A man can swin with speed 6 km/h in still water. If he tries of swim across a river flowing with speed 5km/h, how fast does a leaf floating on water appears to go , to the man ?

Options

A. 6 km/h across the river
B. 5 km/h across the river
C. 6km/h upstream
D. 5 km/h upstream

Answer

To solve the problem, we need to determine how fast a leaf floating on the water appears to move to a man swimming across the river.

### Step-by-Step Solution:

1. **Identify the Speeds**:
- The speed of the man in still water is given as \( V_m = 6 \, \text{km/h} \).
- The speed of the river current is \( V_r = 5 \, \text{km/h} \).

2. **Set Up the Coordinate System**:
- Let's assume the man swims in the positive \( j \) direction (across the river).
- The river flows in the positive \( i \) direction (downstream).

3. **Determine the Velocity Vectors**:
- The velocity vector of the man swimming across the river can be represented as:
\[ \vec{V_m} = 0 \hat{i} + 6 \hat{j} \, \text{km/h} \]
- The velocity vector of the river (and thus the leaf) is:
\[ \vec{V_r} = 5 \hat{i} + 0 \hat{j} \, \text{km/h} \]

4. **Calculate the Relative Velocity**:
- The velocity of the leaf with respect to the man is given by:
\[ \vec{V_{l/m}} = \vec{V_r} - \vec{V_m} \]
- Substituting the values:
\[ \vec{V_{l/m}} = (5 \hat{i} + 0 \hat{j}) - (0 \hat{i} + 6 \hat{j}) = 5 \hat{i} - 6 \hat{j} \]

5. **Interpret the Result**:
- The resulting vector \( \vec{V_{l/m}} = 5 \hat{i} - 6 \hat{j} \) indicates that the leaf appears to move downstream (in the \( i \) direction) and also appears to move in the negative \( j \) direction (which means it appears to be moving away from the man).
- To find the speed of the leaf with respect to the man, we calculate the magnitude of the velocity vector:
\[ |\vec{V_{l/m}}| = \sqrt{(5)^2 + (-6)^2} = \sqrt{25 + 36} = \sqrt{61} \approx 7.81 \, \text{km/h} \]

6. **Conclusion**:
- Therefore, the speed of the leaf as it appears to the man is approximately \( 7.81 \, \text{km/h} \).

### Final Answer:
The leaf appears to go at a speed of approximately \( 7.81 \, \text{km/h} \) to the man.