$1.1.6.$ 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?
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
$$t = \frac{l}{v + u}$$
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
$$t = \frac{{l}'}{v - u}$$
where ${l}'$ is the length of the new “column” after overtaking. Then
$$\frac{{l}'}{v - u} = \frac{l}{v + u}$$
and the length of the new column
$$\fbox{${l}' = \frac{v - u}{v + u}l$}$$
When all athletes turn around, the length of the new column will be equal to ${l}' = \frac{v - u}{v + u}l$