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 $S=5 \mathrm{m}^{2} ; \rho$ is the density of air, $v_{0}$ is the wind speed $v_{0}=6 \mathrm{~m} / \mathrm{c}, v$ is the speed of the sailboat. At some moment in time, the instantaneous power of the wind reaches its maximum value. What is the speed of the sailboat at this moment?

Given:

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

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

$v_{0}=6 \mathrm{~m} / \mathrm{c}$

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

$$
\begin{aligned}
& \left.\begin{array}{l}
F=f(v) \\
N=F \cdot v
\end{array}\right\} \Rightarrow N=f(v) \\
& N=\frac{\operatorname{CS\rho }}{2}\left(v_{0}^{2}-2 v_{0} v+v^{2}\right) v=\frac{\operatorname{CS\rho }}{2}\left(v_{0}^{2} v-2 v_{0} v^{2}+v^{3}\right) \\
& N=N_{\max } \Rightarrow N^{\prime}(v)=0 \\
& N^{\prime}(v)=\frac{\operatorname{CS\rho }}{2}\left(v_{0}^{2}-4 v_{0} v+3 v^{2}\right)=0
\end{aligned}
$$

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

## VARIANT 2