Water is flowing steadjly in a river. P and Q are two layers of water at heights of20 cm and 50 cm from the bottom. The velocity of the layer A is 15 cm/sec. What is the velocity of the layer 8 ?

Options

A. 15cm/s
B. 22.5 cm/s
C. 30 cm/s
D. 37.5 cm/s

Explanation

To solve the problem of finding the velocity of layer Q in a river where water flows steadily, we can use the principles of fluid mechanics, specifically the relationship between velocity and height in a viscous fluid.

### Step-by-Step Solution:

1. **Identify Given Information**:
- Height of layer P (h1) = 20 cm
- Height of layer Q (h2) = 50 cm
- Velocity of layer P (v1) = 15 cm/sec
- We need to find the velocity of layer Q (v2).

2. **Understand the Relationship**:
- According to the Newton's law of viscosity, the shear force (f) acting between layers is proportional to the velocity gradient (du/dy).
- For steady flow in a river, the shear force is constant across layers, which implies that the ratio of height to velocity for two layers is constant.

3. **Set Up the Proportionality**:
- From the relationship established, we can write:
\[ \frac{h1}{h2} = \frac{v1}{v2} \]
- Here, h1 and h2 are the heights of layers P and Q, and v1 and v2 are their respective velocities.

4. **Substitute the Known Values**:
- Plugging in the known values:
\[ \frac{20 \text{ cm}}{50 \text{ cm}} = \frac{15 \text{ cm/sec}}{v2} \]

5. **Cross Multiply to Solve for v2**:
- Cross multiplying gives:
\[ 20 \cdot v2 = 50 \cdot 15 \]
- Simplifying the right side:
\[ 20 \cdot v2 = 750 \]

6. **Isolate v2**:
- Now, divide both sides by 20:
\[ v2 = \frac{750}{20} = 37.5 \text{ cm/sec} \]

7. **Conclusion**:
- Therefore, the velocity of layer Q is **37.5 cm/sec**.