Chapter 6, Problem 12PS

(a)

To determine

Complete the table.

Answer to Problem 12PS

Table is shown below.

Explanation of Solution

Given:

The table is completed as given below.

$\theta$	0.5°	1.0°	1.5°	2.0°	2.5°	3.0°
$\parallel v\parallel \sin \theta$	0.8727	1.7452	2.618	3.49	4.362	5.234
$\parallel v\parallel \cos \theta$	99.996	99.985	99.9657	99.939	99.905	99.863

(b)

To determine

Does an airplane's speed equal the sum of the vertical and horizontal components of its velocity.

Answer to Problem 12PS

The airplane speed is not the sum of the horizontal and vertical components of the velocity.

Explanation of Solution

Given:

The airplane speed is not the sum of the horizontal and vertical components of the velocity. Instead it is

  $\begin{array}{cc}\sqrt{{(|v|\sin \theta )}^2+{(\parallel v\parallel \cos \theta )}^2}& =\parallel v\parallel \sqrt{{\sin }^2\theta +{\cos }^2\theta }\\ & =\parallel v\parallel \end{array}$

Square the two components add them and take the square root.

(c)

To determine

Use the result of part (b) to find the speed of an airplane with the given velocity components.

Answer to Problem 12PS

(i) $\parallel v\parallel \cos \theta =149.909\text{mph}$

(ii) $\begin{array}{cc}\parallel y\parallel & =\sqrt{{10.463}^2+{149.634}^2}\\ & =149.999\text{mph}\end{array}$

Explanation of Solution

Given:

(i) $\Vert v\Vert \sin \theta =5.235$ miles per hour $\Vert v\Vert \cos \theta =149.909$ miles per hour

(ii) $\Vert \text{v}\Vert \sin \theta =10.463$ miles per hour $\Vert v\Vert \cos \theta =149.634$ miles per hour

1) when $\parallel v\parallel \sin \theta =5.235\text{mph}$ and $\parallel v\parallel \cos \theta =149.909\text{mph}$ .

  $\parallel v\parallel \cos \theta =149.909\text{mph}$

2)when $\parallel v\parallel \sin \theta =10.463\text{mph}$ and $\parallel v\parallel \cos \theta =149.634\text{mph}$ .

  $\begin{array}{cc}\parallel y\parallel & =\sqrt{{10.463}^2+{149.634}^2}\\ & =149.999\text{mph}\end{array}$