An ideal fluid flows through a pipe of circular cross-section made of two sections with diameters `2.5 cm` and `3.75 cm`. The ratio of the velocities in the two pipes is

Options

A. `9:4`
B. `3:2`
C. `sqrt(3):sqrt(2)`
D. `sqrt(2):sqrt(3)`

Explanation

To solve the problem of finding the ratio of velocities in two sections of a pipe with different diameters, we can follow these steps:

### Step 1: Understand the Problem
We have a pipe with two sections having diameters \(D_1 = 2.5 \, \text{cm}\) and \(D_2 = 3.75 \, \text{cm}\). We need to find the ratio of the velocities \(V_1\) and \(V_2\) of the fluid flowing through these sections.

### Step 2: Use the Equation of Continuity
According to the equation of continuity for an incompressible fluid, the product of the cross-sectional area and the velocity at any two points in a flow must be equal:
\[ A_1 V_1 = A_2 V_2 \]
where \(A_1\) and \(A_2\) are the cross-sectional areas at points 1 and 2, respectively.

### Step 3: Calculate the Cross-Sectional Areas
The cross-sectional area \(A\) of a circular pipe can be calculated using the formula:
\[ A = \frac{\pi D^2}{4} \]
Thus, we can express the areas \(A_1\) and \(A_2\) as:
\[ A_1 = \frac{\pi D_1^2}{4} \quad \text{and} \quad A_2 = \frac{\pi D_2^2}{4} \]

### Step 4: Substitute Areas into the Continuity Equation
Substituting the areas into the continuity equation gives:
\[ \frac{\pi D_1^2}{4} V_1 = \frac{\pi D_2^2}{4} V_2 \]
We can cancel \(\frac{\pi}{4}\) from both sides:
\[ D_1^2 V_1 = D_2^2 V_2 \]

### Step 5: Rearrange to Find the Ratio of Velocities
Rearranging the equation to find the ratio of velocities:
\[ \frac{V_1}{V_2} = \frac{D_2^2}{D_1^2} \]

### Step 6: Substitute the Values of Diameters
Now we substitute the values of \(D_1\) and \(D_2\):
\[ \frac{V_1}{V_2} = \frac{(3.75)^2}{(2.5)^2} \]

### Step 7: Calculate the Squares
Calculating the squares:
\[ (3.75)^2 = 14.0625 \quad \text{and} \quad (2.5)^2 = 6.25 \]
Thus,
\[ \frac{V_1}{V_2} = \frac{14.0625}{6.25} \]

### Step 8: Simplify the Ratio
Now simplify the ratio:
\[ \frac{14.0625}{6.25} = \frac{14.0625 \div 6.25}{6.25 \div 6.25} = \frac{2.25}{1} = 2.25 \]
To express this as a fraction:
\[ 2.25 = \frac{9}{4} \]

### Final Answer
Thus, the ratio of the velocities \(V_1 : V_2\) is:
\[ \frac{V_1}{V_2} = \frac{9}{4} \]