$2.2.38.$ A water jet boat moves in calm water. Force of water resistance to boat movement $F = k v^2$ . The speed of the ejected water relative to the boat $u$. Determine the steady-state speed of the boat, if the cross-section of the flow of water captured by the engine is $S$, the water density is $\rho$.
Since in steady-state mode remains unchanged, by Newton's first law, the thrust force is compensated by the drag force $$F_\text{resistance}=F_\text{traction}$$ During the time $dt$ the mass $dm$ of escaping water is: $$dm=\rho S v\,dt$$ In this case, relative to the Non-inertial Reference Frame, the water flies out with a velocity $u-v$.
Hence the momentum $dp$ of mass $dm$ $$dp=\rho S v(u-v)\,dt$$ The force of water thrust, we find through the change of momentum $$F=\frac{dp}{dt}=\rho S v(u-v)$$ Let's substitute and rewrite the equilibrium condition $$\rho S v(u-v)=k v^2$$ From where $$\boxed{v=u \frac{\rho S}{\rho S+k}}$$
$$v=\rho Su/(\rho S+k)$$