Q.The reading of pressure meter attached with a closed pipe is $5\times10^{5}$ N m$^{-2}$. On opening the valve of the pipe, the reading of the pressure meter is $4.5\times10^{5}$ N m$^{-2}$. Calculate the speed of the water flowing in the pipe.

Bernoulli's Principle: From Intuition to Precision

Imagine you're standing on a railway platform as an express train roars past. You feel a strange pull — a gentle but definite tug — drawing you toward the train. That's not your imagination. It's Bernoulli's Principle at work in the real world.

The Core Intuition

Here's the simplest way to think about it: fast-moving fluid (air or water) exerts less pressure than slow-moving fluid.

When the train rushes by, it drags the air next to it along. That air moves fast. The air on the other side of you, far from the train, is nearly still. The still air pushes harder than the fast air, so you feel a net push toward the train.

This isn't magic — it's a direct consequence of how energy is conserved in a flowing fluid.

The Precise Statement

Where:

• $P$ = pressure of the fluid
• $\rho$ = density of the fluid
• $v$ = speed of the fluid
• $g$ = acceleration due to gravity
• $h$ = height above a reference level

Bernoulli's Principle states: For an ideal fluid (incompressible, non-viscous, flowing steadily along a streamline), an increase in the fluid's speed occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.

Breaking It Down Piece by Piece

The equation has three terms, each representing a form of "energy per unit volume":

1. $P$ — Pressure energy. Think of it as the "push" the fluid has stored.
2. $\frac{1}{2}\rho v^2$ — Kinetic energy per unit volume. Faster flow means more of this.
3. $\rho gh$ — Gravitational potential energy per unit volume. Higher elevation means more of this.

The sum stays constant along a streamline. So if one term goes up, at least one other must go down.

A Concrete Example: The Garden Hose

Put your thumb over the end of a garden hose. The water shoots out faster — but the pressure at the nozzle drops. You can feel it: the hose feels "softer" near your thumb. The water's speed increased, so its pressure decreased. That's Bernoulli in your hands.

When Does Bernoulli's Principle Apply?

It works perfectly for:

• Ideal fluids — water, air at moderate speeds (below about 0.3 times the speed of sound)
• Steady flow — no turbulence or sudden changes
• Along a single streamline — you can't compare two different streamlines unless they start from the same reservoir