Chapter 8.6, Problem 34E

Solution

To calculate: The vector $v$ that represents the rocket’s velocity relative to the point of takeoff.

  $v=2089.43i+5.66j+11817.69k$

Given information:

A rocket is headed east and is climbing at an angle of $80°$ .

Airspeed is $12,000$ mph.

Wind is out of the southwest at $8$ mph.

Formula used:

The formula to find the velocity relative to the ground is $v=a+w$ , where $a$ is the rocket’s air velocity and $w$ is the wind velocity.

Calculation:

Let $i$ point east $j$ point north and $k$ point up. The rocket’s air velocity is:

  $a=〈12000\cos 80°,0,12000\sin 80°〉$

The wind velocity, which is pointing southwest is:

  $w=〈8\cos 45°,8\sin 45°,0〉$

The velocity relative to the ground is $v=a+w$ , so

  $\begin{array}{c}v\approx 〈2083.78,0,11817.69〉+〈5.66,5.66,0〉\\ \approx 〈2083.78+5.66,0+5.66,11817.69+0〉\\ \approx 〈2089.43,5.66,11817.69〉\\ \approx 2089.43i+5.66j+11817.69k\end{array}$

Hence, the velocity of the rocket relative to the point of takeoff is $2089.43i+5.66j+11817.69k$