Chapter 6.3, Problem 94E

a.

To determine

Draw a diagram that gives a visual representation of the problem.

Answer to Problem 94E

  

Explanation of Solution

Given information:

A commercial jet travels from Miami to Seattle. The jet’s velocity with respect to the air is $580$ miles per hour, and its bearing is $332°$ . The jet encounters a wind with a velocity of $60$ miles per hour from the southwest.

Draw a diagram that gives a visual representation of the problem.

Calculation:

A diagram that gives a visual representation of the problem,

  

Hence the diagram is as shown,

b.

To determine

Write the velocity of the wind as a vector in component form.

Answer to Problem 94E

  $\boxed{\langle -\frac{60}{\sqrt{2}},-\frac{60}{\sqrt{2}}\rangle }$

Explanation of Solution

Given information:

A commercial jet travels from Miami to Seattle. The jet’s velocity with respect to the air is $580$ miles per hour, and its bearing is $332°$ . The jet encounters a wind with a velocity of $60$ miles per hour from the southwest.

Write the velocity of the wind as a vector in component form.

Calculation:

The component form of the velocity of the wind is,

  $\begin{array}{l}{V}_W=\langle \Vert V_W\Vert \cos 225°,\Vert V_W\Vert \sin 225°\rangle \\ =\langle 60\left(-\frac{1}{\sqrt{2}}\right),60\left(-\frac{1}{\sqrt{2}}\right)\rangle \\ =\langle -\frac{60}{\sqrt{2}},-\frac{60}{\sqrt{2}}\rangle \end{array}$

Hence, component form of the velocity of the wind is, $\boxed{\langle -\frac{60}{\sqrt{2}},-\frac{60}{\sqrt{2}}\rangle }$

c.

To determine

Write the velocity of the jet relative to the air in component form.

Answer to Problem 94E

  $\boxed{\langle 272.29,512.11\rangle }$

Explanation of Solution

Given information:

A commercial jet travels from Miami to Seattle. The jet’s velocity with respect to the air is $580$ miles per hour, and its bearing is $332°$ . The jet encounters a wind with a velocity of $60$ miles per hour from the southwest.

Write the velocity of the jet relative to the air in component form.

Calculation:

The component form of the velocity of jet relative to the air is,

  $\begin{array}{l}V=\langle \Vert V\Vert \cos 62°,\Vert V\Vert \sin 62°\rangle \\ =\langle 580\left(0.469\right),580\left(0.883\right)\rangle \\ =\boxed{\langle 272.29,512.11\rangle }\end{array}$

Hence, component form of the velocity of jet relative to the air is, $\boxed{\langle 272.29,512.11\rangle }$

d.

To determine

Find the speed of the jet with respect to the ground.

Answer to Problem 94E

  $\boxed{522.913miles/hour}$

Explanation of Solution

Given information:

A commercial jet travels from Miami to Seattle. The jet’s velocity with respect to the air is $580$ miles per hour, and its bearing is $332°$ . The jet encounters a wind with a velocity of $60$ miles per hour from the southwest.

What is the speed of the jet with respect to the ground?

Calculation:

The the velocity of jet with respect to the ground is,

  $\begin{array}{l}V\text{'}=V+V_W\\ V\text{'}=\langle 272.29,512.11\rangle +\langle -\frac{60}{\sqrt{2}},-\frac{60}{\sqrt{2}}\rangle \\ =229.87,469.68\end{array}$

Now the speed of jet with respect to ground is,

  $\begin{array}{l}\Vert V\text{'}\Vert =\sqrt{{229.87}^2+{469.68}^2}\\ =\sqrt{273437.953}\\ =522.913miles/hour\end{array}$

Hence, the speed of jet with respect to ground is, $\boxed{522.913miles/hour}$ ,

e.

To determine

Find the true direction of the jet.

Answer to Problem 94E

  $\boxed{333.92°}$

Explanation of Solution

Given information:

A commercial jet travels from Miami to Seattle. The jet’s velocity with respect to the air is $580$ miles per hour, and its bearing is $332°$ . The jet encounters a wind with a velocity of $60$ miles per hour from the southwest.

What is the true direction of the jet?

Calculation:

If $\theta$ is the direction of jet velocity with respect to the ground then,

  $\begin{array}{l}\tan \theta =\frac{469.68}{229.87}\\ \theta ={\tan }^{-1}\left(2.043\right)\\ \approx 63.923°\end{array}$

So angle with vertical is

  $\begin{array}{l}90°-63.923°=26.078°\\ 360°-26.078°=333.92°\end{array}$

Hence the true angle of jet is $\boxed{333.92°}$