Chapter 9.4, Problem 78AYU

a.

To determine

Toexpress:the velocity ${\text{v}}_a$ of the A320 relative to the ground.

Answer to Problem 78AYU

  $50\sqrt{2}\left(i-j\right)$ .

Explanation of Solution

Given:

Constant speed of AirbusA320 jumbo jet is $500\text{mi/hr}$

Speed of jet stream is $100\text{mi/hr}$ in the southeasterly direction.

Calculation:

According to the question,

  $\begin{array}{l}{v}_a=-500j\\ v_a=100\left(\cos {315}^{\circ }i+i\sin {315}^{\circ }j\right)\\ v_a=100\left(\frac{\sqrt{2}}{2}i-\frac{\sqrt{2}}{2}j\right)\\ =50\sqrt{2}i-50\sqrt{2}j\\ =50\sqrt{2}\left(i-j\right)\end{array}$ .

Hence, the velocity ${\text{v}}_a$ of the A320 relative to the groundis $50\sqrt{2}\left(i+j\right)$ .

b.

To determine

To find: the velocityof the A320 relatives to the ground.

Answer to Problem 78AYU

  $\left(50\sqrt{2}-550\right)i-50\sqrt{2}j$ .

Explanation of Solution

Given:

Constant speed of Airbus A320 jumbo jet is $500\text{mi/hr}$

Speed of jet stream is $100\text{mi/hr}$ in the southeasterly direction.

Calculation:

According to the question,

  $\begin{array}{l}{v}_g=v_a+v_w\\ =-500j+50\sqrt{2}i-50\sqrt{2}j\\ =\left(50\sqrt{2}-550\right)i-50\sqrt{2}j\end{array}$ .

Hence, the velocity of the A320 relatives to the groundare $\left(50\sqrt{2}-550\right)i-50\sqrt{2}j$ .

c.

To determine

To find: the actual speed and direction of the A320 relatives to the ground.

Answer to Problem 78AYU

  $435.1\text{mgh;}S{80.6}^{\circ }W$ .

Explanation of Solution

Given:

Constant speed of Airbus A320 jumbo jet is $500\text{mi/hr}$ .

Speed of jet stream is $100\text{mi/hr}$ in the southeasterly direction.

Calculation:

According to the question,

  $\begin{array}{l}\Vert v_g\Vert =\sqrt{{\left(50\sqrt{2}-500\right)}^2+{\left(50\sqrt{2}\right)}^2}\\ =\sqrt{189289.3219}\\ =435.1\\ \tan \alpha =\frac{-50\sqrt{2}}{\left(50\sqrt{2}-500\right)}\\ \therefore \alpha ={9.4}^0\end{array}$ .

Therefore, the direction is ${\left(90-9.4\right)}^{\circ }=S{80.6}^{\circ }W$ .

Hence, the actual speed and direction of the A320 relatives to the ground are $435.1\text{mgh;}S{80.6}^{\circ }W$ .