5. The force with which the airflow acts on the sail can be calculated using the formula

$F=\frac{C S \rho\left(v_{0}-v\right)^{2}}{2}$, where $C-$ is the aerodynamic force coefficient, $S-$ is the area of the sail $S=5 \mathrm{~m}^{2} ; \rho$ - density of air, $v_{0}$ - wind speed $v_{0}=6 \mathrm{M} / c, v$ - speed of the sailboat. At some point in time, the instantaneous power of the wind reaches its maximum value. What is the speed of the sailboat at this moment?

Given:

$F=\frac{\operatorname{CS} \rho\left(v_{0}-v\right)^{2}}{2}$

$N\left(t^{\prime}\right)=N_{\max }$

$v_{0}=6 м / c$

$\overline{v\left(t^{\prime}\right)-?}$
Solution:

\[
\left.\begin{array}{l}
F=f(v) \\
N=F \cdot v
\end{array}\right\} \Rightarrow N=f(v)
\]

\[
N=\frac{\operatorname{CS\rho }}{2}\left(v_{0}^{2}-2 v_{0} v+v^{2}\right) v=\frac{\operatorname{CS\rho }}{2}\left(v_{0}^{2} v-2 v_{0} v^{2}+v^{3}\right)
\]

\[
N=N_{\max } \Rightarrow N^{\prime}(v)=0
\]

\[
N^{\prime}(v)=\frac{C S \rho}{2}\left(v_{0}^{2}-4 v_{0} v+3 v^{2}\right)=0
\]

\[
\begin{aligned}
& v_{0}^{2}-4 v_{0} v+3 v^{2}=0 \\
& 3 v^{2}-4 v_{0} v+v_{0}^{2}=0 \\
& v_{1,2}=\frac{4 v_{0} \pm \sqrt{16 v_{0}^{2}-12 v_{0}^{2}}}{6}=\frac{4 v_{0} \pm \sqrt{4 v_{0}^{2}}}{6}=\frac{4 v_{0} \pm 2 v_{0}}{6} \\
& v_{1}=v_{0} \text{ (does not satisfy the condition)} \\
& v_{2}=\frac{v_{0}=2 \mathrm{m} / \mathrm{s}}{3}
\end{aligned}
\]

## VARIANT 2