Water flows steadily through a horizontal pipe of a variable cross-section. If the pressure of the water is p at a point , where the speed of the flow is v. What is the pressure at another point , where the speed of the flow is 2 v ? Let the density of water be `1 g cm ^(-3)`.

Options

A. `p+(3/2)rhov^2`
B. `p-2rhov^2`
C. `p - 3rhov^2`
D. `p- (3/2) rhov^2`

Answer

To solve the problem, we will use Bernoulli's equation, which relates the pressure, velocity, and height of a fluid in a streamline flow. Since the pipe is horizontal, we can ignore the height terms.

### Step-by-Step Solution:

1. **Understand Bernoulli's Equation**:
The equation for a horizontal flow is given by:
\[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \]
where:
- \( P_1 \) is the pressure at point 1,
- \( v_1 \) is the velocity at point 1,
- \( P_2 \) is the pressure at point 2,
- \( v_2 \) is the velocity at point 2,
- \( \rho \) is the density of the fluid.

2. **Assign Known Values**:
From the problem:
- Let \( P_1 = P \) (pressure at point 1),
- Let \( v_1 = v \) (velocity at point 1),
- Let \( v_2 = 2v \) (velocity at point 2).

3. **Substitute Known Values into Bernoulli's Equation**:
Substitute \( P_1 \), \( v_1 \), and \( v_2 \) into the equation:
\[ P + \frac{1}{2} \rho v^2 = P_2 + \frac{1}{2} \rho (2v)^2 \]

4. **Simplify the Equation**:
The term \( (2v)^2 \) simplifies to \( 4v^2 \):
\[ P + \frac{1}{2} \rho v^2 = P_2 + \frac{1}{2} \rho (4v^2) \]
This can be rewritten as:
\[ P + \frac{1}{2} \rho v^2 = P_2 + 2 \rho v^2 \]

5. **Rearranging to Solve for \( P_2 \)**:
Rearranging gives:
\[ P_2 = P + \frac{1}{2} \rho v^2 - 2 \rho v^2 \]
\[ P_2 = P - \frac{3}{2} \rho v^2 \]

6. **Substituting the Density of Water**:
Given that the density of water \( \rho = 1 \, \text{g/cm}^3 = 1000 \, \text{kg/m}^3 \) (for SI units), we can leave it in terms of \( \rho \) for now:
\[ P_2 = P - \frac{3}{2} \rho v^2 \]

### Final Answer:
Thus, the pressure at the second point where the speed of flow is \( 2v \) is:
\[ P_2 = P - \frac{3}{2} \rho v^2 \]