Question

A boat $B$ is moving upstream with velocity $3\text{m/s}$ with respect to ground. An observer standing on boat observes that a swimmer $S$ is crossing the river perpendicular to the direction of motion of boat. If river flow velocity is $4\text{m/s}$ and swimmer crosses the river of width $100\text{m}$ in $50\text{sec}$, then

• A. Velocity of swimmer w.r.t ground is $\sqrt{13}\text{m/s}$
• B. Drift of swimmer along river is zero
• C. Drift of swimmer along river will be $100\text{m}$
• D. Velocity of swimmer w.r.t. ground is $2{\text{ms}}^{-1}$

Answer 1

The correct option is A Velocity of swimmer w.r.t ground is $\sqrt{13}\text{m/s}$

Step 1, Given data and figure

Velocity of the boat = 3 m/s

Velocity of the river = 4 m/s

Width of the river = 100 m

Step 2, Finding the velocity of the swimmer with respect to ground
Here, width of the river $=100\text{m}$,

We can write
Velocity of boat w.r.t ground $\overrightarrow{V_{BG}}=-3\hat{i}\text{m/s}$ (Direction is opposite to +ve $x$-axis)

Velocity of river w.r.t ground $\overrightarrow{V_{RG}}=4\hat{i}\text{m/s}$

From the boat, the swimmer appears to cross the width of the river $\left(100\text{m}\right)$ in

(50\) seconds perpendicular to the direction of boat (along $Y-$ axis).

Hence, $\overrightarrow{V_{SB}}=100/50=2\hat{j}\text{m/s}$
We have,
$\Rightarrow \overrightarrow{V_{SB}}=\overrightarrow{V_{SG}}-\overrightarrow{V_{BG}}$
$\Rightarrow \overrightarrow{V_{SG}}=\overrightarrow{V_{BG}}+\overrightarrow{V_{SB}}=-3\hat{i}+2\hat{j}$
$|\overrightarrow{V_{SG}}|=\sqrt{(3^2+2^2)}$ = $\sqrt{13}\text{m/s}$

Hence the velocity of the swimmer with respect to ground is $\sqrt{13}$ m/s