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| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
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| <p class="author"> | | <p class="author"> |
| 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> |
| </p> | | </p> |
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| <h3 id="back-link"><a href="../../#1.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.1">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.1.6.$ Athletes run in a column of length $l$ at speed $v$. The coach runs towards them with speed $u < v$. Each athlete, when he reaches the coach, turns around and starts running back with the same speed. What is the length of the column when all the athletes turn around? | | $1.1.6.$ Athletes run in a column of length $l$ at speed $v$. The coach runs towards them with speed $u < v$. Each athlete, when he reaches the coach, turns around and starts running back with the same speed. What is the length of the column when all the athletes turn around? |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p class="TxtSolutions"> Let us imagine stopping the column, then the coach besides his speed will have the speed of the column directed in the opposite direction. With this relative velocity $v + u$ he will during time | | <p class="TxtSolutions"> Let us imagine stopping the column, then the coach besides his speed will have the speed of the column directed in the opposite direction. With this relative velocity $v + u$ he will during time |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$t = \frac{l}{v + u}$$ | | $$t = \frac{l}{v + u}$$ |
| </p> | | </p> |
| <p class="TxtSolutions"> will run along the column and equalise with the tail. The head of the column, having turned round, will move with relative velocity $v - u$ and after time | | <p class="TxtSolutions"> will run along the column and equalise with the tail. The head of the column, having turned round, will move with relative velocity $v - u$ and after time |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$t = \frac{{l}'}{v - u}$$ | | $$t = \frac{{l}'}{v - u}$$ |
| </p> | | </p> |
| <p class="TxtSolutions"> where ${l}'$ is the length of the new “column” after overtaking. Then | | <p class="TxtSolutions"> where ${l}'$ is the length of the new “column” after overtaking. Then |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$\frac{{l}'}{v - u} = \frac{l}{v + u}$$ | | $$\frac{{l}'}{v - u} = \frac{l}{v + u}$$ |
| </p> | | </p> |
| <p class="TxtSolutions"> and the length of the new column | | <p class="TxtSolutions"> and the length of the new column |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$\fbox{${l}' = \frac{v - u}{v + u}l$}$$ | | $$\fbox{${l}' = \frac{v - u}{v + u}l$}$$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| When all athletes turn around, the length of the new column will be equal to ${l}' = \frac{v - u}{v + u}l$ | | When all athletes turn around, the length of the new column will be equal to ${l}' = \frac{v - u}{v + u}l$ |
