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?
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<h3> Statement </h3>
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$1.1.6.$ 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?
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<h3>Solution</h3>
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<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
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<p style="text-align: center;">
$$t = \frac{l}{v + u}$$
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<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
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<p style="text-align: center;">
$$t = \frac{{l}'}{v - u}$$
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<p class="TxtSolutions"> where ${l}'$ is the length of the new “column” after overtaking. Then
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<p style="text-align: center;">
$$\frac{{l}'}{v - u} = \frac{l}{v + u}$$
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<p class="TxtSolutions"> and the length of the new column
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<p style="text-align: center;">
$$\fbox{${l}' = \frac{v - u}{v + u}l$}$$
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<h4>Answer</h4>
<p>
When all athletes turn around, the length of the new column will be equal to ${l}' = \frac{v - u}{v + u}l$
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