Water enters through end A with a speed `v_1` and leaves through end B with a speed `v_2` of cylindrical tube AB. The tube is always completely filled with water. In case I the tube is horizontal, in case II it vertical with the end A upward and in case III it is vertical with the end B upward. We have `v_1=v_2`for

Options

A. caseI
B. caseII
C. caseIII
D. each case

Answer

To solve the problem, we need to analyze the three cases of the cylindrical tube filled with water and determine under what conditions the speeds \( v_1 \) and \( v_2 \) are equal. We will use the principle of continuity in fluid mechanics, which states that for an incompressible fluid flowing through a pipe, the product of the cross-sectional area and the fluid velocity is constant along the pipe.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a cylindrical tube with two ends: A (where water enters) and B (where water exits).
- The water enters with speed \( v_1 \) at end A and exits with speed \( v_2 \) at end B.
- The tube is completely filled with water.

2. **Case I: Horizontal Tube**:
- In this case, the tube is horizontal, and the height difference between A and B is zero.
- According to the continuity equation:
\[ A_1 v_1 = A_2 v_2 \]
- Since the cross-sectional area \( A_1 \) at A is equal to the cross-sectional area \( A_2 \) at B (the tube is cylindrical and uniform), we can conclude:
\[ v_1 = v_2 \]

3. **Case II: Vertical Tube with A Upward**:
- Here, the tube is vertical with end A at the top and end B at the bottom.
- The continuity equation still applies:
\[ A_1 v_1 = A_2 v_2 \]
- Again, since \( A_1 = A_2 \), we have:
\[ v_1 = v_2 \]
- The gravitational potential energy does not affect the flow speed directly since the tube is completely filled with water.

4. **Case III: Vertical Tube with B Upward**:
- In this scenario, the tube is vertical with end B at the top and end A at the bottom.
- The continuity equation applies as well:
\[ A_1 v_1 = A_2 v_2 \]
- Since \( A_1 = A_2 \), we again find:
\[ v_1 = v_2 \]
- The orientation does not change the fact that the tube is filled with water.

5. **Conclusion**:
- In all three cases (horizontal, vertical A up, vertical B up), the speeds \( v_1 \) and \( v_2 \) are equal due to the continuity equation and the uniform cross-sectional area of the tube.

### Final Answer:
In all cases (I, II, and III), we have \( v_1 = v_2 \).