Why Does the Factor of 1/2 Appear in Dynamic Pressure Calculations?

[malindenmoyer]

Please refer to this website: http://www.grc.nasa.gov/WWW/K-12/airplane/dynpress.html"

In the "Simplify" step, where does the factor of 1/2 come from? I must be missing something simple...

Thanks for the help!

 

[Gyro]

What is the derivative of d(u^2)/dx? How does that differ from the second term in the 'algebra step'?
This relates to the concept of anti-derivative and integration. Have you studied that yet?

 

[malindenmoyer]

I get it now--I'm not used to a 'physicist's' calculus. Would it be proper to simply multiply each side of the original equation by 'dx' and integrate each side with the respective variables? This leaves (r*u^2)/2=-p -> (r*u^2)/2+p=0

Is there a reason why this method is not preferable?

 

[Gyro]

Maybe to re-inforce that a constant of integration is required to specify an initial condition? The method you used does not account for the 'constant' as you have the static + dynamic pressure equal to zero instead of the 'total pressure".

Remember, indefinite integrals require a constant of integration.

 

[rwisz]

I am happy to provide a response to your question about the derivation of dynamic pressure. The factor of 1/2 in the "Simplify" step comes from the application of Bernoulli's equation, which relates the pressure and velocity of a fluid in motion. In this case, we are looking at the dynamic pressure, which is the difference between the total pressure and the static pressure. When we apply Bernoulli's equation to this situation, we can see that the factor of 1/2 arises as a result of simplifying the equation. This factor is necessary to correctly relate the dynamic pressure to the velocity of the fluid. I hope this helps to clarify the derivation for you. If you have any further questions, please do not hesitate to ask.