Two water pipes of diameters 2 cm and 4 cm are connected with the main supply line. The velocity of flow of water in the pipe of 2 cm

Options

A. 4 times that in the other pipe
B. `(1)/(4)` times that in the other pipe
C. 2 times that in the other pipe
D. `(1)/(2)` times that in the other pipe

Explanation

To solve the problem of finding the velocity of water flow in two connected pipes with different diameters, we can use the principle of continuity. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have two pipes with diameters:
- Pipe 1 (D1) = 2 cm
- Pipe 2 (D2) = 4 cm

We need to find the velocity of water flow in Pipe 1 (V1) given the velocity in Pipe 2 (V2).

### Step 2: Apply the Equation of Continuity
According to the equation of continuity for incompressible fluids, the product of the cross-sectional area (A) and the velocity (V) is constant along the flow. This can be expressed as:
\[ A_1 V_1 = A_2 V_2 \]

### Step 3: Calculate the Areas of the Pipes
The cross-sectional area (A) of a pipe can be calculated using the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the pipe.

- For Pipe 1 (D1 = 2 cm):
- Radius \( R_1 = \frac{D_1}{2} = \frac{2}{2} = 1 \) cm
- Area \( A_1 = \pi (1)^2 = \pi \) cm²

- For Pipe 2 (D2 = 4 cm):
- Radius \( R_2 = \frac{D_2}{2} = \frac{4}{2} = 2 \) cm
- Area \( A_2 = \pi (2)^2 = 4\pi \) cm²

### Step 4: Substitute Areas into the Continuity Equation
Now we can substitute the areas into the continuity equation:
\[ A_1 V_1 = A_2 V_2 \]
\[ \pi V_1 = 4\pi V_2 \]

### Step 5: Simplify the Equation
We can cancel \( \pi \) from both sides:
\[ V_1 = 4 V_2 \]

### Step 6: Conclusion
Thus, the velocity of water flow in Pipe 1 (V1) is 4 times the velocity in Pipe 2 (V2):
\[ V_1 = 4 V_2 \]

### Final Answer
The velocity of flow in the pipe of diameter 2 cm is 4 times that in the pipe of diameter 4 cm.

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