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

$F=\frac{B S \rho\left(v_{0}-v\right)^{2}}{2}$, where $B-$ is the aerodynamic force coefficient, $S-$ is the area of the sail $S$ $=7 \mathrm{M}^{2} ; \rho$ - air density, $v_{0}$ - wind speed $v_{0}=6.3 \mu / c, v-$ 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:

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

$N\left(t^{\prime}\right)=N_{\text {max }}^{2}$

$v_{0}=6.3 \frac{M}{c}$

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

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

$N=\frac{B S \rho}{2}\left(v_{0}^{2}-2 v_{0} v+v^{2}\right) v=\frac{B S \rho}{2}\left(v_{0}^{2} v-2 v_{0} v^{2}+v^{3}\right)$

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

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

$v_{0}^{2}-4 v_{0} v+3 v^{2}=0$

$3 v^{2}-4 v_{0} v+v_{0}^{2}=0$

$v_{1,2}=\frac{4 v_{0} \pm \sqrt{16 v_{0}^{2}-12 v_{0}^{2}}}{6}=\frac{4 v_{0} \pm \sqrt{4 v_{0}^{2}}}{6}=\frac{4 v_{0} \pm 2 v_{0}}{6}$

$v_{1}=v_{0}$ (not satisfied) $v_{2}=\frac{v_{0}}{3}=3.1 \mathrm{M} / \mathrm{c}$

## VARIANT 4