P i = 1 2 ρ Δ V Δ t v 0 2 = ρ A 2 v 0 3 {\displaystyle P_{i}={\frac {1}{2}}\rho {\frac {\Delta V}{\Delta t}}v_{0}^{2}={\frac {\rho A}{2}}v_{0}^{3}}
Immediately before and immediately after the blade
ρ Δ V Δ t = A ( v 0 + v 1 ) 2 {\displaystyle \rho {\frac {\Delta V}{\Delta t}}=A{\frac {(v_{0}+v_{1})}{2}}}
Thus
P e f f = P i − P f = 1 2 ρ Δ V Δ t ( v 0 2 − v 1 2 ) {\displaystyle P_{e}ff=P_{i}-P_{f}={\frac {1}{2}}\rho {\frac {\Delta V}{\Delta t}}(v_{0}^{2}-v_{1}^{2})} = ρ A 4 ( v 0 + v 1 ) ( v 0 2 − v 1 2 ) {\displaystyle ={\frac {\rho A}{4}}(v_{0}+v_{1})(v_{0}^{2}-v_{1}^{2})}
C p = ( v 0 + v 1 ) ( v 0 2 − v 1 2 ) 2 v 0 3 = ( 1 + x ) ( 1 − x 2 ) 2 {\displaystyle C_{p}={\frac {(v_{0}+v_{1})(v_{0}^{2}-v_{1}^{2})}{2v_{0}^{3}}}={\frac {(1+x)(1-x^{2})}{2}}}
d d x C p = 0 → x = 1 3 {\displaystyle {\frac {d}{dx}}C_{p}=0\rightarrow x={\frac {1}{3}}}
C p = % 59 {\displaystyle C_{p}=\%59}
