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<meta name="description" content="A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is 200 m wide?">
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<meta property="og:description" content="A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is 200 m wide?">
<title>A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is 200 m wide?</title>
$1.4.18^*.$ A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is $200$ m wide?
</p>
<h3>Solution</h3>
<p>
<p>Consider the boy's movements with the speed $\vec{v}$, when he is carried away by the river with the current $\vec{u}$</p>
Representation of $\vec{v'}$ as the sum of two vectors
</figcaption>
@@ -67,33 +67,33 @@
</figure>
</center>
<p>Let's write in projections on the horizontal and vertical axes, taking into account that $u = 2v$</p>
−
<p class="exp">$$v_x = 2v - v \cos\alpha$$</p>
−
<p class="exp">$$v_y = v \sin\alpha$$</p>
+
<p class="exp">$$v_x = 2v - v \cos\alpha$$</p>
+
<p class="exp">$$v_y = v \sin\alpha$$</p>
<p>Find the time it takes the boy to swim across the river</p>
<p class="exp">
−
$$ t = \frac{H}{v_y} = \frac{H}{v \sin\alpha}$$
+
$$ t = \frac{H}{v_y} = \frac{H}{v \sin\alpha}$$
</p>
<p>During time $t$ it will be carried along the coast by an amount</p>
<p class="exp">
−
$$ L = v_xt = H\frac{2 - \cos\alpha}{\sin\alpha}\quad(1) $$
+
$$ L = v_xt = H\frac{2 - \cos\alpha}{\sin\alpha}\quad(1) $$
</p>
−
<p>To find the minimum of $L$, it is necessary to find the extremum of the function $f(\alpha) = \frac{2 - \cos\alpha}{\sin\alpha}$ on the interval $\alpha\in (0,\pi)$</p>
+
<p>To find the minimum of $L$, it is necessary to find the extremum of the function $f(\alpha) = \frac{2 - \cos\alpha}{\sin\alpha}$ on the interval $\alpha\in (0,\pi)$</p>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
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<meta name="description" content="A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is 200 m wide?">
<meta name="description" content="A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is 200 m wide?">
<meta property="og:title" content="A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is 200 m wide?">
<meta property="og:title" content="A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is 200 m wide?">
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<meta property="og:description" content="A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is 200 m wide?">
<meta property="og:description" content="A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is 200 m wide?">
<title>A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is 200 m wide?</title>
<title>A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is 200 m wide?</title>
$1.4.18^*.$ A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is $200$ m wide?
$1.4.18^*.$ A boy who can swim at half the speed of a river wants to swim across the river so that he is not carried downstream as much as possible. At what angle to the shore should he swim? How far will it go if the river is $200$ m wide?
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
<p>Consider the boy's movements with the speed $\vec{v}$, when he is carried away by the river with the current $\vec{u}$</p>
<p>Consider the boy's movements with the speed $\vec{v}$, when he is carried away by the river with the current $\vec{u}$</p>
Representation of $\vec{v'}$ as the sum of two vectors
Representation of $\vec{v'}$ as the sum of two vectors
</figcaption>
</figcaption>
@@ -67,33 +67,33 @@
</figure>
</figure>
</center>
</center>
<p>Let's write in projections on the horizontal and vertical axes, taking into account that $u = 2v$</p>
<p>Let's write in projections on the horizontal and vertical axes, taking into account that $u = 2v$</p>
<p class="exp">$$v_x = 2v - v \cos\alpha$$</p>
<p class="exp">$$v_x = 2v - v \cos\alpha$$</p>
<p class="exp">$$v_y = v \sin\alpha$$</p>
<p class="exp">$$v_y = v \sin\alpha$$</p>
<p>Find the time it takes the boy to swim across the river</p>
<p>Find the time it takes the boy to swim across the river</p>
<p class="exp">
<p class="exp">
$$ t = \frac{H}{v_y} = \frac{H}{v \sin\alpha}$$
$$ t = \frac{H}{v_y} = \frac{H}{v \sin\alpha}$$
</p>
</p>
<p>During time $t$ it will be carried along the coast by an amount</p>
<p>During time $t$ it will be carried along the coast by an amount</p>
<p class="exp">
<p class="exp">
$$ L = v_xt = H\frac{2 - \cos\alpha}{\sin\alpha}\quad(1) $$
$$ L = v_xt = H\frac{2 - \cos\alpha}{\sin\alpha}\quad(1) $$
</p>
</p>
<p>To find the minimum of $L$, it is necessary to find the extremum of the function $f(\alpha) = \frac{2 - \cos\alpha}{\sin\alpha}$ on the interval $\alpha\in (0,\pi)$</p>
<p>To find the minimum of $L$, it is necessary to find the extremum of the function $f(\alpha) = \frac{2 - \cos\alpha}{\sin\alpha}$ on the interval $\alpha\in (0,\pi)$</p>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
