If a liquid enters a pipe of diameter 'd' with velocity 'v', then what will be its velocity at the exit if the diameter of the pipe reduces to 0.5d?
- A4vCorrect
- B0.5v
- Cv
- D2v
Solution
Concept
For incompressible flow, discharge Q = A × V stays the same at every section of the pipe.
Formula
A1 V1 = A2 V2; V2 = V1 × (d1/d2)2
Step-by-step solution
1. Continuity equation: A1 V1 = A2 V2 2. Area A = (π/4)d2, so V2 = V1 × (d1/d2)2 3. d1 = d, d2 = 0.5d, so d1/d2 = 2 4. V2 = v × (2)2 = 4v
Explanation
Liquid is incompressible, so the same discharge must pass through every section. When the diameter becomes half, the area becomes one-fourth. So the velocity becomes four times, which is 4v.
Why the other options are wrong
B (0.5v): velocity increases, not decreases, when area reduces. C (v): velocity stays the same only when the area does not change. D (2v): this treats velocity as varying with diameter, but it varies with area (d2).
Shortcut
Velocity is inversely proportional to d2. Half the diameter gives 4 times the velocity.
Common mistake
Using the diameter ratio instead of the area ratio and getting 2v.
Correct answer: A) 4v