Chapter 8.2, Problem 47E

a.

To determine

To find:the time required to travel the distancefrom one bank to other.

Answer to Problem 47E

30 seconds

Explanation of Solution

Given information:

Width of river, $d=150\text{ m}$

Rate of river flowing in south direction, $\overrightarrow{v_C}=1.0\text{ m/s}$

Rate of raft in still water, $\overrightarrow{v_A}=5.0\text{ m/s}$

The figure for the given situation is shown below.

  

Calculation:

The time required will be evaluated as follows:

  $\begin{array}{c}\overrightarrow{v_A}=\frac{d}{t}\\ \text{5 m/s}=\frac{150\text{ m}}{t}\\ t=\frac{150\text{ m}}{5\text{ m/s}}\\ =30\text{ s}\end{array}$

Thus, the time required to travel the distance from one bank to other is 30 seconds.

b.

To determine

To find: the distance where the raft will land.

Answer to Problem 47E

30 m

Explanation of Solution

Given information:

Time, $t=30\text{ s}$

Width of river, $d=150\text{ m}$

Rate of river flowing in south direction, $\overrightarrow{v_C}=1.0\text{ m/s}$

Rate of raft in still water, $\overrightarrow{v_A}=5.0\text{ m/s}$

The figure for the given situation is shown below.

  

Calculation:

Let $x$ be the distance where the raft will land. The value of $x$ will be evaluated as follows:

  $\begin{array}{c}\overrightarrow{v_C}=\frac{x}{t}\\ 1.0\text{ m/s}=\frac{x}{30\text{ s}}\left[\text{Substitute the values}\right]\\ x=\left(1.0\text{ m/s}\right)\left(30\text{ s}\right)\\ =30\text{ m}\end{array}$

Thus, the distance where the raft will land is 30 m.

c.

To determine

To find:velocity of raft relative to the shore.

Answer to Problem 47E

  $5.1\text{ m/s}$

Explanation of Solution

Given information:

Time, $t=30\text{ s}$

Width of river, $d=150\text{ m}$

Rate of river flowing in south direction, $\overrightarrow{v_C}=1.0\text{ m/s}$

Rate of raft in still water, $\overrightarrow{v_A}=5.0\text{ m/s}$

The figure for the given situation is shown below.

  

Calculation:

The velocity of raft relative to the shore will be a vector sum of $\overrightarrow{v_A}\text{ and }\overrightarrow{v_C}$ . The magnitude of $\overrightarrow{v_A}\text{ and }\overrightarrow{v_C}$ will be resolved into $\widehat{i}\text{ and }\widehat{j}$ components. Let the direction towards the other shore and downstream be $+x$ and $+y$ respectively.

The velocity of raft relative to the shore will be evaluated as follows:

  $\begin{array}{c}\left|\overrightarrow{r}\right|=\left|\overrightarrow{v_A}+\overrightarrow{v_C}\right|\\ =\sqrt{1^2+5^2}\\ =\sqrt{26}\\ =5.1\text{ m/s}\end{array}$

Thus, the velocity of raft relative to the shore is $5.1\text{ m/s}$ .