Bernoulli’s Theorem Principle and Its Derivation

In this post, we will discuss in detail the Explanation of Bernoulli’s principle, its statement, and the proof of its Formula through Derivation. Bernoulli’s theorem is an important topic in the Class 11 fluid chapter. Bernoulli’s theorem always occurs in half-yearly or yearly examinations. So let’s get started with its statement:-

Bernoulli’s principle is one of the most important concepts in physics and mathematics, helping students understand how pressure, velocity, and energy behave in moving fluids. In this article, we will clearly explain Bernoulli’s theorem, its real-life meaning, and a step-by-step Bernoulli theorem proof in a simple and easy-to-follow way.

Along with this, you will also learn about the Bernoulli differential equation, its standard form, and how it is solved in mathematics. Whether you are preparing for Class 11, Class 12, or competitive exams, this guide will help you build strong conceptual understanding with clear explanations, formulas, and practical examples.

What is Bernoulli’s Principle?

According to Bernoulli’s Principle, the sum of kinetic energy, potential energy, and pressure energy at unit volume remains constant at every point of the flow is called Bernoulli’s Equation.

Bernoulli’s equation is derived from the principle of conservation of energy.

Bernoulli’s Principle Formula

The formula for Bernoulli’s principle is given as follows:

P+1/2ρv²+ρgh = Constant

Where v is the velocity of the fluid, ρ is the density of the fluid, p is the pressure exerted by the fluid, h is the height of the container, and the sum of all is constant.

Bernoulli’s Equation Derivation

Work = Force x Displacement
Work at per unit time
Work = Force x Displacement/time
Work = Force x Velocity Equation -1

Pressure = Force/Area
Force = Pressure x Area
Put in Equation 1
Work = Pressure x Area x Velocity

Change in work = Wf – Wi
= P₁ A₁ V₁ – P₂ A₂ V₂
ΔW = P₁ A₁ V₁ – P₂ A₂ V₂ Equation –2
According to Eqn of Continuity-
A₁ V₁ = m/ρ
ΔW = P₁ m/ρ – P₂ m/ρ
ΔW = m/ρ ( P₁ – P₂)

Change in Potential Energy-
ΔP = Pf – Pi
ΔP = mgh₁ – mgh₂
ΔP = mg (h₁ – h₂)

Change in Kinetic Energy-
ΔK = Kf – Ki
ΔK = 1/2mv₁² – 1/2mv₂ ²
ΔK = 1/2m(v₁² – v₂ ²)

Change in Work = Change in Kinetic and Potential Energy
m/ρ ( P₁ – P₂) = mg (h₁ – h₂) + 1/2m(v₁² – v₂ ²)
Cut all the Masses
1/ρ ( P₁ – P₂) = 1/g (h₁ – h₂) + 1/2(v₁² – v₂ ²)
Rho will move to the RHS
P₁ – P₂ = ρgh₁ – ρgh₂ + (1/2ρv₁² – 1/2ρv₂ ²)
P + ρgh + 1/2ρv² = Constant
Hence Proved

Example of Bernoulli’s Principle

1. Flying of an Aeroplane Wing

When an aeroplane moves forward, air flows faster over the curved upper surface of the wing than below it.

• Faster air → lower pressure
• Slower air → higher pressure
  This pressure difference creates an upward lift force, allowing the plane to fly.

2. Perfume Spray or Atomiser

When you press a perfume spray, air moves fast across the top of the tube. This creates low pressure, which pulls the liquid up and sprays it out as fine droplets.

Other Physics Content

• Parallelogram Law of Vector Edition and its Derivation
• Work Energy Theorem for Constant and Variable Force
• Newton’s Three Laws of Motion – Physics
• Johannes Kepler’s Law of Planetary Motion
• Class 11 Physics Chapter 1 Notes on Unit and Measurement