# Derivation of Dynamic Pressure

• arithhuh
In summary, the dynamic pressure formula of 1/2PV^2, where P is density and V is velocity, comes from looking at Newton's Second Law for a fluid particle along a streamline. This is the result of integrating the term \rho V \frac{\partial{V}}{\partial{s}}. This formula can be explained by considering the change in air velocity and the resulting change in momentum, leading to a force that is directly proportional to v^2.
arithhuh
Dear All, I need your help with a present problem -

I am a student pilot, and want to see the derivation for the dynamic pressure formula - which is 1/2PV^2 where P - Density, V-Velocity

I was asked an interesting question by my a fellow student: Why is dynamic pressure directly proportional to v^2 and not just V?

Kindly explain to me these points.

Thank you,

It comes from looking at Newton's Second Law for a fluid particle along a streamline. In a nutshell, the term is the result of an integration of the term
$$\rho V \frac{\partial{V}}{\partial{s}}$$

How much detail do you want to get into?

Here is a simple argument why the pressure is proportional to v^2:

The object flying through air must remove air in front of it. If you assume that the change of air velocity is proportional to the speed of the object, then the change of momentum of air (pushed away in a certain time) is proportional to v^2 (because the mass current is also proportional to v). Force equals the change of momentum/change of time, so force is also proportional to v^2.

## 1. What is dynamic pressure?

Dynamic pressure is a measure of the kinetic energy of a fluid or gas in motion. It is caused by the movement of the fluid, such as air flow around an object.

## 2. How is dynamic pressure calculated?

Dynamic pressure is calculated by taking the density of the fluid, multiplying it by the velocity squared, and dividing by two. The formula for dynamic pressure is:
Dynamic Pressure = 1/2 * density * velocity2

## 3. What is the difference between dynamic pressure and static pressure?

The main difference between dynamic pressure and static pressure is that dynamic pressure is caused by the movement of a fluid, while static pressure is caused by the weight of a fluid. Static pressure does not take into account the velocity of the fluid.

## 4. What are some real-world applications of dynamic pressure?

Dynamic pressure is commonly used in aerodynamics and fluid mechanics to calculate the forces acting on an object in motion, such as an airplane or car. It is also used in weather forecasting to measure wind speeds and in HVAC systems to determine air flow rates.

## 5. How does dynamic pressure affect lift and drag?

Dynamic pressure plays a key role in determining the lift and drag forces on an object. It is directly related to the airspeed of the object, and as the dynamic pressure increases, so does the lift and drag forces. This is why airplanes need to reach a certain speed for takeoff and why aerodynamic designs are important for reducing drag and increasing lift.

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