Question

Water flows through a horizontal pipe of varying cross-section at the rate of \( \pi \times 10^{-1} \, \text{m}^3/\text{s} \). The velocity of water at a point where the radius of the pipe is 10 cm is \( (\pi = 3.14) \)

• A. 0.1 m/s
• B. 1 m/s
• C. 10 m/s
• D. 100 m/s

Hint:

The continuity equation relates the flow rate, cross-sectional area, and velocity of a fluid. For incompressible fluids, the flow rate must remain constant at any point in the pipe.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We use the equation of continuity, which states that the volume flow rate is the product of the cross-sectional area and the fluid velocity.
Step 2: Key Formula or Approach:
Flow Rate \( Q = A \cdot v \), where \( A = \pi r^2 \).
Step 3: Detailed Explanation:
Given: \( Q = \pi \times 10^{-1}\text{ m}^3\text{/s} \) and \( r = 10\text{ cm} = 0.1\text{ m} \).
Area \( A = \pi (0.1)^2 = 0.01\pi\text{ m}^2 \).
Equating the flow rate:
\[ \pi \times 10^{-1} = (0.01\pi) \cdot v \]
\[ 0.1\pi = 0.01\pi \cdot v \]
\[ v = \frac{0.1}{0.01} = 10\text{ m/s} \]
Step 4: Final Answer:
The velocity of water is 10 m/s.