Question

A swimmer is capable of swimming at 0.60 m/s in still water. (a) If she aims her body directly across a 45 m wide river whose current is 0.50 m/s, how far downstream (from a point opposite to her starting point) will she land? (b) How long will it take her to reach the other side?

Short answer

(a) The distance traveled by the swimmer while moving downstream is 38 m. (b) The swimmer will take 75 s to reach the other side.

Step-by-step solution

Step 1. Meaning of velocity

Velocity may be described as the rate at which an object is displaced from one location to another. Its value is altered inversely to the value of the time taken by the particle.

Step 2. Draw a vector diagram

Here, ${\overrightarrow{v}}_{pw}$is the velocity of the person relative to the water, ${\overrightarrow{v}}_{ws}$is the velocity of the water relative to the shore, and ${\overrightarrow{v}}_{ps}$is the velocity of the person relative to the shore.

Step 3. Given information

Given data:

The velocity of the water relative to the shore is${\overrightarrow{v}}_{ws}=\left(0.50\mathrm{m}/\mathrm{s}\right)\hat{j}$.

The velocity of the person relative to the water is${\overrightarrow{v}}_{pw}=\left(0.60\mathrm{m}/\mathrm{s}\right)\hat{i}$.

The width of the river is $w=45\mathrm{m}$.

Step 4. Calculate the magnitude of the velocity of the person relative to the shore

(a)

The expression for the relative velocity vector of the velocity of the person relative to the shore is given by

$\begin{array}{l}{\overrightarrow{v}}_{ps}={\overrightarrow{v}}_{pw}+{\overrightarrow{v}}_{ws}\\ {\overrightarrow{v}}_{ps}=\left(0.60\mathrm{m}/\mathrm{s}\right)\hat{i}+\left(0.50\mathrm{m}/\mathrm{s}\right)\hat{j}\end{array}$

The magnitude of the velocity of the person relative to the shore can be calculated as

$\begin{array}{l}\left|{\overrightarrow{v}}_{ps}\right|=\sqrt{v_x^2+v_y^2}\\ \left|{\overrightarrow{v}}_{ps}\right|=\sqrt{{\left(0.60\mathrm{m}/\mathrm{s}\right)}^2+{\left(0.50\mathrm{m}/\mathrm{s}\right)}^2}\\ \left|{\overrightarrow{v}}_{ps}\right|=0.78\mathrm{ms}\end{array}$

Step 5. Calculate the time taken by the swimmer to travel the total width of the water

The time taken by the swimmer to travel the total width of the water can be calculated as

$\begin{array}{l}t=\frac{w}{{\overrightarrow{v}}_{pw}}\\ t=\frac{\left(45\mathrm{m}\right)}{\left(0.60\mathrm{m}/\mathrm{s}\right)}\\ t=75\mathrm{s}\end{array}$

Step 6. Calculate the downstream distance the swimmer has to travel

The downstream distance the swimmer has to travel can be calculated as

$\begin{array}{l}d={\overrightarrow{v}}_{ws}t\\ d=\left(0.50\mathrm{m}/\mathrm{s}\right)\left(75\mathrm{s}\right)\\ d\approx 38\mathrm{m}\end{array}$

Thus, the downstream distance the swimmer has to travel is $38\mathrm{m}$.

Step 7. Calculate the time taken by the swimmer to travel the total width of the river

(b)

The time taken by the swimmer to travel the total width of the river can be calculated as

$\begin{array}{l}t=\frac{w}{{\overrightarrow{v}}_{pw}}\\ t=\frac{\left(45\mathrm{m}\right)}{\left(0.60\mathrm{m}/\mathrm{s}\right)}\\ t=75\mathrm{s}\end{array}$

Thus, the time taken by the swimmer to travel the total width of the river is $75\mathrm{s}$.