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| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
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| <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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? | | $1.5.14^*.$ 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? |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
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| <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> |
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| <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> |
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| <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> |
| <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> |
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| <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> |
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| <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> |
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