Chapter 6, Problem 9PS

a.

To determine

Write the vectors $u$and $v$ in the component form.

Answer to Problem 9PS

  $\boxed{u=-120j}$

  $\boxed{v=40i}$

Explanation of Solution

Given information:

A skydiver is falling at a constant downward velocity of $120$ miles per hour. In the figure, vector $u$ represents the skydiver’s velocity. A steady breeze pushes the skydiver to the east at $40$ miles per hour. Vector $v$ represents the wind velocity.

Write the vectors $u$and $v$ in the component form.

Calculation:

Magnitude of vector $u$ is $120$ .

Magnitude of vector $v$ is $40$ .

Hence, $\boxed{u=-120j}$ and $\boxed{v=40i}$ .

b.

To determine

Use the figure to sketch $s$ .

Answer to Problem 9PS

  

Explanation of Solution

Given information: A skydiver is falling at a constant downward velocity of $120$ miles per hour. In the figure, vector $u$ represents the skydiver’s velocity. A steady breeze pushes the skydiver to the east at $40$ miles per hour. Vector $v$ represents the wind velocity.

Let

  $s=u+v$

Use the figure to sketch $s$ .

Calculation:

From the figure,

  $s=u+v$

  $=40i-120j$

  

Hence, the result.

c.

To determine

Find the magnitude of $s$ and information about the skydiver’s fall.

Answer to Problem 9PS

  $\boxed{126.49}$ miles per hour.

The magnitude gives the actual rate of the skydiver’s fall.

Explanation of Solution

Given information:

A skydiver is falling at a constant downward velocity of $120$ miles per hour. In the figure, vector $u$ represents the skydiver’s velocity. A steady breeze pushes the skydiver to the east at $40$ miles per hour. Vector $v$ represents the wind velocity.

Find the magnitude of $s$ . What information does the magnitude give you about the skydiver’s fall?

Calculation:

Magnitude of $s$ is given by,

  $\Vert s\Vert =\Vert 40i-120j\Vert =\sqrt{{40}^2+{120}^2}=126.49$ mph.

Hence, the effect of wind is negligible in the skydiver’s fall.

d.

To determine

Find angle to the ground.

Answer to Problem 9PS

  $\boxed{{71.57}^{\circ }}$

Explanation of Solution

Given information:

A skydiver is falling at a constant downward velocity of $120$ miles per hour. In the figure, vector $u$ represents the skydiver’s velocity. A steady breeze pushes the skydiver to the east at $40$ miles per hour. Vector $v$ represents the wind velocity.

If there were no wind, then the skydiver would fallin a path perpendicular to the ground. At what angle to the ground is the path of the skydiverwhen affected by $40$ mile-per-hour-windfrom due west?

Calculation:

Consider the figure below,

  

The angle $\theta$ is given by

  $\begin{array}{l}\tan \theta =\frac{120}{40}=3\\ {\tan }^{-1}3={71.57}^{\circ }\end{array}$

Hence, angle to the ground is the path of the skydiverwhen affected by $40$ mile-per-hour-windfrom due west $\boxed{{71.57}^{\circ }}$ .

e.

To determine

Draw a neat figure that gives a visual representation of the problem and find the skydiver’s new velocity.

Answer to Problem 9PS

  

  $\boxed{123.69}$

Explanation of Solution

Given information:

A skydiver is falling at a constant downward velocity of $120$ miles per hour. In the figure, vector $u$ represents the skydiver’s velocity. A steady breeze pushes the skydiver to the east at $40$ miles per hour. Vector $v$ represents the wind velocity.

The skydiver is blown to the west at $30$ miles per hour. Draw a neat figure that gives a visual representation of the problem and find the skydiver’s new velocity.

Calculation:

The diagram is given below,

  

The skydiver’s new velocity is $s=30i-120j$ .

Magnitude of $s$ is given by,

  $\Vert s\Vert =\Vert 30i-120j\Vert =\sqrt{{30}^2+{120}^2}=123.69$ mph.

Hence, the skydiver’s new velocity is $\boxed{123.69}$ mph.