A horizontal pipe of area of cross-section a and 3a respectively, then the ratio of velocity of flow at two different cross-section (If flow is streamline) is

Options

A. `1:9`
B. `9:1`
C. `1:3`
D. `3:1`

Answer

To solve the problem of finding the ratio of the velocities of flow at two different cross-sections of a horizontal pipe with areas of cross-section A and 3A, we can use the principle of continuity for fluid flow. Here’s the step-by-step solution:

### Step 1: Understand the Principle of Continuity
The principle of continuity states that for an incompressible fluid flowing in a pipe, the product of the cross-sectional area (A) and the velocity (V) at any two points along the pipe must be constant. This can be expressed mathematically as:
\[ A_1 V_1 = A_2 V_2 \]
where:
- \( A_1 \) is the area of the first cross-section,
- \( V_1 \) is the velocity at the first cross-section,
- \( A_2 \) is the area of the second cross-section,
- \( V_2 \) is the velocity at the second cross-section.

### Step 2: Assign Values to Areas
In this problem, we have:
- \( A_1 = A \) (the area of the first cross-section),
- \( A_2 = 3A \) (the area of the second cross-section).

### Step 3: Apply the Continuity Equation
Using the continuity equation:
\[ A_1 V_1 = A_2 V_2 \]
Substituting the values of \( A_1 \) and \( A_2 \):
\[ A V_1 = 3A V_2 \]

### Step 4: Simplify the Equation
We can cancel \( A \) from both sides (assuming \( A \neq 0 \)):
\[ V_1 = 3 V_2 \]

### Step 5: Find the Ratio of Velocities
To find the ratio of the velocities \( V_1 \) and \( V_2 \):
\[ \frac{V_1}{V_2} = 3 \]

### Conclusion
Thus, the ratio of the velocity of flow at the two different cross-sections is:
\[ \frac{V_1}{V_2} = 3:1 \]