A caravan of camels, 1 km long, moves uniformly across the desert. A rider rode from the end of the caravan to the beginning and returned to the end of the caravan. During this time, the caravan traveled 1 km. What distance did the rider travel if his speed was constant?
Let the speed of the caravan be a unit, and the speed of the rider be denoted as $v$. Then the time spent traveling from the end of the caravan to the beginning is $\frac{1}{v-1}$, and the time spent traveling from the beginning to the end is $\frac{1}{v+1}$. According to the condition, $\frac{1}{v-1}+\frac{1}{v+1}=1$. This equation can be transformed into

$v^{2}-2 v-1=0$. From this, $v=1+\sqrt{2}$, which means that in the time it takes for the caravan to travel 1 km, the rider will travel the distance specified in the answer.

## Answer

$(1+\sqrt{2})$ km.