Question

A swimmer crosses the river along the line making an angle of ${45}^{\circ }$ with the direction of flow. Velocity of the river water is $5\text{m/s}$. Swimmer takes 6 seconds to cross the river of width $60\text{m}$. The magnitude of velocity of the swimmer with respect to river will be

• A. $10\text{m/s}$
• B. $12\text{m/s}$
• C. $5\sqrt{5}\text{m/s}$
• D. $10\sqrt{5}\text{m/s}$

Answer 1

The correct option is C $5\sqrt{5}\text{m/s}$

Let
${\overrightarrow{V}}_{r,g}=$ Veloctiy of river w.r.t ground
${\overrightarrow{V}}_{m,g}=$ Veloctiy of swimmer w.r.t ground
​${\overrightarrow{V}}_{m,r}=$ Veloctiy of swimmer w.r.t river

Vertical component of ${\overrightarrow{V}}_{m,g}$, $V_y=\frac{\text{Distance}}{\text{Time}}=\frac{60}{6}=10\text{m/s}$
Since ${\overrightarrow{V}}_{m,g}$ makes an angle of ${45}^{\circ }$ with the direction of river, its horizontal and vertical component will be same.
$\therefore {\overrightarrow{V}}_{m,g}=10\hat{i}+10\hat{j}$
${\overrightarrow{V}}_{r,g}=5\hat{i}$
Relative velocity of swimmer w.r.t river,
${\overrightarrow{V}}_{m,r}={\overrightarrow{V}}_{m,g}-{\overrightarrow{V}}_{r,g}$
${\overrightarrow{V}}_{m,r}=10\hat{i}+10\hat{j}-5\hat{i}=5\hat{i}+10\hat{j}$
Magnitude, $|{\overrightarrow{V}}_{m,r}|=\sqrt{5^2+{10}^2}=5\sqrt{5}\text{m/s}$