Wind Turbine Power Calculation Equations

[recreated]

Glad it's Friday? I will be if you can help me with this :D :

I'm trying to understand an equation about calculating the power extracted by the blades of a wind turbine. I have spent a couple of hours going over it but still not making much more sense.

I'm trying to find out why P = 1/2*m*(v² - v²_d). Why the 1/2? Why the '-' sign?? And why the ² ? I assume it is to do with F=ma, so the v (velocity) is squared to show acceleration but still I can't figure it all out.

Any tips on how they got to the last equation (18.5) shown above would be amazing!

Thank you! Cannot wait for this to make sense

 

[CWatters]

https://www.raeng.org.uk/education/diploma/maths/pdf/exemplars_advanced/23_Wind_Turbine.pdf

 

[RTW69]

Does K.E.= 1/2 X M X V² look familiar?

 

[recreated]

It does now, RTW69