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<meta name="description" content="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
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<meta property="og:title" content="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
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<meta property="og:description" content="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 $\rho$.">
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<meta property="og:description" content="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
<title>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 $\rho$.</title>
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<title>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
$2.2.38.$ 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>
<center>
<figure>
<img src="draw.png"
loading="lazy" width="230" />
<figcaption>
For problem $2.2.38$
</figcaption>
</figure>
</center>
<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 $u-v$.
</p><p>
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
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
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<meta name="description" content="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 $\rho$.">
<meta name="description" content="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
<meta property="og:title" content="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 $\rho$.">
<meta property="og:title" content="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.">
<meta property="og:image" content="img/logo.png">
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<meta property="og:description" content="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 $\rho$.">
<meta property="og:description" content="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
<title>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 $\rho$.</title>
<title>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
$2.2.38.$ 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 $\rho$.
$2.2.38.$ 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>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="draw.png"
<img src="draw.png"
loading="lazy" width="230" />
loading="lazy" width="230" />
<figcaption>
<figcaption>
For problem $2.2.38$
For problem $2.2.38$
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
Since in steady-state mode remains unchanged, by Newton's first law, the thrust force is compensated by the drag force
Since in steady-state mode remains unchanged, by Newton's first law, the thrust force is compensated by the drag force
$$F_\text{resistance}=F_\text{traction}$$
$$F_\text{resistance}=F_\text{traction}$$
During the time $dt$ the mass $dm$ of escaping water is:
During the time $dt$ the mass $dm$ of escaping water is:
$$dm=\rho S v\,dt$$
$$dm=\rho S v\,dt$$
In this case, relative to the Non-inertial Reference Frame, the water flies out with a velocity $u-v$.
In this case, relative to the Non-inertial Reference Frame, the water flies out with a velocity $u-v$.
</p><p>
</p><p>
Hence the momentum $dp$ of mass $dm$
Hence the momentum $dp$ of mass $dm$
$$dp=\rho S v(u-v)\,dt$$
$$dp=\rho S v(u-v)\,dt$$
The force of water thrust, we find through the change of momentum
The force of water thrust, we find through the change of momentum
$$F=\frac{dp}{dt}=\rho S v(u-v)$$
$$F=\frac{dp}{dt}=\rho S v(u-v)$$
Let's substitute and rewrite the equilibrium condition
Let's substitute and rewrite the equilibrium condition
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
