Relation Between Pressure and Velocity: Formula, Bernoulli's Principle and Applications

The relation between pressure and velocity explains how the pressure of a moving fluid changes as its speed changes. This concept is important in fluid mechanics and helps explain the working of many everyday devices, such as aeroplane wings, spray bottles, and water pipes.

This article explains what is the relation between velocity and pressure, the formulas used to describe it, Bernoulli's principle, and its real-life applications in simple and easy-to-understand language.

Table of Contents

• What is the Relation Between Pressure and Velocity
• Bernoulli's Equation for the Relation Between Pressure and Velocity
• Laplace's Formula for the Relation Between Pressure and Velocity
• Applications of the Relation Between Pressure and Velocity
• Frequently Asked Questions on Relation Between Pressure and Velocity

What is the Relation Between Pressure and Velocity

Pressure and velocity are two important physical quantities that describe the behavior of moving fluids and gases. Before learning how they are connected, it is helpful to know what each term means. 

What is Pressure

Pressure is the force acting per unit area on a surface. In simple words, it tells us how much force is applied over a given area. Sharp objects, such as knives and nails, create more pressure because the force acts on a smaller area.

The formula for pressure is

$P=\frac{F}{A}$

Where, P is Pressure (Pa), F is Force (N) and A is Area (m²)

What is Velocity

Velocity is the rate of change of an object's position in a specific direction. Since it has both magnitude and direction, velocity is a vector quantity.

$v=\frac{\text{Displacement}}{\text{Time}}$

Where, v is Velocity (m/s), Displacement is Change in position (m) and Time is Time taken (s)

Relation between Pressure and Velocity

The relation between pressure and velocity depends on the situation being studied.

• In fluid mechanics, pressure and velocity are inversely related.
• In the motion of sound waves, pressure and velocity are directly related.

The fact is that different physical conditions lead to different relationships between these two quantities.

The relationship between them is explained mainly by two equations:

• Bernoulli's Equation (for fluid flow)
• Laplace's Formula (for the velocity of sound)

Bernoulli's Equation for the Relation Between Pressure and Velocity

The most common explanation of the relation between pressure and velocity is given by Daniel Bernoulli through Bernoulli's Principle. For an incompressible and non-viscous fluid, the relation between pressure and velocity is given by Bernoulli's equation,

$P+\frac{1}{2}\rho v^2+\rho gh=\text{Constant}$

Where,

• Pis pressure of the fluid (Pa)
• ρ is density of the fluid (kg/m³)
• v is velocity of the fluid (m/s)
• g is acceleration due to gravity (m/s²)
• h is height above the reference level (m)

According to this equation, when the velocity of a fluid increases, its pressure decreases if the height remains constant. Similarly, when the velocity decreases, the pressure increases.

This inverse relationship helps explain the flow of fluids through pipes, nozzles, and aircraft wings.

Laplace's Formula for the Relation Between Pressure and Velocity

Pierre-Simon Laplace improved Newton's equation for the velocity of sound by considering that sound travels through the air as an adiabatic process, where no heat is exchanged with the surroundings.

The formula is,

$v=\sqrt{\frac{\gamma P}{\rho}}$

Where,

• v is the velocity of sound (m/s)
• P is the pressure of the medium (Pa)
• γ is the adiabatic constant
• Ρ is the density of the medium (kg/m³)

In this case, pressure and velocity are directly related.

• Higher pressure increases the velocity of sound.
• Lower pressure decreases the velocity of sound.

Applications of the Relation Between Pressure and Velocity

The relation between pressure and velocity has many practical uses in everyday life and engineering. According to Bernoulli's Principle, when the velocity of a fluid increases, its pressure decreases. This principle is used in several devices and technologies.

• Carburetors: In vehicles, carburetors use the pressure difference created by fast-moving air to mix fuel with air before it enters the engine.
• Venturimeters: These instruments measure the flow rate of liquids in pipes by comparing changes in pressure and velocity.
• Spray Bottles and Atomizers: When liquid passes through a narrow nozzle, its velocity increases and pressure decreases. This breaks the liquid into a fine spray.
• Water Supply Systems: Engineers use this principle while designing pipelines and pumping systems to ensure smooth and efficient fluid flow.
• Medical Nebulizers: Nebulizers use fast-moving air to convert liquid medicine into a fine mist, making it easier to inhale.

These applications show how this relation between pressure and velocity plays an important role in science, engineering, and many devices used in daily life.

In this article, we have learned about the relation between pressure and velocity and how these two physical quantities are connected in different situations. We also studied Bernoulli's equation, which explains why pressure decreases as the velocity of a fluid increases, and Laplace's formula, which describes the relationship between pressure and the velocity of sound.