A boy sitting on a boat pulls on a rope with a constant force $F$ over a duration of time $t$. The other end of the rope is either tied to a bridge or to another freely floating boat of equal mass. Does the boy do more work in the case of the bridge, or in the case of the other freely floating boat?
Here's my attempt: Work is defined as $\int F dx$. Since $F$ remains constant for both cases, we only need to analyze the difference in displacement. This is where I am confused. In the case of the bridge, only the boy is moving (the bridge is fixed). In the second case, the boy is moving towards the other boat, and the other boat is also moving towards the other boy. The relative speed of approach would seem to be greater in second case (exactly twice), and since $x = \int v dt$, the boy does more work in the second case.
Is my analysis sound? Is there a better way to deduce the result?