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<title>Four turtles are located at the vertices of a square with side a. They start moving simultaneously at a constant modulo velocity v. Each turtle moves clockwise in the direction of its neighbor. Where will the turtles meet and after what time?</title>
$1.5.14^*.$ Four turtles are located at the vertices of a square with side $a$. They start moving simultaneously at a constant modulo velocity $v$. Each turtle moves clockwise in the direction of its neighbor. Where will the turtles meet and after what time?
<p>Consider the change in the coordinates of the turtles over a short period of time $dt$</p>
<p>Over time $dt$ the distance between neighboring turtles changed from $a$ to $a'$</p>
<p>Express $a'$ using the Pythagorean theorem</p>
$$ a' = \sqrt{(a-dx)^2 - d^2x}$$
<p>Considering the smallness of the value $dx$</p>
$$ a' = \sqrt{a^2 - 2a\, dx}$$
$$ a' = a\sqrt{1 - \frac{2dx}{a}}$$
<p>We will use the formula for small quantities $(1+x)^\alpha\approx 1+\alpha x$, where $x\rightarrow 0$:</p>
$$ a' = a - dx $$
<p>Thus, the increment of the coordinate $a$ is</p>
$$ da = a' - a = dx $$
<p>Hence, the rate of change of distance between the turtles is</p>
$$ u = \frac{da}{dt} = -\frac{dx}{dt}=-v $$
<p>From this it follows that after $a=0$, after a period of time</p>
$$ t = a/v $$
<p>From the symmetry of the problem, it follows that all turtles will go the same way and end up in the center of the square.</p>
<p>NO: It would be interesting to know what would happen if it were not a square? If there were not $4$ turtles, but $n$ ones? A more detailed version of the problem can be found in <a href="https://belphol.github.io/books/LongProblemsPart1.pdf" target="_blank">"Very Long Physics Problems"</a> by A.I. Slobodyanyuk (Problem 1)</p>
</p>
<h4>Answer</h4>
<p>
<p>At the center of the square after time $t = a/v$.</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>
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<meta name="description" content="Four turtles are located at the vertices of a square with side a. They start moving simultaneously at a constant modulo velocity v. Each turtle moves clockwise in the direction of its neighbor. Where will the turtles meet and after what time?">
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<meta property="og:description" content="Four turtles are located at the vertices of a square with side a. They start moving simultaneously at a constant modulo velocity v. Each turtle moves clockwise in the direction of its neighbor. Where will the turtles meet and after what time?">
<meta property="og:description" content="Four turtles are located at the vertices of a square with side a. They start moving simultaneously at a constant modulo velocity v. Each turtle moves clockwise in the direction of its neighbor. Where will the turtles meet and after what time?">
<title>Four turtles are located at the vertices of a square with side a. They start moving simultaneously at a constant modulo velocity v. Each turtle moves clockwise in the direction of its neighbor. Where will the turtles meet and after what time?</title>
<title>Four turtles are located at the vertices of a square with side a. They start moving simultaneously at a constant modulo velocity v. Each turtle moves clockwise in the direction of its neighbor. Where will the turtles meet and after what time?</title>
$1.5.14^*.$ Four turtles are located at the vertices of a square with side $a$. They start moving simultaneously at a constant modulo velocity $v$. Each turtle moves clockwise in the direction of its neighbor. Where will the turtles meet and after what time?
$1.5.14^*.$ Four turtles are located at the vertices of a square with side $a$. They start moving simultaneously at a constant modulo velocity $v$. Each turtle moves clockwise in the direction of its neighbor. Where will the turtles meet and after what time?
<p>Consider the change in the coordinates of the turtles over a short period of time $dt$</p>
<p>Consider the change in the coordinates of the turtles over a short period of time $dt$</p>
<p>Over time $dt$ the distance between neighboring turtles changed from $a$ to $a'$</p>
<p>Over time $dt$ the distance between neighboring turtles changed from $a$ to $a'$</p>
<p>Express $a'$ using the Pythagorean theorem</p>
<p>Express $a'$ using the Pythagorean theorem</p>
$$ a' = \sqrt{(a-dx)^2 - d^2x}$$
$$ a' = \sqrt{(a-dx)^2 - d^2x}$$
<p>Considering the smallness of the value $dx$</p>
<p>Considering the smallness of the value $dx$</p>
$$ a' = \sqrt{a^2 - 2a\, dx}$$
$$ a' = \sqrt{a^2 - 2a\, dx}$$
$$ a' = a\sqrt{1 - \frac{2dx}{a}}$$
$$ a' = a\sqrt{1 - \frac{2dx}{a}}$$
<p>We will use the formula for small quantities $(1+x)^\alpha\approx 1+\alpha x$, where $x\rightarrow 0$:</p>
<p>We will use the formula for small quantities $(1+x)^\alpha\approx 1+\alpha x$, where $x\rightarrow 0$:</p>
$$ a' = a - dx $$
$$ a' = a - dx $$
<p>Thus, the increment of the coordinate $a$ is</p>
<p>Thus, the increment of the coordinate $a$ is</p>
$$ da = a' - a = dx $$
$$ da = a' - a = dx $$
<p>Hence, the rate of change of distance between the turtles is</p>
<p>Hence, the rate of change of distance between the turtles is</p>
$$ u = \frac{da}{dt} = -\frac{dx}{dt}=-v $$
$$ u = \frac{da}{dt} = -\frac{dx}{dt}=-v $$
<p>From this it follows that after $a=0$, after a period of time</p>
<p>From this it follows that after $a=0$, after a period of time</p>
$$ t = a/v $$
$$ t = a/v $$
<p>From the symmetry of the problem, it follows that all turtles will go the same way and end up in the center of the square.</p>
<p>From the symmetry of the problem, it follows that all turtles will go the same way and end up in the center of the square.</p>
<p>NO: It would be interesting to know what would happen if it were not a square? If there were not $4$ turtles, but $n$ ones? A more detailed version of the problem can be found in <a href="https://belphol.github.io/books/LongProblemsPart1.pdf" target="_blank">"Very Long Physics Problems"</a> by A.I. Slobodyanyuk (Problem 1)</p>
<p>NO: It would be interesting to know what would happen if it were not a square? If there were not $4$ turtles, but $n$ ones? A more detailed version of the problem can be found in <a href="https://belphol.github.io/books/LongProblemsPart1.pdf" target="_blank">"Very Long Physics Problems"</a> by A.I. Slobodyanyuk (Problem 1)</p>
</p>
</p>
<h4>Answer</h4>
<h4>Answer</h4>
<p>
<p>
<p>At the center of the square after time $t = a/v$.</p>
<p>At the center of the square after time $t = a/v$.</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>
