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 Problem
The swimmer crosses a river that flows horizontally. The swimmer swims at an angle of 45 degrees to the flow of the river. The width of the river is 60 meters, and the velocity of the river is 5 m/s. The swimmer takes 12 seconds to cross the river.

### Step 2: Calculate the Velocity of the Swimmer in the Y-Direction
The width of the river (the distance the swimmer needs to cross) is 60 meters, and the time taken to cross is 12 seconds. We can calculate the swimmer's velocity in the y-direction (the direction across the river) using the formula:

\[ \text{Velocity in y-direction (Vy)} = \frac{\text{Distance}}{\text{Time}} = \frac{60 \, \text{m}}{12 \, \text{s}} = 5 \, \text{m/s} \]

### Step 3: Determine the Components of the Swimmer's Velocity
Since the swimmer swims at a 45-degree angle to the flow of the river, the components of the swimmer's velocity with respect to the ground can be expressed as:

\[ V_y = V_x \]

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

\[ V_x = 5 \, \text{m/s} \]

### Step 4: Write the Velocity of the Swimmer with Respect to the Ground
The velocity of the swimmer with respect to the ground can be represented as a vector:

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

### Step 5: Write the Velocity of the River
The velocity of the river is given as:

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

### Step 6: Calculate the Velocity of the Swimmer with Respect to the Water
To find the swimmer's velocity with respect to the water, we use the concept of relative velocity:

\[ \vec{V}_{\text{swimmer (water)}} = \vec{V}_{\text{swimmer (ground)}} - \vec{V}_{\text{river}} \]

Substituting the values we have:

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

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

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

### Final Answer
The velocity of the swimmer with respect to the water is **5 m/s**.

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