Question

4 mark for review two boats cross a river that has a current of velocity (v_r), as shown. the boats have identical speeds with respect to the water. boat 1 heads at an angle upstream such that it reaches the opposite shore directly across from its starting point. boat 2 heads in a direction perpendicular to the current and reaches a position on the opposite shore downstream of its starting point. which boat takes less time to cross the river, and what reasoning supports this claim? a boat 1 takes less time because it travels a shorter distance than boat 2. b boat 1 takes less time because its velocity with respect to the water and the current velocity add together, giving boat 1 a greater resultant velocity. c boat 2 takes less time because its velocity has a greater y - component with respect to the ground than boat 1. d boat 2 takes less time because the current increases the boats speed relative to the ground.

Explanation:

Step1: Analyze velocity components

Let the speed of each boat with respect to the water be $v$. For Boat 1, to reach the opposite - shore directly across, its velocity vector is angled upstream. Its component of velocity perpendicular to the banks (the component that determines the time to cross the river) is $v_{1y}

Step2: Use time - distance formula

The time to cross the river $t=\frac{d}{v_y}$, where $d$ is the width of the river and $v_y$ is the component of the boat's velocity perpendicular to the banks. Since $d$ is the same for both boats and $v_{2y}>v_{1y}$, using the formula $t=\frac{d}{v_y}$, we have $t_1=\frac{d}{v_{1y}}$ and $t_2=\frac{d}{v_{2y}}$, so $t_2 < t_1$.

Answer:

C. Boat 2 takes less time because its velocity has a greater $y$-component with respect to the ground than Boat 1.