Question

Water is flowing steadily in a river. A and B are the two layers of water at heights $40 \text{ cm}$ and $90 \text{ cm}$ from the bottom. The velocity of the layer A is $12 \text{ cm/s}$. The velocity of the layer B is

• A. $15 \text{ cm/s}$
• B. $21 \text{ cm/s}$
• C. $27 \text{ cm/s}$
• D. $36 \text{ cm/s}$

Hint:

Velocity gradient ($dv/dy$) is constant for ideal steady river flow.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In the steady, laminar flow of a liquid like a river over a horizontal bed, the velocity of the fluid layers increases with height from the bottom.
For typical textbook problems involving river flow, a linear velocity profile is assumed unless stated otherwise.
This means the velocity of a layer is directly proportional to its vertical distance from the stationary bottom.
Step 2: Key Formula or Approach:
Assuming a constant velocity gradient, velocity $v$ is proportional to height $h$ ($v \propto h$).
We can set up a ratio for the two layers: $\frac{v_A}{h_A} = \frac{v_B}{h_B}$.
Step 3: Detailed Explanation:
We are given the height of layer A, $h_A = 40 \text{ cm}$.
The velocity of layer A is $v_A = 12 \text{ cm/s}$.
The height of layer B is $h_B = 90 \text{ cm}$.
Using the direct proportionality relationship: \[ \frac{v_A}{h_A} = \frac{v_B}{h_B} \] Substitute the known values into the equation: \[ \frac{12}{40} = \frac{v_B}{90} \] Simplify the left side of the equation: \[ 0.3 = \frac{v_B}{90} \] Multiply both sides by 90 to solve for $v_B$: \[ v_B = 0.3 \times 90 \] \[ v_B = 27 \text{ cm/s} \] Step 4: Final Answer:
The velocity of layer B is $27 \text{ cm/s}$.