat starts moving in the river, the velocity of the river pushes the boat towards the right.
Hence the boat should make an angle θ towards the left of the vertical, so that it reaches the exact opposite point to the point of start.
For the drift to be zero, the magnitude of $x$ component of the velocity of the boat w.r.t. river should be equal to the magnitude of velocity of the river. Hence,
$40\sin \theta =20\sqrt{3}\Rightarrow \theta ={60}^{\circ }$
Hence $y$ component of the velocity $v_y=v_b\cos {60}^o$.
Time taken by boat to cross the river $=t=\frac{s}{\left(v_b\cos {60}^o\right)}$
$s=\text{width of river}=4\text{km}$
$v_b=40\text{km/h}t=\frac{4}{\left(40\cos {60}^o\right)}=\frac{1}{5}h=12\text{minutes}$
Hence the correct option is (d).