Solving the Boat Catch-Up Problem

The problem involves a boat needing to catch a float drifting downstream. We are given the distance between them and the boat's speed in still water.

• Distance downstream ($d$): $10\text{ m}$
• Boat speed in still water ($v_b$): $10\text{ m/minute}$
• Boat direction: Downstream

Understanding Relative Speed

When the boat travels downstream, its speed relative to the ground is its speed in still water plus the speed of the river current ($v_r$). The float drifts at the speed of the river ($v_r$).

• Boat speed relative to ground: $v_{boat, ground} = v_b + v_r$
• Float speed relative to ground: $v_{float, ground} = v_r$

The speed at which the boat closes the distance to the float is the relative speed between them:

$v_{relative} = v_{boat, ground} - v_{float, ground} = (v_b + v_r) - v_r = v_b$

Notice that the river's speed ($v_r$) cancels out. The relative speed is simply the boat's speed in still water.

Calculating Catch-Up Time

The time taken to catch up is the distance divided by the relative speed.

Time ($t$) = Distance ($d$) / Relative Speed ($v_{relative}$)

Substituting the given values:

Therefore, the boat will take exactly 1 minute to catch up with the float.