A swimmer crosses the river along the line making an angle of `45^@` with the direction of flow. Velocity of the river water is `5(m)/(s)`. Swimmer takes 12 seconds to cross the river of width 60 m. The velocity of the swimmer with respect to water will be:

Options

A. `10(m)/(s)`
B. `5(m)/(s)`
C. `5sqrt5(m)/(s)`
D. `5sqrt2(m)/(s)`

Explanation

To solve the problem, we need to determine the velocity of the swimmer with respect to the water. Let's break down the solution step by step.

### Step 1: Understand the Given Information
- The swimmer is crossing a river at an angle of \(45^\circ\) to the direction of the river flow.
- The velocity of the river water is \(5 \, \text{m/s}\).
- The width of the river is \(60 \, \text{m}\).
- The time taken to cross the river is \(12 \, \text{s}\).

### Step 2: Calculate the Velocity of the Swimmer in the Y-Direction
The swimmer crosses the river of width \(60 \, \text{m}\) in \(12 \, \text{s}\). The velocity in the y-direction (the direction across the river) can be calculated using the formula:

\[ V_y = \frac{\text{Width of river}}{\text{Time taken}} = \frac{60 \, \text{m}}{12 \, \text{s}} = 5 \, \text{m/s} \]

### Step 3: Determine the Components of the Swimmer's Velocity
Since the swimmer is swimming at an angle of \(45^\circ\) to the direction of the river flow, we can denote the swimmer's velocity as \(V\). The components of the swimmer's velocity can be expressed as:

\[ V_x = V \cos(45^\circ) \quad \text{and} \quad V_y = V \sin(45^\circ) \]

Given that \(V_y = 5 \, \text{m/s}\), we can find \(V\):

\[ V \sin(45^\circ) = 5 \, \text{m/s} \implies V \cdot \frac{1}{\sqrt{2}} = 5 \implies V = 5\sqrt{2} \, \text{m/s} \]

### Step 4: Calculate the Components of the Swimmer's Velocity
Now we can find the x-component of the swimmer's velocity:

\[ V_x = V \cos(45^\circ) = 5\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 5 \, \text{m/s} \]

### Step 5: Combine the Velocities
The swimmer's velocity vector can be represented as:

\[ \vec{V}_{\text{swimmer}} = V_x \hat{i} + V_y \hat{j} = 5 \hat{i} + 5 \hat{j} \, \text{m/s} \]

### Step 6: Calculate the Velocity of the Swimmer with Respect to Water
The velocity of the river current is:

\[ \vec{V}_{\text{water}} = 5 \hat{i} \, \text{m/s} \]

To find the velocity of the swimmer with respect to the water, we subtract the velocity of the water from the velocity of the swimmer:

\[ \vec{V}_{\text{swimmer, water}} = \vec{V}_{\text{swimmer}} - \vec{V}_{\text{water}} = (5 \hat{i} + 5 \hat{j}) - (5 \hat{i}) = 5 \hat{j} \, \text{m/s} \]

### Step 7: Find the Magnitude of the Velocity
The magnitude of the swimmer's velocity with respect to the water is:

\[ |\vec{V}_{\text{swimmer, water}}| = 5 \, \text{m/s} \]

### Final Answer
The velocity of the swimmer with respect to water is \(5 \, \text{m/s}\).

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