distance travelled

Question

A troop 5 metres long starts marching. A soldier at the end of the file steps out and starts marching at a higher speed. On reaching the head of the column , he immediately turns around and marches back at the same speed. As soon as he reaches the end of the file, the troop stops marching and is found to have travelled 5 metres exactly. What distance has the soldier travelled?

Answer 1

Suppose the troop marches a distance $d$ before the soldier turns around. Then the soldier has marched a distance $5+d$, so he is marching $\frac{5+d}{d}$ times faster. Having turned around the troop marches a distance $5-d$ before he gets back to his position and he marches a distance $d$. So $\frac{d}{5-d}=\frac{5+d}{d}$. Hence $d=\frac{5}{\sqrt2}$. The soldier marches a total distance $5+2d=5(1+\sqrt2)$.