An incompressible fluid flows steadily through a cylindrical pipe which has radius 2 R at point A and radius R at point B farther along the flow direction. If the velocity at point A is v, its velocity at point B is

Options

A. `2 v`
B. `v`
C. `v//2`
D. `4 v`

Explanation

To solve the problem, we will use the principle of conservation of mass, which is expressed through the continuity equation for incompressible fluids. The continuity equation states that the product of the cross-sectional area and the fluid velocity at two points in a flow must be equal.

### Step-by-Step Solution:

1. **Identify the Areas**:
- At point A, the radius of the pipe is \(2R\). The cross-sectional area \(A_1\) can be calculated using the formula for the area of a circle:
\[ A_1 = \pi (2R)^2 = \pi (4R^2) = 4\pi R^2 \]
- At point B, the radius of the pipe is \(R\). The cross-sectional area \(A_2\) is:
\[ A_2 = \pi R^2 \]

2. **Apply the Continuity Equation**:
The continuity equation states:
\[ A_1 V_1 = A_2 V_2 \]
where:
- \(V_1\) is the velocity at point A (given as \(v\)),
- \(V_2\) is the velocity at point B (which we need to find).

3. **Substitute the Areas and Velocities into the Equation**:
Substituting the values we found for \(A_1\) and \(A_2\) into the continuity equation:
\[ (4\pi R^2) v = (\pi R^2) V_2 \]

4. **Simplify the Equation**:
We can cancel \(\pi R^2\) from both sides of the equation (since they are non-zero):
\[ 4v = V_2 \]

5. **Solve for \(V_2\)**:
Thus, the velocity at point B is:
\[ V_2 = 4v \]

### Final Answer:
The velocity at point B is \(4v\).

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