Chapter 3, Problem 19Q

Solution

The rower who reaches the opposite side first.

Solution:

The rower who heads upstream at an angle.

Explanation:

Given:

Rower 1 heads straight but is pulled downstream by the current.

Rower 2 heads upstream at an angle and reaches the exactly at the opposite end of the starting point.

Calculation

Case 1: When rower heads straight but is pulled downstream by the current.

  

  $\begin{array}{l}{\overrightarrow{v}}_{b,g}={\overrightarrow{v}}_{b,w}+{\overrightarrow{v}}_{w,g}\\ {\overrightarrow{v}}_{b,g}=\text{velocity of boat with respect ground}\\ {\overrightarrow{v}}_{b,w}=\text{velocity of boat with respect to water}\\ {\overrightarrow{v}}_{w,g}=\text{velocity of water with respect to ground}\end{array}$

  $\begin{array}{l}{v}_{b,g}=\sqrt{v_{b,w}{}^2+v_{w,g}{}^2}\\ v_{b,w}=\sqrt{v_{b,g}{}^2-v_{w,g}{}^2}\end{array}$

Let the distance between the opposite sides be: d

Then time taken by rower 1 to cross the river is:

  $\begin{array}{l}{t}_1=\frac{d}{v_{b,w}}\\ t_1=\frac{d}{\sqrt{v_{b,g}{}^2-v_{w,g}{}^2}}\end{array}$

Case 2: When rower heads upstream at an angleand reaches the exactly at the opposite end of the starting point.

  

  $\begin{array}{l}{\overrightarrow{v}}_{b,w}={\overrightarrow{v}}_{b,g}+{\overrightarrow{v}}_{w,g}\\ {\overrightarrow{v}}_{b,g}=\text{velocity of boat with respect ground}\\ {\overrightarrow{v}}_{b,w}=\text{velocity of boat with respect to water}\\ {\overrightarrow{v}}_{w,g}=\text{velocity of water with respect to ground}\end{array}$

  $v_{b,g}=\sqrt{v_{b,w}{}^2-v_{w,g}{}^2}$

The distance between the opposite sides isd .

Then time taken by rower 2 to cross the river is:

  $\begin{array}{l}{t}_2=\frac{d}{v_{b,g}}\\ t_2=\frac{d}{\sqrt{v_{b,w}{}^2-v_{w,g}{}^2}}\end{array}$

  $\begin{array}{l}\because v_{bg}>\sqrt{v_{b,g}{}^2-v_{w,g}{}^2}\\ \therefore t_2<t_1\end{array}$

Conclusion:

Hence, the rower heads who upstream at an angle and reaches the exactly at the opposite end of the starting point will reach before the rower who heads straight but is pulled downstream by the current.