Chapter 9.1, Problem 65E

To determine

To Evaluate: Expression of velocity of boat relative to water in component form, speed of the water and true speed of the boat.

Answer to Problem 65E

Expression of velocity of boat relative to water in component form is $22.8i+7.4j$. Speed of water is $7.4\text{mi/h}$ and speed of boat is $22.8\text{mi/h}$.

Explanation of Solution

Given:

The speed of boat is $24\text{mi/h}$ and direction of boat is $\text{N}\text{72}°\text{E}$.

The speed of boat relative to water is $24\text{mi/h}$ and direction of boat is $\text{N}\text{72}°\text{E}$. The water is flowing towards south direction.

Formula used:

If velocity of boat is vector u which makes angle $\theta$ with positive $x$-axis, then boat velocity is,

$u=\left|u\right|\cos \theta i+\left|u\right|\sin \theta j$ (1)

If velocity of water is vector v which makes angle $\theta$ with positive $x$-axis, then velocity of water is,

$v=\left|v\right|\cos \theta i+\left|v\right|\sin \theta j$ (2)

Calculation:

The speed of boat is $24\text{mi/h}$ and direction of boat is $\text{N}\text{72}°\text{E}$.

The angle made by boat velocity with positive $x$ axis is,

$\begin{array}{c}\theta =90°-72°\\ =18°\end{array}$

Velocity vector of boat u is shown on coordinate axis in Figure (1).

Figure (1)

Substitute $24$ for $\left|u\right|$, $18°$ for $\theta$ in equation (1).

$\begin{array}{c}u=24\cos 18°+24\sin 18°\\ =22.82i+7.41j\end{array}$

Thus, component form of river velocity u is $22.82i+7.41j$.

The speed of boat relative to water is $24\text{mi/h}$ and direction of boat is $\text{N}\text{72}°\text{E}$ and the water is flowing towards south direction. Vector v represents velocity of water as shown in figure (1).

Figure (1)

As the water is flowing towards south, angle made by vector v is $270°$ with positive $x$-axis or east.

Substitute $270°$ for $\theta$ in equation (2).

$\begin{array}{c}v=\left|v\right|\cos 270°i+\left|v\right|\sin 270°j\\ =-\left|v\right|j\end{array}$

If vector p represents the true velocity of boat then vector p will be resultant sum of vector u and vector v.

$p=u+v$

From section (a), velocity of boat vector u is $22.82i+7.41j$.

As calculated above vector v is $-\left|v\right|j$.

Substitute $22.82i+7.41j$ for vector u and $-\left|v\right|j$ for vector v.

$\begin{array}{c}p=22.82i+7.4j+\left(-\left|v\right|j\right)\\ =22.82i+\left(7.4-\left|v\right|\right)j\end{array}$

As the true velocity of boat is along east direction, therefore vertical component of vector p is zero.

$7.4-\left|v\right|=0$

Solve for $\left|v\right|$.

$\left|v\right|=7.4$

Thus, speed of water given by $\left|v\right|$ is $7.4\text{mi/h}$.

Vector p is in east direction. Vertical component of vector p is zero

$\begin{array}{c}p=22.82i+0j\\ =22.82i\end{array}$

$\begin{array}{c}\left|p\right|=\sqrt{{22.82}^2+0^2}\\ =22.82\end{array}$

Thus speed of boat is $22.82\text{mi/h}$.

Therefore, speed of water is $7.4\text{mi/h}$ and speed of boat is $22.8\text{mi/h}$.