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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?"> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" 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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| + | <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> | ||
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| + | <header style="text-align:center;"> | ||
| + | <h2>Solutions of Savchenko Problems in Physics</h2> | ||
| + | <p class="author"> | ||
| + | Aliaksandr Melnichenka <br/> | ||
| + | October 2023 | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $1.5.14^*.$ 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> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.5.14/draw.png" | ||
| + | loading="lazy" width="150" /> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <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>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> | ||
| + | </p> | ||
| + | |||
| + | |||
| + | <footer class="row container"> | ||
| + | <br> | ||
| + | <p> | ||
| + | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | ||
| + | </p> | ||
| + | <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> | ||
| + | </p> | ||
| + | </footer> | ||
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| + | |||
| + | </html> | ||
| @@ -0,0 +1,103 @@ | |||
| <!DOCTYPE html> | |||
| <html lang="en"> | |||
| <head> | |||
| <meta charset="utf-8"> | |||
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
| <meta http-equiv="content-language" content="en"> | |||
| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | |||
| <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?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" 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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| <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> | |||
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| }); | |||
| </script> | |||
| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <h2>Solutions of Savchenko Problems in Physics</h2> | |||
| <p class="author"> | |||
| Aliaksandr Melnichenka <br/> | |||
| October 2023 | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $1.5.14^*.$ 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> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.5.14/draw.png" | |||
| loading="lazy" width="150" /> | |||
| </figure> | |||
| </center> | |||
| <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>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> | |||
| </p> | |||
| <footer class="row container"> | |||
| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </p> | |||
| <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> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||