Question

At what upstream angle must the swimmer in Problem 46 aim if she is to arrive at a point directly across the stream? (b) How long will it take her to do so?

Short answer

(a) The upstream angle must be $56°$.

(b) It will take her 134.3 s.

Step-by-step solution

Step 1. Meaning of relative velocity

Relative velocity may be defined as the vector difference between the velocities of two particles. Its value can be positive, negative, or zero.

Step 2. Draw the vector diagram

Here, ${\overrightarrow{v}}_{ws}$is the velocity of the water relative to the shore, ${\overrightarrow{v}}_{sw}$is the velocity of the swimmer relative to the water, and ${\overrightarrow{v}}_{ss}$is the velocity of the swimmer relative to the shore.

Step 3. Components of the velocities

(a)

The x component of the velocity of the swimmer relative to the water is

${\left(v_{sw}\right)}_x=0$.

The y component of the velocity of the swimmer relative to the water is

${\left(v_{sw}\right)}_y=0.60\mathrm{m}/\mathrm{s}$.

The x component of the velocity of the water relative to the shore is

${\left(v_{ws}\right)}_x=0.50\mathrm{m}/\mathrm{s}$.

The y component of the velocity of the water relative to the shore is

${\left(v_{ws}\right)}_y=0$.

Step 4. Calculate the velocity vector for the velocity of the swimmer relative to the shore

The velocity of the swimmer relative to the shore can be calculated as

$\begin{array}{l}{\overrightarrow{v}}_{ss}={\overrightarrow{v}}_{sw}+{\overrightarrow{v}}_{ws}\\ {\overrightarrow{v}}_{ss}=\left(0,0.60\mathrm{m}/\mathrm{s}\right)+\left(0.50\mathrm{m}/\mathrm{s},0\right)\\ {\overrightarrow{v}}_{ss}=\left(0.50\mathrm{m}/\mathrm{s},0.60\mathrm{m}/\mathrm{s}\right)\end{array}$

Step 5. Calculate the direction of the velocity of the swimmer relative to the shore

The direction of the velocity of the swimmer relative to the shore can be calculated as

$\begin{array}{l}\theta ={\sin }^{-1}\left(\frac{v_x}{v_y}\right)\\ \theta ={\sin }^{-1}\left(\frac{0.50\mathrm{m}/\mathrm{s}}{0.60\mathrm{m}/\mathrm{s}}\right)\\ \theta \approx 56°\end{array}$

Thus, the upstream angle must be $56°$.

Step 6. Calculate the speed of the swimmer relative to the shore

The speed of the swimmer relative to the shore can be calculated as

$\begin{array}{l}{v}_{ss}={\left(v_{sw}\right)}_y\cos \theta \\ v_{ss}=\left(0.60\mathrm{m}/\mathrm{s}\right)\cos \left(56°\right)\\ v_{ss}=0.335\mathrm{m}/\mathrm{s}\end{array}$

Step 7. Calculate the required time 

(b)

The required time can be calculated as

$\begin{array}{l}t=\frac{w}{v_{ss}}\\ t=\frac{\left(45\mathrm{m}\right)}{\left(0.335\mathrm{m}/\mathrm{s}\right)}\\ t=134.3/\mathrm{s}\end{array}$

Thus, the required time is $134.3/\mathrm{s}$.