Water coming out of the mouth of a tap and falling vertically in streamline flow forms a tapering, column, i.e., the area of crosssection of the liquid column decreases as it moves down. Which of the following is the most accurate explanation for this

Correct Answer: The mass of water flowing past any cross-section must remain constant. Also, water is almost incompressible. Hence, the rate of volume flow must remain constant. As this is equal to velocity × area, the area decreases as velocity increases.

To understand why a falling stream of water tapers as it moves down, we can break down the physics principles involved step-by-step:

1. Principle of Conservation of Mass and Incompressibility
Water is an incompressible fluid, which means its density remains constant throughout the flow. According to the law of conservation of mass, the mass of water entering any cross-section of the stream per unit time must equal the mass of water leaving it. For an incompressible fluid, this translates to the conservation of volume flow rate (also known as the Equation of Continuity).

2. The Equation of Continuity
The volume flow rate of a fluid passing through a cross-sectional area is given by the product of the cross-sectional area and the velocity of the fluid at that point. Mathematically, it is expressed as:

$Q=A\times v=\text{constant}$

where:
• $Q$ is the volume flow rate,
• $A$ is the cross-sectional area of the water column, and
• $v$ is the velocity of the falling water at that section.

3. Effect of Gravity on velocity
As the water leaves the tap, it falls vertically under the influence of gravity. Gravity accelerates the falling water, meaning its downward speed increases continuously as it moves lower. If $v_0$ is the initial velocity of water at the tap mouth, the velocity $v$ at a depth $h$ below the tap is given by the equation of motion:

$v^2=v_0^2+2gh$

Since the acceleration due to gravity $g$ is acting downwards, $v$ increases as $h$ increases.

4. Conclusion
Since the product of the area and velocity must remain constant to satisfy the continuity equation, the cross-sectional area must decrease to compensate for the increase in velocity:

$A\propto \frac{1}{v}$

Therefore, as the velocity of the water increases during its fall, the cross-sectional area of the stream decreases, resulting in the characteristic tapering column of water.