Assertion: The stream of water flowing at high speed from a garden hose pipe tends to spread like a

Q: Why doesn't the water stream just stay the same width as the nozzle?

Because gravity changes the speed of the water after it leaves the nozzle. Since the water is in free fall (or free rise), its speed must change. To satisfy the conservation of mass (Equation of Continuity), the cross-sectional area must adjust inversely to the speed. If it stayed the same width, mass would either accumulate or disappear, which is physically impossible!

Q: Does surface tension play a role in this narrowing and spreading?

While surface tension helps keep the stream cohesive and eventually causes a very thin falling stream to break into droplets (Plateau-Rayleigh instability), the primary macroscopic narrowing and spreading behavior is fully governed by gravity and the Equation of Continuity.

Q: What happens if the flow is not steady or the fluid is compressible?

If the fluid were highly compressible (like a gas), the density $\rho$ would change, meaning $\rho A v = \text{constant}$ would allow density changes to offset some of the area changes. However, for liquids like water, compressibility is negligible, so $Av = \text{constant}$ holds strictly.

Q: Why does the upward stream eventually break into droplets instead of widening infinitely?

As the upward velocity approaches zero, the continuity equation would theoretically require the area to become infinite. In reality, as the stream slows down and widens, surface tension and air resistance destabilize the wide sheet of water, causing it to break up into individual droplets, creating the classic fountain spray.

Q: Is the volume flow rate really constant even if there is air resistance?

Yes, the volume flow rate of the water stream itself remains constant along the continuous portion of the stream because water cannot be compressed or destroyed. Air resistance may slightly reduce the acceleration (or increase deceleration), but at any cross-section, the product of the actual velocity and area will still remain constant.

Question

**Assertion**: The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down.

**Reason**: In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant.

Options:

(A) If Assertion is true, Reason is true; Reason is the correct explanation for Assertion

(B) If Assertion is true, Reason is true; Reason is not a correct explanation for Assertion

(C) If Assertion is true; Reason is false

(D) If Assertion is false; Reason is true

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are watering a garden.

When you point the hose straight up, the water shoots out in a tight stream but quickly spreads out, forming a wide, beautiful fountain-like shape.

Conversely, when you point the hose straight down from a balcony, the water stream starts wide at the nozzle but becomes incredibly thin and narrow as it falls.

Why does this happen?

Is it some mysterious force, or is it a beautiful consequence of the conservation of mass?

Let us dive deep into the fluid mechanics that govern this everyday phenomenon.

### The Master Equation: Continuity

To understand this behavior, we must look at the **Equation of Continuity**, which is a direct mathematical statement of the **conservation of mass** for fluids.

For any steady, streamline flow of an incompressible fluid (like water), the mass of fluid entering any cross-section per unit time must equal the mass of fluid leaving it.

Mathematically, this is written as:

$$	ext{Mass flow rate} = ho A v = 	ext{constant}$$

Since water is incompressible, its density $ho$ is constant throughout the flow.

This simplifies our master equation to the volume flow rate:

$$A v = 	ext{constant}$$

Where:
- $A$ is the cross-sectional area of the fluid stream.
- $v$ is the velocity of the fluid at that specific cross-section.

This simple equation reveals a profound inverse relationship:

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

This means that **area and velocity are inversely proportional**.

If the velocity of the fluid increases, the cross-sectional area must decrease to maintain a constant flow rate.

If the velocity decreases, the area must increase.

### Case 1: Flowing Vertically Upwards

Let us apply this principle to the first scenario: pointing the hose vertically upwards.

As water leaves the nozzle with an initial velocity $u$, it moves upwards against the pull of gravity.

According to the equations of kinematics under gravity:

$$v^2 = u^2 - 2gh$$

As the height $h$ increases, the velocity $v$ of the water must decrease.

Now, let us bring in our continuity relationship, $A \propto \frac{1}{v}$.

Because the velocity $v$ is decreasing as the water climbs higher, the cross-sectional area $A$ of the stream must increase to keep the volume flow rate constant.

This increase in area is why the stream spreads out, eventually breaking apart and behaving like a fountain.

Thus, the first part of the Assertion is **true**.

### Case 2: Flowing Vertically Downwards

Now, let us look at the second scenario: pointing the hose vertically downwards.

As water leaves the nozzle and falls downwards, gravity acts in the same direction as the motion, accelerating the water.

Using kinematics:

$$v^2 = u^2 + 2gh$$

As the fall distance $h$ increases, the velocity $v$ of the water stream increases.

Applying the continuity relationship $A \propto \frac{1}{v}$ once again:

Since the velocity $v$ is increasing as the water falls, the cross-sectional area $A$ must decrease to maintain a constant flow rate.

This reduction in area causes the stream to narrow down as it descends.

Thus, the second part of the Assertion is also **true**.

### Evaluating Assertion and Reason

Let us evaluate both statements:

1. **Assertion**: "The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down." 
   - This is **True**, as we have physically and mathematically verified both cases.

2. **Reason**: "In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant."
   - This is **True**, as it is the definition of the Equation of Continuity ($Av = 	ext{constant}$).

Since the Equation of Continuity is the exact physical principle used to explain why the stream spreads or narrows, the Reason is the **correct explanation** for the Assertion.

Therefore, the correct option is **(a)**.

Select Answer:

Visualized Solution

Visualizing the Flow

The Equation of Continuity

Inverse Relationship between Area and Velocity

Case 1: Flowing Vertically Upwards

Spreading of the Upward Stream

Case 2: Flowing Vertically Downwards

Narrowing of the Downward Stream

Evaluating Assertion and Reason

The Sigma Insight: Flow of Fluid

Analyzing the Setup

The Master Equation

Case 1

Case 2

Evaluating Assertion and Reason

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