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 ho.

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

    <h3 id="back-link"><a href="/#2.2">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $2.2.38.$ 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 $\rho$.
    </p>

    <h3>Solution</h3>
    <p>
        </p>

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 $$\boxed{v=u \frac{\rho S}{\rho S+k}}$$

    <h4>Answer</h4>
    <p>
        $$v=\rho Su/(\rho S+k)$$
    </p>


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